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DYNAMICS OF A MODEL THREE SPECIESPREDATOR-PREY SYSTEM WITH CHOICEDouglas MagomoUniversity of Southern Mississippi

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The University of Southern Mississippi

DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM

WITH CHOICE

by

Douglas Magomo

A Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

Approved:

August 2007

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

C o p y r ig h t b y

D o u g l a s M a g o m o

2007


The University of Southern Mississippi


WITH CHOICE

by

Douglas Magomo

Abstract of a Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

August 2007


ABSTRACT


WITH CHOICE

by Douglas Magomo

August 2007

Studies of predator-prey systems vary from simple Lotka-Volterra type to nonlinear

systems involving the Holling Type II or Holling Type III functional response functions.

Some systems are modeled to represent a simple food chain, while others involve mutual

ism, competition and even switching of predator-prey roles. In this study, we investigate

the dynamics of a three species system in which the principle predator has a choice of two

prey, while the prey species change their behavior from being prey to predator and vice

versa.

Biological and mathematical conditions for the existence of equilibria and local stabil

ity are given. A proof to show the nonexistence of periodic solutions in the corresponding

two species system is also given. Global stability of the coexistence equilibrium in the

top predator-prey system is demonstrated through numerical simulations. The resulting

quintic polynomial through full symmetry analysis provide for conditions for a cusp bifur

cation using resultants theory. Various numerical simulations to illustrate the population

dynamics of the corresponding two species system as well as the three species system

model are given.

ii


ACKNOWLEDGMENTS

I w ould like to express my sincere thanks to the professors who have agreed to be part

o f this research struggle, in their capacities as advisors and com m ittee mem bers.

M y special acknow ledgem ents are reserved for my supervisors D r Sherry H erron, Dr.

Louis Rom ero and Dr. Joseph Kolibal.

D r Herron allow ed m e to conduct such research in the m athem atics education program

dom ain and I thank her for her encouragem ent.

Dr. Louis Rom ero is a researcher with Sandia N ational Laboratories, New M exico,

A lberqueque. His contribution is very unique in that w e never met physically and no one

had refered me to him for supervision. Here is a selfless professor who was w illing to

be bothered in the m iddle o f the night through phone calls of academ ic enquiry from a

stranger. He displayed som e strange patience with m e and I can only thank God who

opened his heart towards the success o f this research endeavor.

Dr. Joseph Kolibal is well known am ong students as a strict and detailed professor.

He would sit down with me to shape the research and correct m ost o f the write-up.

Thank you professors and I hope that in som e way, I would becom e the person you

collectively produced and be a representation o f each one o f you through your respective

contribution.

iii


TABLE OF CONTENTS

A B ST R A C T ................................................................................................................ ii

ACKNOWLEDGEMENTS.............................................................................................iii

LIST OF ILLUSTRATIONS.................................................................................... vi

LIST OF T A B L E S .................................................................................................. viii

NOTATION AND GLOSSARY................................................................................ ix

1 Background........................................................................................................... 1

1.1 Predator-prey Models 1

2 The Model Three Species S y s te m ..................................................................... 10

2.1 Mathematical Model 10

3 Two-Species S y ste m ............................................................................................. 14

3.1 Invariant Set 14

3.2 Local Stability Analysis 18

3.3 Symmetry and bifurcation in 2D 29

3.4 Other Corresponding Two Species Systems 35

4 THREE SPECIES S Y S T E M ........................................................................... 42

4.1 Invariant Set 42

4.2 Symmetry-Breaking Bifurcation 57

5 Discussions and Conclusions...................................................................................59

iv


5.1 General observations 59

A Computational Systems A n a ly s is ...................................................................... 63

A.l Matlab Code for Sensitivity of One Parameter 63

A.2 Matlab Code for Time Series 63

A.3 Matlab Code for Quintic Eigenvalue Problem 65

BIBLIOGRAPHY..................................................................................................... 68

v


LIST OF ILLUSTRATIONS

Figure

2.1 Three sp e c ie s ........................................................................................................ 10

2.2 An increase of bij in the response function a ,j means increased interaction. . 12

2.3 An increase of c,y in the response function a i;- means a decreased but pro

longed interaction................................................................................................... 12

3.1 F l o w i n g ........................................................................................................... 17

3.2 Phase flow near zero ........................................................................................... 22

3.3 Time evolution in 2 D ........................................................................................... 25

3.4 Stability regions in the corresponding two species system.................................. 25

3.5 Transcritical........................................................................................................... 27

3.6 Single species stabilization.................................................................................. 28

3.7 Spiral and a sa d d le .............................................................................................. 29

3.8 Two spirals and a saddle ..................................................................................... 30

3.9 Low p r e y .............................................................................................................. 30

3.10 High predator........................................................................................................ 31

3.11 Saddle form ation................................................................................................. 31

3.12 Single coexistence .............................................................................................. 32

3.13 Environmental carrying capacity l im it ............................................................... 32

3.14 Branching and Limit points.................................................................................. 35

3.15 Cubic r o o t s ........................................................................................................... 38

3.16 Large d ................................................................................................................. 40

3.17 Small d ................................................................................................................. 40

3.18 Spiral and a sa d d le .............................................................................................. 41

3.19 Zoom near z e r o .................................................................................................... 41

vi


4.1 Pitchfork .............................................................................................................. 49

4.2 Projection of cusp ................................................................................................. ^8

vii


LIST OF TABLES

Table

1.1 Variables descriptions and response functions...................................................... 3

1.2 S ign ificance o f the re sp o n se fu n c tio n ................................................................................... 4

1.3 Parameter meaning for Ruxon’s system................................................................ 6

3.1 2D equilibria........................................................................................................ 21

3.2 Parameters and variables describing the predator-prey system (3.28)................ 37

3.3 Equilibrium solutions for the two species system................................................ 39

4.1 Assigning values to certain parameters, k and d vary........................................... 46

4.2 Block matrices ..................................................................................................... 55

4.3 Critical population density at critical parameter values....................................... 57

viii


NOTATION AND GLOSSARY

G e n era l U sage a n d T erm inology

The notation used in this text represents fairly standard m athem atical and com putational

usage as used in M athem atical B iosciences publications. In m any cases these fields tend

to use different preferred notation to indicate the same concept, and these have been

reconciled to the extent possible, given the interdisciplinary nature o f the m aterial. In

particular, the notation for partial derivatives varies extensively, and the notation used is

chosen for stylistic convenience based on the application. W hile it w ould be convenient to

utilize a standard nom enclature for this im portant sym bol, the m any alternatives currently

in the published literature will continue to be utilized.

The rate o f change o f a population, that is, the rate o f increase or decrease o f pop

ulation size, where the population species are distinguished by the variables x ,y and z

is denoted by and ^ or x ,y and z respectively. The blackboard fonts are usedat a t at

to denote standard sets o f numbers: R for the field o f real num bers, C for the com plex

field, Z for the integers, and Q for the rational num bers. The capital letters, A , B , • • • are

used to denote m atrices, including capital G reek letters, e.g., A for a diagonal matrix.

The Jacobian matrix is denoted by the letter J. Functions which are denoted in bold

face type typically represent vector valued functions, and real valued functions usually

are set in lower case Rom an or G reek letters. Caligraphic letters, e.g., 7 , are used to

denote param eter spaces, w hile lower case letters such as i , j , k , l , m ,n . r , som etim es with

subscripts, r \ , r 2 ,ri,,bj,kj represent rates, environm ental carrying capacities and other pa

ram eters w hich influence the dynam ics o f the three species system. M atrices are typeset

in square brackets or parenthesis. In general the absolute value o f num bers is denoted

using a single pairs o f lines, e.g., | • |. Single pairs o f lines around m atrices indicates the

determ inant o f the matrix.

ix


Chapter 1

Background

1.1 P re d a to r-p re y M odels

The study o f predator-prey system s dates back to the 1950s with attem pts to m odel real

life system interactions through discrete and continuous m athem atics. The fam ous Lotka-

Volterra equations (1.1) assum ed that in a sim ple predator-prey system, predation would

linearly depend on the availability o f the prey species. The growth rate o f each species

was considered to be a function o f a single param eter. The rate at which predation oc

curred was also a linear com bination o f the interacting variables, in this case, the predator

variable and the prey variable. It was then within the limits o f available technology and

m athem atical analysis skills that such a system gained infirmity. Thus, for an n-species

system we w ould have the system

^

- ± = Xj (b /+ Y , au xj ) , 1, ; '= ,n , (1.1)

d x mwhere for each i, —2 denotes the rate o f growth o f species x,, w hile /;, represented the

atintrinsic population growth or decline in the absence o f o ther species; a,j is the predation

rate, positive if it is for the predator, and negative if the equation is representing the prey

population, [6],

A num ber o f criticism s arose out o f system (1.1), [11], [15] [25], [4], [6], [1] . First,

the linear functional response o f predation was not realistic. Predators kill w hen there is

need and not because o f abundance and availability o f prey population, and prey species

m ortality occurs for reasons other than predation only. Factors such as com petition for

food am ong prey and diseases definitely affect the rate o f increase o f prey population.

C onstant rates expressed as param eters, (1.1), and b, are not representative o f a m ore

realistic situation as these could be functions that express environm ental changes which

affect the growth o f any population, such as the hunting strategies and hiding strategies

o f both predator species and prey species respectively, [11]. Yet, som e m odification on

these param eters define sym biosis and m utualism in population dynam ics [29], [34], [41 ].

Therefore m odifications are being m ade such that these param eters are actually functions

which respond to environm ental changes.

1


CHAPTER 1. BACKGROUND 2

By considering response functions such as the Holling Type II function, Holling

[1965], [27], allowed for the tim e needed by a predator to kill or consum e each prey,

as well as the lapse in tim e before it is hungry again. Therefore, the response function

1 + m x

expresses the lim iting factor in the predation term o f the predator species where m > 0

is the environm ental carrying capacity o f the predator population x, [25]. O thers, [4],

regard the Holling Type II function as one that describes consum ption by predation as an

increasing function o f prey, which saturates at higher prey densities.

Therefore a simple two species system taking into account the above considerations

would be represented as follows:

d x ,~jt = f { t ) x - g ( t ) x y ,

di = -* « )> • +dt 1 + my

where the functions f ( t ) , h(t) and g(t) are positive definite. Unfortunately, system s that

are explicit as well as non-linear are not easily analyzed. Equations in explicit form

are often expressed im plicity even though the resultant system becom es more non-linear.

Also, unfortunately, the Holling Type II function does not describe decreasing predation

at lower prey densities, [4], First introduced by H olling (1959), and subsequently used by

other researchers, [3], [26], [27], the H olling Type III sigm oidal-shaped function

2ax(1.2)

1 + bx2

not only describes decreasing predation at lower prey densities, but also describes predator

switching to different prey as well as accounting for predator hunting experience and

gam ing strategies. The three species m odel w e will consider in this research utilizes this

type o f response function.

Studies o f predator-prey system s with ’’sw itching” were made after it was discovered

that in certain circum stances species which initially act as predators becom e prey under

alm ost sim ilar environm ental conditions. W hether this switch is due to population size

fluctuations or slight environm ental changes, rem ains an open question. H ernandez and

Barradas [11], provided a two species predator-prey system with a switching effect. They



determ ined that the equations representing this sw itching phenom enon be m odeled as:

d x ( x \ x y_ = n , ^ - _ j + a , 2_

d y (■> y \_ =r 2 > , h _ _ j + a 2 , _ .

where

« i 2 (*,y) = y2 and g 2i M = ^ X2 - (1.4)\ + c \ y l 1 + C 2XZ

Table 1.1 describes the param eters as well as the trophic functions that govern the system

(1.3).

Table 1.1: Variables descriptions and response functions.

Symbol Description

X population density for species x

y population density for species y

r\ intrinsic rate o f increase o f species x

r 2 intrinsic rate o f increase o f species y

k \ environm ental carrying capacity o f species x

k-2 environm ental carrying capacity o f species y

a.\2 interaction coefficient, expressing effect o f interaction between species x and y

«21 interaction coefficient, expressing effect o f interaction between species y and x

In their paper, H ernandez and Barradas [11] desired to model this sw itching behavior

by considering the quadratic-ratio functional response form to be as described in (1.4).

Their m athem atical choice was designed to account for saturation effects, that is, that

predation cannot go beyond what the environm ent can sustain and that food availability



is limited. Also by saturation, we m ean to express that population densities do not grow

without bound. These assum ptions are m ore realistic biologically. Viewed differently, the

functional response function oc;y in (1.4) is a m odification o f the H olling Type III function.

We sum m arize the significance and properties o f the trophic function , /, j = 1,2, / / j

as described in [11]. As a result o f this choice, the m odel two dim ensional system is such

Table 1.2: Significance o f the response function.

Sign or m agnitude o f a Significance

a X2 > 0, ( a 2\ < 0) a positive contribution to the density-dependent factor in the per capita growth rate o f species x

a 2\ > 0, {a,\2 < 0) a positive contribution to the density-dependent factor in the per capita growth rate o f species x

a ij m easure o f intensity o f contribution

that there w ould be no periodic solutions expected. Therefore, the only possible solutions

are equilibrium solutions, no limit cycles, since the only unstable steady state w ould be a

saddle. By the Poincare-Bendixon theorem , limit cycles are possible if there are unstable

foci or nodes. Variable outcom es were noted due to the use o f the a -function and variation

o f certain sensitive param eters. The association or interaction o f one species with another

prom oted an increase in the growth rate and equilibrium density o f the other, so m uch that

when a critical size o f the form er is reached a reverse effect is observed.

W hat we seek to accom plish by this study is to understand the dynam ics o f a sim ilar

system but one which includes a third force, a predator that influences the dynam ics of

these variable outcom es. The top predator is assum ed to have a choice o f food, hence the

use o f the response function defined by (1.2). This understanding would include know l

edge o f w hether in the presence o f the top predator, the ’’sw itching” behavior betw een the

prey species would continue, and in the event o f predation by the top predator, w hether

the lower predator continues to prey on the bottom species independent o f the population

sizes.

The system (1.3) is also considered to be one that models mutual dynam ics, [16],

These functions = 1,2 instead are regarded as the com petition coefficient for



which different qualitative long term dynam ic behavior m ay occur depending on whether

the interaction is considered weak or strong. M utual benefit and strength according to

Kooi, et. al, [16] was dependent on w hether the product a \ 2 • « 2 i was less than one,

for weak interaction, or greater than one, for strong interaction. The m agnitude o f this

product expressed the strength o f m utualism between the species. The w ork by Barradas,

et. al, [11] considered the sam e system w here these com petition coefficients were in fact

response functions (1.4) that provided the sw itching o f prey-predator roles.

Studies o f chaos in population system s date back to the 1970’s w here difference-

equation m odels o f a single population exhibited chaotic behavior, M ay (1995). Rai and

Kumar, [27] m ade som e extensive analysis o f why chaos is rarely observed in population

dynam ics and G ilpin [7], and Iyengar et. al. [37] observed quasi-cyclical behavior o f tra

jectories in ecological system s and defined this behavior as spiral chaos . They concluded

that ecological system s with high levels o f nonlinearities exhibited chaotic behavior and

notable tim e delays. As described above, nonlinearities are necessary to account for sat

uration effects. Tim e delays are due to m aturation o f organism s or the flow o f nutrients

through ecosystem s. All these are characteristic ingredients of chaotic dynam ics in eco

logical m odels.

Ruxon [30] modified a model by M cCann and Yodzis, [25] by introducing a term that

defines the im m igration o f a constant rate o f individuals into the resource population as

a fraction o f the corresponding environm ental carrying capacity. His m odel included a

population floor effect on a continuous-tim e population model. The m odel

described the rates o f change o f the predator P, the consum er C, and the resource popu

lation R over time. Table 1.3 describe the param eters that influence the dynam ics o f this

system. To include a situation where there w ould always be some resource that is im mune

to exploitation, Rs, exploitable resource H was defined as a piece-w ise function

With a 5% size o f irremovable resource, R uxon’s model exhibits chaotic behavior under

certain param eter values. Ruxon concluded that with im position o f population floors,

xpy pPC

ct{C + Co)x pypPC x cy cCH

a ( C + C0) a ( H + R 0yx cy cC H

a ( H + Ro) a ’

R - R s R > R S.

0 otherwise.(1.5)


CHAPTER 1. BACKGROUND

Table 1.3: Param eter m eaning for R uxon’s system.

6

Param eter Significance

x p m ass specific m etabolic rate o f species P m easuredrelative to production-to-biom ass ratio o f resource population R

x c mass specific m etabolic rate o f species C m easuredrelative to production-to-biom ass ratio of resource population R

y P m easure o f ingestion rate per unit m etabolic o f species P o r C

P(). Ro and C0 are initial population sizes

a resource population carrying capacity

H exploitable resource relative to production-to-biom ass ratio o f resource population x

Rs resource individuals im m une to exploitation

continuous-tim e predator-prey m odels show chaotic behavior. An overall look at this

three species system leaves us wondering w hat dynam ics can com e out o f a system where

the resource R and the consum er C som etim es interchange roles, while the predator P is

free to choose on which to feed, given the abundance or availability o f either o f the two.

W ith this in mind, one three species system worth noting concerns the predator-prey

relationship betw een lion, zebra and w ildebeest in which the lion had a food choice be

tween the two, depending on the population densities. Fay and Greeff, [6] produced a

model that tries to fit the census data o f these anim als at the Kruger National Park, South



Africa as reported during the 1970s. Their system,

dxdt

dzdt

0.125(xy + yz)

= —g(t) ( l - -0 .8 1 x y + 0.015xz,

= 0.34z (1 - - 0 .15yz + 0.02xz,

contains the functions g(t) and v(t) , m odifications that account for seasonal calving of

w ildebeest and cropping o f lion respectively. The m ixed quadratic term s in the system

define the interaction term s where the corresponding coefficients are the predation rates.

A positive sign on the m ixed term m eans a positive contribution towards the per capita

growth rate o f the corresponding species, w hile a negative sign produces the opposite

effect.

The overall system resem bles the general three species system w here the lion species

equation incorporates a H olling Type II functional response function, the other two prey

species having the logistic part and the predation contribution terms. W hile this model

system is explicitly responding to the available data, the authors did not explore the gen

eral dynam ics or variable interactions between the species. Very specific param eter val

ues such as predation rates, growth rates and environm ental carrying capacity values were

found, values that provide the best fit to available data. Also in this m odel system , it is

assum ed that the lion population had equal chances o f m aking a kill on either the zebra

or the wildebeest. Positive term s reflecting m utual benefit between zebra and w ildebeest

characterize this model system. A particular case sim ilar to this system could provide

extensions o f analysis o f our system where instead o f fixed param eters, we w ould wish to

consider a num ber o f possible scenarios.

Predator-prey interactions determ ine m any aspects o f population dynam ics and com

m unity structure o f estuarine and m arine ecosystem s. In a M aster’s thesis on a simple

three species food chain, M agom o [22] m odeled the dynam ical system along the Leslie-

Gower, [17] schem e and the resulting behavior in certain param eter regions was chaos

through period doubling bifurcations. S im ilar system s that were o f Kolm ogorov type also

produced interesting dynam ics such as H opf-bifurcation and chaos, [5], [35], [9]. Letel-

lier and A ziz-A laoui, [17] considered a sim ilar three species system but regarded the last

equation o f the above system to take the form

( 1.6)



Their original predator equation was o f the logistic form, but they considered instead a

is directly proportional to the num ber o f m ales as well as females o f the z species. M any

researchers consider that the growth rate takes into account this proportionality, [17], [5],

[11], In their study, bifurcation param eters w ere defined and chaos was observed through

variation o f the intrinsic growth rate o f the prey species x. We are therefore m otivated

into considering the top predation equation either to take this form (1.6) or the one (1.7)

in which predator dynam ics is represented by a logistic model with carrying capacity

proportional to the num ber o f prey species x and y, in this case, carrying capacity for the

predator species z , k = k^xy:

The study by Song and X iang, [42] revealed that variation o f sensitive param eters resulted

in period doubling cascades leading to chaos. How ever their m odels on pest control

produced periodic solutions where the interaction term s are m ostly linear. This study is

com plicated in that we include these response functions a,y that give rise to predator-prey

role interchange under certain param eter conditions, while the top predator has a choice

o f the seem ingly com peting species.

The organization o f our research is as follows: In C hapter 2 we define our model sys

tem in relation to the biological hypothesis provided for by Barradas, [11], We explain

why it is im portant to analyze this three species predator-prey system and how we de

sire to analyze this system. The equations that model this system are explained in their

relationship to biological realities.

In C hapter 3 we analyze the corresponding two species system. We prove that given

the choice o f the trophic functions em ployed in this m odel, response functions that m an

age the sw itching phenom enon, there are no periodic solutions in the positive half plane.

We obtain equilibrium solutions and analyze local stability o f these equilibria. In trying

to sim plify the highly nonlinear system, we analyze the system under symmetry. U nder

certain param eters, we observe som e exchange o f stability between certain critical points

and show that some equilibrium solutions undergoes a transcritical bifurcation. We fur

ther show phase portraits o f the two species interactions. Phase diagram s are shown to

reveal bifurcation o f equilibria as som e sensitive param eter, nam ely the environm ental

carrying capacity, is varied.

In C hapter 4 we further break down the three species system into other two species

systems. Phase portraits o f the resulting two species system reveal the nature o f stability

squared term a ^ z 2 to justify the fact that the m ating frequency o f the predator population

(1.7)



o f the resulting equilibrium solutions. Analysis o f the two species system through sym

m etry produce a pitchfork bifurcation. Analysis o f the three species system, again by

considering symmetry, lead towards obtaining a cusp bifurcation using resultant theory.

C hapter 5 provides a discussions and conclusion. We briefly review the results o f the

model analysis in relation to possible ecological situations.


Chapter 2

The Model Three Species System

2.1 M ath em a tica l M odel

The snail-lobster model that was studied by B arradas et. al, [11] is the ecological basis for

the three species predator-prey system under this study. Both lobsters and snails are prey

to dogfish. This principal predator preys on either o f the two depending on the availability

or abundance o f the prey species.

We therefore gain the m otivation to study the three species system in which lobsters

and snails have this sw itching effect while the two rem ain prey to dogfish.

Figure 2.1: Relationship betw een food chain species. Reverse arrow on the bottom two species predicts switching.

It is this unique setup that m otivates us into the need to analyze the dynam ics o f this

system,

predator

lobster

(2 .1)

— = r3z - z f i i 23 (x ,y ,z), at

10


CHAPTER 2. THE MODEL THREE SPECIES SYSTEM 11

where the loss in the predator population z is proportional to the reciprocal o f per capita

availability o f the two prey species x and y. This loss, represented by the function /3i 23

could be represented by either o f the two forms:

ZP m = — , m ultiplicative saturation form.

(2 .2)P \23 = ----------------- additive DeAngelis interference form , [24].

q + m{x + y)

The environm ental carrying capacity for the top predator z is therefore represented by

some prey ratio-dependent expression o f the m ultiplicative form or the D eAngeles inter

ference form (2 .2 ).

The response functions a \ 2 and a 2\ are as described by the equation (1.4) and the

system param eters are as described in Table 1.1, [11]. The param eters r(, i — 1,2,3

are the intrinsic growth rates o f species i. The carrying capacity o f the environm ent for

species are represented by the param eter kj, / = 1,2. In the absence o f predation, each

individual species grows to its corresponding carrying capacity. The sign o f a ,7, see Table

1 .2 , determ ines which species, betw een species x and y, assum es the role o f predator or

prey. If a \ 2 > 0 then the species x preys on species y. If a 2 \ > 0 then the species y preys

on species x.

Increasing b,j, the param eter which can be described as the intensity interaction pa

rameter, m eans an increase in reaction o f the species interaction, see Fig. 2.2 on page

12. Increasing the param eter c (/ produces a negative response, it prolongs the interac

tion, hence a dum ping effect, see Fig. 2.3 on page 12. The H olling Type III functional

response function that describe the predation on species x and y by the top predator z is

used;

P n { x ,z ) = J 0'* Z2 and /323(>’, z) = \ • (2.3)a 1 + x z d 2 + y L

We observe that the param eters £0;, i = 1,2 are the predation rates; d 2 is the value o f y at

which the per capita removal rate o f y becom es half the predation rate (0 2 . The same can

be said o f the param eter d\ in relation to the prey species x.

Com bining these term s, the system (2.1), under consideration takes the form

dx / x \ ( b \ y — y2\ x y C0\x2z = r\x 1 +

d t \ k\ J \ \ + c \ y 2 J k \ \ + d \ x 2 '

d y ( . y \ ( b 2x - x 2 \ x y o>2 y 2z— = r2y 1 - 7 - +d t \ k2 J \ 1 + C2X 2 J k2 1 + d 2y 2 ’

dz ( 1 z= Oz[ 1 -

dt ' \ k2xy



a

0 02

0 1 0 2 0 3 0 4

b0.5

Figure 2.2: An increase o f bjj in the response function a tj means increased interaction.

a

0 2 0 3

c0 5

Figure 2.3: An increase o f c,7 in the response function (Xjj m eans a decreased but p ro longed interaction.

ordxdt

dydt

dzdt

= r xx |^1

= r2y 1

x \ ( b \ y — y 2

k\

y_k2

+

+b 2 X — x 2

q + m (x + y )

4y _ (0{X2Zk\ d\ + x 2

xy a>iy2z

h d2 + y 2

(2.4a)

(2.4b)

(2.4c)



depending on whether w e require a quadratic-ratio dependent predator environm ental car

rying capacity proportional to the product o f the prey density, K = k^xy or linear com bina

tion o f the prey densities equal to the carrying capacity o f the predator, K = q + m {x + y ) .

We concentrate on the system defined in equations (2.4a)-(2.4c) in which we consider

an additive saturation term for both species x and y. The third equation o f the above sys

tem contains a m odified Holling Type II functional response function which characterize

interference between the two prey species w hile com petition for food am ong the predator

species experience this saturation effect o f the prey species. The param eters d\ and di

quantify the extent to which the environm ent provides protection to prey species x and

y respectively. We note that the predator equation, (2.4c) is the logistic type equation

whereby the carrying capacity is proportional to the additive com bined prey abundance,

m (x + y) , [36], The param eter q, norm alizes the residual reduction in the predator popu

lation z because o f severe scarcity o f prefered food.

The last equation, (2.4c), also shows that in the absence of both prey species x and y,

then the predator species goes to extinction as long as q < r$, otherw ise this w ould grow

unboundedly to infinity if this inequality is reversed, which is not acceptable biologically.

2.1.1 D iscussion

O ur model (2.1) represents a generalized three species predator-prey system. The spe

cific m odel represented by the equations (2.4a)-(2.4c) is a build-up from the two species

system that exhibited a sw itching phenom enon and which was presented by H ernandez

and Barradas, [11], We have chosen to consider the response function with the functional

response characteristics o f not only providing a choice o f food by the top predator, but

also giving the predator the trait o f not killing prey even when it exists in abundance. In

the same way, the prey species attains the ability to hide and evade being killed.

The nature o f the logistic function for the top predator is that saturation is density-

dependent. This m eans that the characteristic sigm oidal shape levels off with increasing

time to the additive linear sum o f the prey species. Com petition am ong predators controls

an otherw ise exponential population rise and we assum ed that the environm ental carrying

capacity was expressed as a prey-density-dependent linear form. In the proceeding chap

ters, w e analyze this system in stages o f corresponding two species to the three species

system.


Chapter 3

Two-Species System

3.1 In v a r ia n t Set

The system defined by equations (2.4a)-(2.4c) reduces to the corresponding two species

system in the absence o f the top predator species z. In that case, the equations governing

the corresponding two species system are described in [11], Barradas et. al determ ined

the dynam ics o f this m odel through com putational num erical sim ulations. In this chapter

we reconsider the corresponding two species system w hose equations model the switching

phenom enon that was docum ented in [11],

The variables x(t) and y( t) represent population densities. It is essential that solutions

with x ( t) > 0 and y ( t) > 0 initially, rem ain in the positive quadrant for all time. The

param eters involved are as defined in Table 1.1.

D efin ition 3.1.1. In v a r ia n t set: An invariant set for a system o f coupled differential

equations is a region D , o f the phase space with the property that any solution starting

within T> rem ains within it for all time. Note that the phase space o f a dynam ical system

is the space o f species interaction w here all the possible states are represented.

(3.1)

T h eo rem 3.1.1. (Invariant Set) The set

D = {(x,>’) e R |0 < x < k\ , 0 < y < k f ) ,

is an invariant set where any solution o f (3.1) starting inside T) will remain inside D f o r

all time.

Proof: The system (3.1) is equivalent to

x (t) = ;c(to)exp' _ r \x{s) + y ( s ) a i 2{s)

(3.2a)

14


CHAPTER 3. TWO-SPECIES SYSTEM 15

y( t) = y(f0)expr2y ( s ) x ( s ) a 2i( s ) l

/"2------ ;-------1-------- :-------- dS (3.2b)

If species x is the prey species, then it is because the response function a \ 2 is negative

while OC21 > 0. For this sign definite o f the response functions we require that

• b\ < y ( t ) < k2,

• 0 < x(t) > b2 < k\.

Substituting the m axim um possible values o f x(t) and y( t) into equations (3.2a)-(3.2b),

we observe that

• j ( 0 0 as t —> °° and

• x( t) —> 0 as t —>• °°.

The assertion o f the theorem follows im m ediately for all t > to. A sim ilar approach is

arrived at if the species x becom es the predator w hile y species is the prey in the two

dim ensional system (3.1). □

Definition 3.1.2. The solution o f (3.1) is said to be ultim ately bounded if there exists

B > 0 such that, for every solution (x ( t ) ,y ( t ) ) o f (3.1), there exists T > 0 such that

||x ( t) ,y (r) || < B for all t > to + T, where B is independent o f the particular solution,

while T may depend on the solution.

Theorem 3.1.2. Let e > 0. I f max {M f } < /?, < k ^ , i = 1,2, then the set Te defined by

is positively invariant with respect to the system, (3.1) where

M f = k\ T £, M f = k2 ~\~ £•

and £ > 0 is sufficiently small.

Proof: The condition b\ < k \r \ ensures that k\r \ term is much larger than a \ 2 for any

r e = j(x,y) £ M |0 < x < M f , 0 < y < Mf j,



forward tim e t > to. Therefore

x(t ) < x(t) n - ^ ( 0

< rM > )k\k\ + e —x(t)

= M f - x ( r )

A standard com parison argum ent shows that

0 < x(to) < M f ==> x( t ) < M f , t > to.

The p roof is com plete if we apply the sam e reasoning for the solution y(t ) . □

The choice o f (Xjj as explained in C hapter 1 provides that the dynam ics o f the system

for w hatever choice o f param eters w ould only yield sinks, saddles and spiral solutions

with no periodic solution in the positive quadrant as alluded to by the following lem m a

3.1.5. However before we give this lem m a, we m ention without proof, B endixson’s N eg

ative Criterion, (see p roof in [14]) in the form o f a theorem . Re-define system (3.1) with

equationsx = f { x , y ) ,

y = g(x,y) .

T h eo rem 3.1.3. Let the two species system be defined by equations (3.3). There are no

closed paths in a sim ply-connected dom ain o f the phase plane on which d f / d x + d g / d y

is o f one sign.

(3.3)

The B endixson’s Negative Criterion is extended to give rise to the so-called D ulac’s test:

T h eo rem 3.1.4. Let the two species system be defined by equations (3.3) and let the

function p ( x , y ) be one that adm its continuous firs t partia l derivatives. Then there are no

closed paths in a sim ply-connected dom ain o f the phase p lane on which

d (p (x, y ) f ( x , y )) j d x + d (p (x, y )g (x, y )) / d y is o f one sign.

The two theorem s 3.1.3 and 3.1.4 for the B endixson’s test lead us to our lemma.



— i---------- 1---------- 1---------- 1---------- 1---------- 1---------- 1---------r i i

le- ; j j ; j

i i k ]

= ! \ i s \ \

i *

i I i ! i i ! i^ 0 5 10 15 20 25 30 35 40 45 50

XFigure 3 .1 : Phase portrait o f the flow in the half plane. Notice the direction o f arrows towards equilibrium solutions w hich are not periodic, in the positive half plane, the invariant region.

L em m a 3.1.5. There are no periodic solutions to the system (3.1) in the positive h a lf

plane R 2 .

Proof: Since R 2 is sim ply connected, we define the C 1 function p ( x , y ) = — whichxy

adm its continuous first partial derivatives in R 2 .

A pplying D ulac’s Test [14], [40], [32], to system (3.1), then

P ( x .y ) f { l ,y ) = i ( t - f ) ^ b ' y - y 2y \ k \ ) \ 1 + c i y 2/ k\ ’

n( X , 1 o f , y \ , ( b2 x - x 2 \ 1p( x , y ) g ( x , y ) = — 1 - —

Therefore

x \ k2 ) \ 1 + c2x J k2

d ( p ( x , y ) f { x , y ) ) = _ n _ d x k \ y '

d ( p{ x , y ) g ( x , y ) ) r2

c)y k2x

d { p { x , y ) f ( x , y ) ) + d(p ( x , y ) g{ x , y ) ) = _ / _n_ + < Qso that

d (P (*d x ' d y V^O7 b2x /

for all G R + , r \ , r 2 G R +. Therefore there are no periodic solutions in the positive

half plane R “L see Fig. 3.1. □



3.2 L ocal S tab ility A nalysis

The corresponding two species system, although studied by H ernandez and B arradas [ 11 J,

is here re-visited for som e thorough m athem atical calculations that include stability of

equilibria. The following theorem s and definitions are necessary to understand the subse

quent m athem atical analysis.

D efin ition 3.2.1. E q u ilib r iu m so lu tion C onsider a general autonom ous vector field,

that is, a vector field associated with the first order autonom ous differential equation,

x = f ( x ) , x G R ” . (3.4)

An equilibrium solution o f (3.4) is a point x e R" such that

/ ( * ) = 0,

that is, the solution o f (3.4) does not change in time.

In ecology, the balance o f nature can be divided into three categories:

1. the claim that natural populations have a m ore or less constant num ber or individu

als,

2 . the claim that natural system s have a more or less constant num ber o f species,

3. and the claim that com m unities o f species m aintain a delicate balance o f relation

ships, w here the removal o f one species could cause the collapse o f the w hole com

munity.

In this study, an equilibrium solution is that state in a dynam ical system where the net rate

o f change in population density is zero.

D efinition 3.2.2. Stability Let x( t ) be any solution o f the system (3.4). This solution is

stable if solutions starting close to x(t ) at a given tim e remain close to x(t ) for all later

times. It is asym ptotically stable if nearby solutions actually converge to x(t ) as t —> °°.

In our study o f species interaction, stability is understood to m ean that the interacting

species m aintain relatively steady population sizes within a given param etric region, for

increasing time.



In order to understand the stability o f solutions to our system (3.1), w e seek to under

stand the nature o f solutions near an equilibrium solution. A perturbation o f the equilib

rium solution jc(?) so that

x = x ( t ) + y ,

and substituting into (3.4), then Taylor expanding leads to

x = x ( t ) + y = f { x ( t ) ) + D f ( x ( t ) ) y + Q(\y\2), (3.5)

where D f is the derivative o f / and | • | denotes the norm on M". Since x( t ) = f ( x ( t ) )

equation (3.5) simplifies to

y = D f ( x ( t ) ) y + 0( \ y \ 2). (3.6)

Equation (3.6) describes the evolution o f orbits near the equilibrium solution x(t) . U n

derstanding the dynam ics o f the system (3.4) require understanding the dynam ics near

the equilibrium solution. Locally, the analysis o f the system (3.4) can be com plete if we

consider the linear part as defined by (3.6) w ithout the higher order terms. Therefore

y = D f ( x ( t ) ) y (3.7)

is the corresponding linearization to the system (3.4). In general, D f ( x ( t ) ) defines the

linearized matrix, called the Jacobian o f the system (3.4). In species interactions, the

eigenvalues o f this Jacobian carry im portant inform ation about the stability nature o f equi

librium solutions. A few o f the theorem s here will clarify local dynam ics and hence global

dynam ics o f the two-species system (3.1).

For stability o f equilibrium solutions o f (3.1), the following steps are necessary:

1. determ ine if trivial solution y = 0 o f (3.7) is stable.

2. show that the stability or instability o f the solution y = 0 o f (3.7) im plies the stability

or instability o f (3.4).

To determ ine the stability o f y = 0, the following theorem is necessary:

Theorem 3.2.1. Suppose all the eigenvalues o f D f ( x ) have negative real parts. Then the

equilibrium solution x = x o f the nonlinear vector fie ld (3.4) is asym ptotically stable .

Definition 3.2.3. The phase portrait o f a dynam ical system (3.4) is a partitioning o f the

state space into orbits.

By looking at the phase portrait, one can determ ine the num ber and types o f asym p

totic states to which the system tends as t —> °° (and as t —> — if the system is invertible).



Theorem 3.2.2. I f the linearization m atrix 3 has no zero or purely imaginary eigenval

ues, then the phase portra it fo r the nonlinear system near an equilibrium po in t is sim ilar

to the phase portra it o f its linearization.

This theorem , cited as the H artm an-G robm an Theorem in W iggins [40], provides

a m icrocosm ic understanding o f the nonlinear dynam ics of the system. In particular,

coupled with the following definitions, one gets a clearer sense o f the justification behind

analysis o f corresponding linear system.

Definition 3.2.4. Guckenheimer and Holmes [8] Two C vector fields, /'. g are said to

be Gk equivalent (k < r) if there exists a Qk diffeom orphism h which takes orbits <j>f (x) of

/ to orbits (jc) o f g, preserving senses but not necessarily param etrization by time. If h

does preserve param etrization by time, then it is called a conjugacy.

Definition 3.2.5. Guckenheimer and Holmes [8]: Structural Stability A C vector

field / is structurally stable if there is an e > 0 such that all C1, £ pertubations o f / are

topologically equivalent to / .

The need to establish topological equivalence betw een vector fields stem s from the

fact that the m odel system under scrutiny has nonlinearities which m ake it hard to analyze.

However if there is topological equivalence or structural stability betw een the vector field

and its corresponding linear field, then the dynam ics o f one infers the dynam ics o f the

other, at least locally.

Definition 3.2.6. A n equilibrium solution is called hyperbolic if there are no eigenvalues

with zero real parts.

Theorem 3.2.3. The phase portraits o f system (3.1) near two hyperbolic equilibria ( x q .v o )

and (xi ,} '|), are locally topologically equivalent i f and only i f these equilibria have the

same num ber n_ and n + o f eigenvalues with Re(X < Oj and with Re(X > 0), respectively.

Therefore, if the eigenvalues o f the Jacobian matrix at the equilibrium solution have

nonzero real parts, then the solution o f this system will not only yield asym ptotic behavior,

but, by H artm an’s theorem and the stable m anifold theorem , it will also provide the local

topological structure o f the nonlinear system.


CHAPTER 3. TWO-SPECIES SYSTEM

Table 3.1: Equilibrium solutions for the two species system.

21

The two species will die out with increased time and is not interesting.

The x species persists w hile the y species eventually disappear due to predation.

The y species persists w hile the x species eventually disappear due to predation.

The nonzero coexistence equilibrium solution.

By equating the right hand side o f (3.1) to zero, and solving the system

b \ y - y 2 'r \ x ^1 -

r2y ( 1 -

+

+

xy+ c \ y 2 ) k\

b2 X — x 2 \ x y

k 21 + c2 x 2

w e obtain the equilibrium solutions shown in Table 3.1.

= 0 ,

= 0 .

L em m a 3.2.4. The equilibrium solution (0 ,0 ) o f (3.1) is unstable.

Proof: Let

f = «x[

g = n y

which are the right hand side o f (3.1).

Since

2k2

b \ y - y 2

1 + c \ y 2

b2x — x 2

1 + c2x 2

xy

xy

~ki

d_d x

d_dy

d_

d x

d_dy

r\x

r \x

1 - *

k\+

- i - ) +

n y i - +

r2y 1 - Lki

+

b \ y - y 2

1 + C[_y2

b \ y - y 2

1 + c i y 2

b2x — x 2

1 + c2 x 2

b2x — x 2

1 + c2 x 2

xy_

k\

xyk\

xy_

kixy

k2

= r l -

-

k\

- ^ k 2

= r2 -

2 r \ x y a 12

k\ k\ '

2b\ - 3y 2 c, (b iy 2 - y 3)1 + c i y 2

2b2 — 3x

1 + c2x 2

2 r2y x a 2\ k 2 k2

(1 + c \ y

2 c 2 (b2 x 2

2 \ 2

(1 +C2X:2 \ 2

(3.8)

(3.9)



2

y

0

Figure 3.2: Phase portrait o f solutions starting near the origin. N otice that the flow is away from the origin, the zero solution is unstable (k = 3, r\ = r2 = 0.3, b\ = b 2 = 5, c\ = c2 = 0.5).

0.2 0.4 0.6 1.2 1.4 1.6

then let

2 r xx y c tn r xx y a n r \xJ\ \ = r \ ----------- = r i ----------------------------- ,

ki k\ ^ ^ ki

h i =xy ( 2bi — 3y 2c 1 ( b ty 2 - y 3)ki V 1 + c i y 2 (1 + c i >’2 ) 2

xy f 2b2 — 3x 2 c2 (b2 X2 — x3'•^21 = T

k 2 V 1 + C2X

2 r2y x a 2Xh i = r2 - — h ——

K2 K2

(1 + c 2 x 2 ) 2

r2y x a 2ir 2 ~ T ~ + ~ T ~ k i k 2

r iyk i

d =

/ i f d p dx ~5y

1\ dx d y ,

J\\ h i h \ h i

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

From equations (3.10) to (3.13) we observe that at the equilibrium solution (0 ,0 ), 7] 1 =



r \ , J 12 = 0, J 2 \ = 0 and J 2 2 = r2. Therefore

J \\ h i h \ -h i (0 ,0 )

n 0

0 r2(3.15)

The eigenvalues o f the Jacobian m atrix (3.15) are r\ and r2 . Since biologically these

param eters are the growth rates o f the two respective populations, these are always posi

tive param eters. It follows that the equilibrium solution (0 ,0) is always unstable. □

Define two param eters

&2X = r 1k\

P = r2 + ^ - a 2 \{k\ ) . k i

Lemma 3.2.5. The equilibrium solution ( k \ .0) is stable asym ptotically i f X < 0 and un

stable i f X > 0 .

Proof: Equations (3.8) can be re-w ritten as

n 1 - H +

r2 1 — f +

b i y - y 2

1 + c i y 2

b 2 X — x 2

1 + c2x2

k\x

ki

= 0 ,

= 0,

so that for nonzero x and y then

r 1 -r \ x , y a 2 1

k\ k\a,2 \xn y .

k 2

= 0 ,

= 0 .(3.16)

W hile J \ 2 = J21 = 0 at the equilibrium solution (&i, 0), the other Jacobian entries J\ \ and

J2 2 are sim plified by (3.16) to get

•̂ 11 J 12J 21 J 2 2 (*1,0)

- n 0

0 X(3.17)

Since r\ is a strictly positive value that represents the growth rate o f the x population, we

observe that the eigenvalues o f the Jacobian (3.17) have negative real parts if X < 0. Under

these conditions the equilibrium solution (&i,0) is asym ptotically stable. O therw ise if

X > 0 then (fci, 0) is unstable. □

Lemma 3.2.6. The equilibrium solution (0. A2) is stable asym ptotically i f p < 0 and un

stable i f p > 0.



Proof: Sim ilar to Lem m a 3.2.5 on page 23, the entries J \ 2 = 72i = 0- By (3.16), J 2 2

simplifies to —r2 while the entry J\ \ simplifies to p . Thus

J 11 J \ 2

h \ J22 (*1,0)P 00 — r2

(3.18)

The eigenvalues o f the Jacobian m atrix in this case are fJ. and —r2 with strictly positive

param eters. The equilibrium solution (0, k f) is asym ptotically stable if ju < 0 and unstable

if ju > 0. □

The non-trivial equilibrium solution (x*,y*), (x* f 0 and y* 0) as obtained from

(3.8) is im portant for coexistence o f the two sw itching species. S tability analysis o f this

coexistence equilibrium is im portant. Equations (3.16) will help sim plify the entries J\ \

a n d 7 22 in the Jacobian matrix, such that (3.10) and (3.13) become

r \ xJ \ 1 =

k\, r2y

J22 = — k-2

respectively. Substituting (3.19) into (3.14), we obtain

(3.19)

V ,v * ) -

/ n x*k\

\ h \

J 12

r y f k2 /

(3.20)

L em m a 3.2.7. The coexistence equilibrium solution (x*.y*) is asym ptotically stable i f the

inequality$ ̂r \r 2 X y

J \ 2h \ <k \k 2

(3.21)

is satisfied.

Proof: The Jacobian matrix in (3.20) has trace t and determ inant d given by

t = -r \x * r2 y*k\ k2

, , r \x r2yand d = ------ J n J2\.

k\ x2

Since trace is always negative, for asym ptotic stability we require that the determ inantf\x* r2y*

be positive. This is possible i f --------------> J \ 2J 2 \ - It m ust be m entioned that for two-k\ k 2

dim ensional system s, the necessary and sufficient condition for local stability o f a fixed

point is that the trace o f the Jacobian m ust be strictly negative and the determ inant o f the

Jacobian m ust be strictly positive. □



B

7.5

7

6.5

6

5.5

5

4.5

4

3.5

3

x stabilizes at about 5.5

1,5

).5

—r ~ r t i l l

>• stabilizes at about 0.4

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

1 t

Figure 3.3: Time series plots o f populations x( t ) and y( t ) stabilizing to the coexistence equilibrium solution with increasing time.

Since the trace o f the Jacobian m atrix is always negative, we then present a schem atic

view o f all the possible equilibria and the nature o f stability in the trace-determ inant plane,

(Fig. 3.4).

D e t .Stablespiral

Stable node4Det

T r a c e J

Saddle

Figure 3.4: Stability regions in the corresponding two species system.



A pplying the H artm an-Grobm an theorem for each equilibrium point and observing

the hyperbolicity o f the particular point, one can make conclusions about the long term

behavior o f the system near the equilibrium point. The problem arising from applying this

tool is w hen the equilibrium solution is nonhyperbolic, which m eans that the real part of

the eigenvalue corresponding to the m atrix jacobian at that equilibrium solution is zero.

In this case we w ould consider higher order term s’ contribution to the system behavior.

In some cases norm al fo rm analysis will determ ine which higher order term s contribute

m ore towards system behavior about the equilibrium point.

A bifurcation is a qualitative change in the behavior o f solutions as one or m ore param

eters are varied. The param eter values at which these changes occur are called bifurcation

points. If the qualitative change occurs in the neighborhood o f a fixed point or periodic

solution, then it is called a local bifurcation, otherw ise it is global. For m ore explanations

and descriptions about bifurcations, see [31], [40], [32], [8].

We include diagram s o f bifurcation analysis o f equilibria as we vary sensitive param

eters, in this case the the carrying capacity k o f the species environm ent. We note that the

equilibrium density o f the species x change, (bifurcate) as we vary this parameter.

Lemma 3.2.8. I f X — 0 then the equilibrium solution to the system (3.1) undergoes a

transcritical bifurcation.

Proof: The coordinate transform ations k \ , y ^ y , X ^ X, preserve the dynam ics

o f the original system , which we write as

x — F ( x , X ) , x €

where x = (x.y).

For this we verify that

• F(0,0) =0.

# c>F(0 , 0 ) Qd x

The two fixed points of

are given by

and

x — F ( x , X ) , x e l 2.

x = 0

x = X.



Hence for A < 0 there are two fixed points; x = 0 which is unstable and x = A which is

stable. These two coalesce at A = 0. For A > 0,x = 0 is stable and x = A is unstable, see

fig. 3.5. This m eans we have a transcritical bifurcation. □

Figure 3.5\ Exchange o f stability occurs at A = 0, a bifurcation known as transcritical bifurcation. The dotted line is the unstable manifold.

A plot o f one species density against its corresponding carrying capacity provides

some graphical way o f observing hysteresis as the carrying capacity param eter k is in

creased. Hysteresis is the lack o f reversibility o f system structure or com position as a

param eter is varied.

The disappearance o f the top nodal sink as it fuses with the non-trivial saddle equi

librium, further confirming another bifurcation o f equilibria, provided som e favorable

predation conditions at the expense o f a friendly environm ental co-existence habitat. If

unchecked, drastic changes occur leading to a rapid loss o f predator population followed

by a subsequent rise in prey population, possibly explaining the reverse effect that was

w itnessed in the South A frican Islands, [11], By further increasing the sensitive param eter

m, the predator population disappears. N otice the characteristic role switch exhibited by

the exchange o f species interaction in the form ation o f saddle points between Fig. 3.7 and

Fig. 3.11. The transition o f exchange o f equilibria from a low coexistence equilibrium

o f one species, through a saddle and two foci, to a low coexistence, then disappearance

o f coexistence equilibria is sim ulated num erically through phase plane diagram s o f the

phase portraits o f the two dim ensional system, (Fig. 3.6 to Fig. 3.13). These variations

are due to changes in the carrying capacity o f one o f the species. This approach not only



7

6

S

4

3

2

1

0

0.5 2.5 3 3.5 40 1 1.5 2X fcj

Figure 3 .6 : U nder certain conditions, the single species system stabilizes at the respective environm ental carrying capacity.

shows the trajectory o f the system , but also indicates the type o f equilibrium in term s of

its stability state.

It can observed that depending on choices o f param eters, one obtains som e variations

o f population interactions. Hysteresis is biologically im portant in that som etim es contin

ued increase o f a sensitive param eter m ay result in som e abrupt or sharp changes in the

dynam ics o f the w hole system. It w ould require a lot o f energy to force the system back

to its original form. Hysteresis m ay as well lead to extinction o f som e species. The upside

down s-shaped form o f the solution curve is typical o f hysteresis, with upper and lower

branches com prising stable equilibria w hile the m iddle branch is unstable, Fig. 3.8. At

the turning points o f this curve, small changes in k m ay provoke som e catastrophic jum p

between stable branches.

We have discussed the stability conditions o f solutions o f the corresponding two

species system. We have observed that for single species equilibria, there exists a critical

param eter value at which the solution becom es nonhyperbolic, that is, that eigenvalues of

the Jacobian for the linearized system have zero real parts. In this case we conclude that

the system becom es structurally unstable so m uch that the dynam ics near the equilibrium



9

a

7

e

4

3

2

1

0

0 ’ 2 3 4 5 6 7X

Figure 3.7: There is a critical /: value at which a saddle point is born while the stable spiral persists.

solution cannot be the sam e as those o f the corresponding nonlinear system when the

sensitive param eter is zero. A sum m ary o f these observations is thus, as follows:

1. Orbits and therefore vector fields are structurally stable in the vicinity o f hyperbolic

fixed points, Thm. 3.2.2.

2. In the neighborhood o f bifurcation points, the orbits and therefore the associated

vector field is structurally unstable.

3.3 Symmetry and bifurcation in 2D

The two species system (3.1) can be described from a sym m etric view point.

1. Full sym m etry is defined when all the corresponding param eters are the same, that

is, when we haver\ = n = r,

k\ = k 2 = k,

b\ = b 2 = b,

Cl = c2 = c,

i i i ................................ 1 ------------------- 1

i

/

i 1 /

T T ' 7 ................i, i

i * i1

T - i - 7 ...................

f i r

V. i ________ . .

5

■



9

7

6

5

4

3

2

00 1 2 5 6 74

X

Figure 3 .8 : Increasing & from 2 to 3 leads to three equilibrium solutions, two stable spirals with a saddle between them.

7

6

5

4

3

2

1

00 0. 5 1 15 2 2. 5 3.5 4. 5 54

X

Figure 3 .9 : For certain param eter choices, a spiral focus with low x-species density and high y-species density em erges, (k\ = 2,&2 = 5 ,c i = 0.5, C2 = 3).


CHAPTER 3. TWO-SPECIES SYSTE M 31

7

6

5

4

3X

2

0~ 1- —

3.50.50 1.5 2 2.5 3 4.5 54

Figure 3 JO: Phase portraits showing a stable spiral focus with low ^-species density and high y-species density and a saddle.

3.5

2.5

0.5

0 0. 5 1.5 2 2. 5 3

Figure 3 .11 : Increasing k\ from k\ — 3 to k\ = 5 leads to the disappearance o f one spiral solution through a saddle form ation.



4

5

>5

2

1.5

1

).5

01.5 2 2.5 S0 0.5 1

X

Figure 3 .12 : A single coexistence equilibrium solution exists for k\ > 4.5.

2

1,8

15

y1.4

1.2

^2 1

08

0.8

0.4

0.2

0

Figure 3 .13 : No coexistence in certain param eter space, each species exists independent o f the other and is lim ited by the environm ental carrying capacity.

o 0.2 0.4 0.6 1 1.4 15 2

k\ x



so that (3.1) becom es

d x ( x \ f b y — y 2 \ x y

* =rxV-l) + \ T T ^ ) l d?=r>,(i-r) + (bx-xl'\v

(3.22)

dt y V k J ' \ 1 + c x 2 J k

2. Efficient asymmetry, [38] is when the predation rate differs from the assim ilation

rate. Excessive efficient asym m etry is understood to mean that the predator is hav

ing problem s in capturing its prey. Therefore in our model

b\ ^ b2 either b\ > £>2 or b 2 > b\ ,

c \ 7̂ c 2 either c\ > c2 or C2 > c\.

3. Driving asym m etry is when the environm ental carrying capacities and the growth

rates are different so m uch that one o f the species dom inate the other. This usually

results in extinction o f one with the persistence o f the other.

Given these points, a perfectly sym m etric system (3.22) would be ideal for consideration

given the sw itching behavior as described in [11]. It is through variation o f a sensitive

param eter such as the environm ental carrying capacity that one obtains different equilibria

with different stability conditions.

From system (3.1) we already observed that (3.8) lead to equilibria such that there is

an equilibrium at x = 0 for all param eter values. Now if we consider the point (k\ ,0 ) as

the bifurcating point, then, near this point

t ( \ r,X l y ( X l 2 nfi(x,y) = n ~ — + — = 0,(3 23)r2y x a 2i

h ( x , y ) = r2 ~ — + = 0.ki k 2

Equilibrium solutions are obtained by solving the system (3.24) simultaneously,

x f \ ( x , y ) = 0 ,(3.24)

y f 2 ( x , y ) = 0 .

N on-zero solutions o f equation (3.24) had to be solved numerically, m ore so, under sym

m etry considerations. Let r\ — r2 — r, k \ — k 2 — k, and note that the sensitive param eter

is k.

As the param eter k is varied, two fixed points approach each other, collide, and are

destroyed. U sually this phenom enon occurs when we have what is referred to as a saddle-

node bifurcation or fo ld . The following conditions are necessary, but not sufficient for a

fold bifurcation to occur:



• F ( x c, kc) — 0.

• D XF m ust have precisely one zero eigenvalue, w hile the other eigenvalue has a

nonzero real part at (xc, kc).

In our case, the function F ( x , k ) is the two dim ensional right hand side o f (3.24)

for w hich D XF defines the Jacobian o f the linearized system at the critical equilibrium

(xc,yc) and the critical sensitive param eter kc. However because o f symmetry, we can

only expect either a transcritical bifurcation or a pitchfork bifurcation. To distinguish

between the two, we w ould require that d 3 F / d x 3 / 0 for a pitchfork bifurcation. In

this case, the sign o f this third partial derivative determ ines w hether the branching is

subcritical, (d 3 F / d x 3 > 0) or supercritical, (d 3 F / d x 3 < 0 ) .

Figure 3.14 shows num erical results at the bifurcation point, i.e. we have for b\ =

= 3, r\ = r2 — 0 .3, then

xc = 3.0737, y c = 3.0737, kc = 3.0000,

correct to four decim al places. We verify the num erical results by substituting x = y in

(3.24), and substituting param eter values into the resulting polynom ial

rx bx2 — jc3r 1 = 0

k k( 1 + cx2)

and sim plifying we get

x3 — 3x2 + 0.05x — 0.85 = 0.

The general solution o f a cubic polynom ial equation o f the form

ax 3 + bx2 + cx + d = 0

is given by

x = \/ —ft3 be y 2 7 a 3 ^ 6 a 2

be27a3 6 a 2

d 2 a

c ft2 3 a 9a 2

+ \/ —ft3 be \ 2 7 a 3 6 a 2

d_ 2 a

3 ( ~ b 3 beV V 27a3 6 a2d_

2 a+

c3 a

iF_ 9a2

b_ 3 a

(3.25)

By substituting a = l ,f t = —3 ,c = 0 .0 5 , d = —0.85, we obtain that x = 3.0737. This

m eans that the equilibrium solution at the branching point is the coexistence equilibrium

(3 .0737,3 .0737).



6

5

4

BPx 3

2

1

BE0 L-10 •5 0 5 10 15 20 25 X 35 4C

k

Figure 3.14: Variation o f param eter k with x showing BP-branching point, LP-fold bifurcation and H P-H opf bifurcation.

3.4 Other Corresponding Two Species Systems

In the previous section we analyzed the two species system that exhibited sw itching be

havior. We observed that variation o f either the interaction param eter b or the environ

m ental carrying capacity under symmetry, produced dynam ics that include bifurcation

o f equilibria in certain param eter regions. This m odel (3.1) was betw een the two prey

species in the absence o f the top predator. There is a critical carrying capacity value kc

where we observed that beyond this value, one o f the species disappeared. This m ay or

m ay not be the reality should there be the existence o f the top predator, which during

the period o f low density o f one due to predation, the top predator would prey on the

dom inant prey species, thus reversing the gain.

W hen one species is absent or occurs at low density due to predation we look at

the dynam ics o f the rem aining system. In that case we have either o f the following two

species system s, whose interaction can be understood by analyzing one o f the two system s



(3.26) and (3.27),

(3.26)

or(oy2z

d + y 2 ’(3.27)

Equations (3.26) and (3.27) are equivalent in that the reduced system is one o f a

predator-prey relationship.

We define the equivalent system to be

and observe that this system is a sim ple two species predator prey system. The predation

term is one incorporating the Holling Type III response function. In Table 3.2 we ex

plain the m eaning o f the term s and the corresponding param eters that are involved in this

system. The predator species equation is logistic. The positive contribution in the growth

rate o f the predator is achieved if there is m inim um com petition am ong predators given

the food abundance and choices that they enjoy.

For equilibrium solutions we equate to zero the right hand side o f (3.28) and solve the

resulting system.

(3.28)

(3.29)

s \U• If u — 0 then u s \ —

iOuv= 0 and v = 0 or v = q.

k d + u2

If v = 0 then for u ^ 0 we have that u = k.

• If both m / 0 and v ^ 0 then v — q + mu and

2 oCl .su + u ( q ( 0 + m(D + — ) — sd = 0.

K(3.30)



Table 3 .2 : Param eters and variables describing the predator-prey system (3.28).

Symbol Description

u population density for the prey species u

V population density for the predator species v

Si intrinsic rate o f increase o f prey species u

S2 intrinsic rate o f increase o f predator species v

k environm ental carrying capacity o f the prey species u

q and m constants defining environm ental carrying capacity o f species v

d predation saturation term

Equation (3.30) is solved again using the form ula developed by Cardano [1545], for

cubic polynom ial equations w here, in this case, a = s / k , b = —s , c = q ( 0 + m ( 0 + s d / k , d =

—s d .

Before solving this system, we w ish to recall D escartes’ Rule o f Sign for polynom ials:

L em m a 3.4.1. The num ber o f positive roots o f a polynomial, counting multiplicity, is

equal to the num ber o f sign changes o f the polynom ial or that num ber decreased by an

even integer.

Since all param eters are positive, the polynom ial equation (3.30) alternates in sign.

Therefore we expect either three positive roots, or one positive root depending on the

choice o f param eters.

Substituting the coefficients into (3.25) we obtain the coexistence equilibrium solution

o f the form (u*,q + mu*). Specific param eter values w ould result in num erical solutions

to the cubic polynom ial. We here choose

• an environm ental carrying capacity, k = 5,



/ ( « ) 60 f ( u ) = m3 — 5 u2 +5u — 5 >—J

40

I///

/

20

//

u = 4.0739/7

4 /-2 -1 0 _____________2 3 5 6

^ ---- ft-2 0

/ - 4 0 -

Figure 3.15: D epending on param eter choices, we get either three or one real roots, in this case for q = 1 — d .m — (O = 0 .2,5] — S2 = 0.3, £ = 5 we have a single coexistence equilibrium solution.

• the growth rate o f the prey species, 5] = 0.3,

• the predation saturation factor, d = q = 1,

• the predation rate, ( 0 = 0.2,

• and the predator com petition control factor, m — 0.2.

These values lead to the cubic polynom ial 0.06w3 — 0.3w2 + 0 .3u — 0.3 = 0, so that

dividing throughout by 0.06 the equation reduces to

m3 — 5m2 + 5m — 5 = 0. (3.31)

The graph o f the cubic function f ( u ) = u 3 — 5u2 + 5u — 5 has a single positive zero

which occurs at u = 4.0739, using N ew ton’s M ethod. So, the corresponding value for

v that results in the coexistence equilibrium is v = 1 + 0 .2 x 4.0739. The coexistence

equilibrium solution is (4 .0739 ,1 .8148), given the selected choice o f param eter values.

O ther param eter values choice could result in three coexistence equilibrium solution. This

m eans that by keeping all the param eters but one constant and varying that param eter,

equilibrium solutions bifurcate from one to three.


CHAPTER 3. TWO-SPECIES SYSTEM

Table 3.3: Equilibrium solutions for the two species system.

39

The two species will die out over time.

The u species persists while the v species eventually disappear due to predation.

The v species persists while the u species eventually disappear due to predation.

The nonzero coexistence equilibrium solution.

In summary, Table 3.3 shows the type o f equilibrium solutions that is obtained under

this model.

We state w ithout p roof the following lem m a showing through phase portraits, the

nature o f stability o f equilibria.

L em m a 3.4.2. The coexistence equilibria i f existing, com prise globally asym ptotically

stable spirals and unstable saddle points.

The coexistence equilibrium solution for this two species system explains the fact

that with low density o f one prey source for the top predator, the predator is now finding

it hard to search for food and m ake a kill. In the process, this leads to a decrease in

population size o f the predator population. M eanw hile the prey species struggles to regain

as there also is less food source for survival. In this case, the other prey species begins to

grow in size because o f heavy presence o f the top predator against its im m ediate threat.

The follow ing figures, (Fig. 3.16 to Fig. 3.19) show phase portraits o f som e coexistence

equilibrium solutions, (saddles and spirals), depending on certain param eter variations.

These phase portraits are obtained as the carrying capacity of the prey species is varied

while other param eters rem ain fixed.



v

40

20

5 62 4 6 7O t

U

Figure 3 .16 : For large carrying capacity d — 5 o f the predator population, the only coexistence eauilibrium is the stable SDiral.

6

S

4

2

0

0 0.2 0.4 0& f .2 14 21 I d

U

Figure 3.17: There is a critical d (d=5.45) value at w hich a saddle point betw een two spirals is bom .



6

5

4

3

2

1

0

0 6 1.2 1.4 1.6 20 0 4

Figure 3 .18 : There is a critical (= 3.25) value at w hich a saddle point is born w hile the stable spiral persists.

“ I------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1------------1T

' I f f I I I................ i.............. V0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 3 .19 : By zoom ing near the origin, the two spirals and a saddle persists for very small d (= 0 .15).


Chapter 4

THREE SPECIES SYSTEM

4.1 Invariant Set

The system given by equations (2.4a)-(2.4c) provides meaningful solutions for all

(x, y,z) G R 3 such th a tx > 0 ,y > 0 ,z > 0. The variables x ,y and z are population den

sities o f the interacting species, therefore it is essential that solutions o f (2.4a)-(2.4c),

initially positive, will remain positive for all time.

Lemma 4.1.1. The non-negative region

^ 3 = {(*,;y,z) £ R + : x < k \ , y < k 2 , z < q + m(k\ + k2)}

is positively invariant fo r system (2.4a)-(2.4c).

Proof: In TIt, we have the following inequalities

x = 0 > 0 when x = 0 or Jc > 0 when 0 < x < k \ ,

y — 0 > 0 when y = 0 or y > 0 when 0 < y < k 2 ,

z > r^z for z < q + m(k\ + k 2).

Thus the nonnegative region Q. is positively invariant for the system (2.4a)-(2.4c). □

Lemma 4.1.2. A ll nonnegative solutions o f the system (2.4a)-(2.4c) are ultim ately uni

fo rm ly bounded in fo rw ard time, and thus they exist fo r all positive time. Moreover, the

system is dissipative.

Ideas o f the p roof o f this lem m a require that under certain environm ental conditions

where the predator-prey relationship is well defined, the response function that should

be em ployed m ust definitely determ ine which species is prey and which is predator for

the corresponding two species system. However we note that all the three species are

governed by the logistic part o f the equations, thus in forw ard time

x < n * ( l —x / k \ ) ,

y < r2y(\ - y / k 2),z < r 3z (l - z / h ) w here £3 > q + m{k\ + k 2).

42


CHAPTER 4. THREE SPECIES SYSTEM 43

This leads to

x ( t ) < xoexp(C t) for som e constant C,

y( t) < yo exp (D t ) for som e constant D, £>2 < y < ^2 ,

z ( t ) < zoexp(r3 t) w here £3 > q + m(k\ + £2 )-

4.1.1 Symmetry

Equations (2.4a)-(2.4c) are to be considered for determ ination o f equilibria. Equilibrium

solutions and their stability analysis are necessary to understand the dynam ics o f the sys

tem under consideration. We observe that little w ould be obtained from this nonlinear

system w ithout considering special structural conditions that m ay influence the dynam

ics o f the system. We therefore begin by considering the system to have a sym m etric

structure.

Definition 4.1.1. A system is said to be sym m etric in the biological sense o f interactions

if all the interacting species have the same environm ental conditions and their growth

rates are the same.

For equilibria, the right hand side o f each o f the equations (2.4a)-(2.4c) is equated to

zero. To determ ine equilibrium solutions under symmetry, that is, by consideration o f the

transform ation (x , y , z ) — *• (y ,x,z) , we first equate the right hand side o f (2.4a)-(2.4c) to

zero. This m eans that the third equation (2.4c) simplifies to

w = q + 2 mx, (4.1)

while the first equation, under symmetry, becom es

/ x \ f b \ x — x 2\ x 2 (Ox2(q + 2 mx)

<42)

Sim plifying (4.2), by substituting (4.1) into (4.2), we obtain a quintic polynom ial in

the variable x and param eters b , c , d , k , m , q , r and ( 0 (ignoring subscripts):

dkr — x(rd + k(Oq) + x2 (kr + krcd + bd — 2 km(o)

— x 2(r + d + kcq ( 0 + cdr) + x 4(ckr + b — 2ckm(o) — x5(l + cr), (4.3)

whose solutions will give rise to the x and y com ponents o f equilibrium solutions o f the

form (x*,y*,z*). Stability o f equilibria under sym m etry is an im portant analysis.



4.1.2 Real Roots under Symmetry

From (4.3) let r — 2m ( 0 = T. Then the quintic polynom ial can be w ritten as

—do u5 + a\U4 — (32 M3 + — CI4 U + £?5 (4.4)

where

ao = 1 + rc ,

d\ = k c z + b,

(32 — r + d + red + kcqco,

(33 = k t + fcrcd + bd,

d4 — rd + kq(0 ,

«5 = krd.

Since all the param eters are positive, we recall that

• k- is the param eter defining the environm ental carrying capacity o f the prey species.

• c- is the param eter that provides the dum ping effect o f the predation reaction.

• d- is the param eter that saturates the prey population due to predation.

• b- and ( 0 are the param eters that increase or decrease predation.

• q- is the param eter that norm alizes the residual reduction in the predator population

due to scarcity o f favorite food.

• m- is the param eter that saturates the predator population with food abundance.

• r- is the prey population growth rate.

It is therefore expected that for coexistence or persistence o f the three species we

require that r > 0 .

Lemma 4.1.3. The condition T > 0 is d necessdry but not sufficient condition f o r the

quintic polynomiol to hdve 5 ,3 or 1 red I roots.

Proof: By D escartes’s sign rule on (4.4) we observe that all the a,-, i — 0 , . . . 5 are

positive, provided T > 0, that is m ( 0 < § . There are therefore five sign changes, m eaning



that we w ould get a m axim um of 5 positive rational roots, 3 possible rational roots or 1

rational root. □

Param eter variations will always be key to the determ ination o f the num ber o f positive

real roots o f the quintic polynom ial (4.3). The choice to fix some o f these param eters is

often guided by such pre-determ ined analysis o f possible outcom es through sensitivity

analysis. Obviously, there are so m any param eters in the polynom ial and some have to

be fixed w hile the overall dynam ics o f the system under sym m etry depend on certain

particularly sensitive param eters.

4.1.3 Tinning Points

Given the conditions for existence o f positive rational roots, we choose some param eter

values that m eet these conditions. Table 4.1 lists the param eter values for the model. To

decrease the com plexity o f this model, all param eters except those that directly increase or

decrease population interaction or predation, were held constant. It should be m entioned

that the param eters k and d are not necessarily unique in generating dynam ics that lead to

chaos in som e param etrization o f the model. B arradas and H ernandez [11] in their two

species system analysis that resulted in hysteresis, considered the environm ental carrying

capacity as one o f the sensitive param eters. E isenberg and M aszle [4] regarded their

sensitive param eter to be the saturation param eter which tends to stabilize interaction in

predation.

The corresponding quintic polynom ial (4.3) that results due to sym m etry w ould there

fore be w ritten as

p(x) = —1.2x5 + (0.1£ + 5)x4 + ( - 0 . 4 - \ . 2 d - 0 A k ) x 3

+ (0.2 k + 5 d + 0.2 kd ) x 2 + ( - 0 . 4 d - 0.2k)x + OAkd. (4.5)

The turning points for this equation are determ ined by solving the system

p{x) = 0 ,

p'{x) = 0,

wherep{x) — aox5 T a \ x 4 + a2 x 2 + a^x2 + a^x + a$ ,

p' (x ) = 5<2o* 4 + 4a i x 3 + 3a2 x 2 + 2 a^x + <34 ,

p' (x) denote the derivative with respect to x o f the polynom ial p(x ) , and the a j , i = 0 . . . 5

are as in Table 4.1. Using the resultant o f the quintic equation (see [33], [39], [2], [15],


CHAPTER 4. THREE SPECIES SYSTEM

Table 4.1: A ssigning values to certain param eters, k and d vary.

46

Param eter Value

b 5

c 0.5

m 0.5

q 1

r 0.4

( 0 0 .2

T 0 .2 (calculated, t = r — 2 mat)

ao 1 .2

a\ 0 A k + 5

(32 0 .4 + \ .2d + 0 . \ k

(33 0.2k + 5d + 0 .2kd

(34 0 A d + 0.2k

(35 OAkd

[19]), we obtain that the discrim inant with respect to the two polynom ial equations (4.6)



for the quintic polynom ial (4.5) is

ao a\ a 2 <23 (24 <35 0 0 0

0 ao a\ a 2 (23 (24 a 5 0 0

0 0 ao a\ (22 a-i (24 a5 0

0 0 0 ao a\ a 2 <33 (24 <35

bo b\ bi h b4 0 0 0 0

0 bo b\ bi h b4 0 0 0

0 0 bo b\ b2 bj. b4 0 0

0 0 0 bo b\ b2 b~i b4 0

0 0 0 0 bo b\ b 2 b3 b4

We observe that in this case the i = 0 . . . 4 are m ultiples o f the a,- i — 0 . . . 4 such

that bo = 5ao,b{ — 4 a \ , b 2 — 3a 2 , b 3 = 2a3 and b4 = <34. We require that for a non-trivial

solution to be obtained, this determ inant m ust be zero;

^■(p ,p') = 0-

Ideally, the resultant condition w ould determ ine param etric conditions for the sensitive param eters k , d that guarantees the existence o f turning points. They could also provide the param eter region where there are these turning points. However the determ inant o f this m atrix is an odd m ixture o f sum m ation o f term s involving these two param eters:

rR(/,y ) = 10252.08253d5 + 79.76225893Ar5d4 - 0.15704064? - U 6d 6

+ 0.1849344? — 2k1 + 136.3582648d4 +415.1209820W 3 - 17329.86828£2d3

+ 185.6106455fc3d + .817275648fc6d4 + 3022.876625k4d4 - 182.5052844ArV

+ .340328448(fc2d - 1385.453962d4£ + 265880.1060d4fc2 + 50222.36202ArV

— 0.26515968? - I k ’d 2 +0.2013696? - I k ’d4 - 14.69000908ksd 5 + 1227.224383d 6

- .361961472&6d5 - 504461.2110d5k - 113330.1009d5k2 - 8459.885367k3d 5

+ .152262912£6 - 0 . 3772416c - I k ’d 5 + ,680656896/W2 + 0.6144? - 5ks

- 1,721032704fc5d 6 - 1.782616320fc6d - 2.403071424£6d2 - 3.381784320fc6d3

+ .453771264d3 +437.7058714£2d 2 - 0.89088? - 4£8d2 - 0.4608? - 5k%d4

-0 .12288? - 4k*d5 + 36.75547238d 1 - 0.31544832? - 1 k1 d 5 - 0.26781696? - 1 k7d

- . 127401984fc4d7 - 1815Al%212d1k - 293.852676Id7*2 - 9.55514880Ar3d7

+ 0.56721408?- U 3 + 27.51232512>t4 - 6195.408007Ar3d2 - 229.4885099£4d

- 1342.401032/t4d 2 - 35696.95305Ar3d 3 -4 8 0 2 .1 29833fc4d 3 + 4.0042629\2k5

-355.1150589k4d 5 -39 .18777139A:5d - 94.45107687k5d2 - 0.98304? - 4k8 d 3

- 0.98304? - 4k*d - 81.93927172k4d6 - 60428.31572d 6k - 17214.25804d6fc2

- 1789.955851A:3d6.



The critical values o f k and d at w hich this resultant is zero have to be obtained.

However given the two param eter polynom ial it is w ise to fix one o f these param eters and

then solve the resultant equation to obtain the values o f the other such that the com bination

would result in the resultant to be zero, [19]. For these critical values, the fact that the

resultant equals zero is a necessary and sufficient condition guaranteeing the existence o f

the turning points.

4.1.4 Bifurcations

We have tried to determ ine the dynam ics o f the three species system through sym m et

ric analysis where w e considered the transform ation (x , y , z ) — > (y , x , z )• The resulting

sim plifications led to the one-dim ensional vector field

dx— = p ( x ,d , k ) , i g R , G R, (4.7)dt

where p is a C r function on som e set R x R . This general vector (4.7) field can further

be reduced to a one param eter, one-dim ensional vector field such that for som e critical

param etric value Ao the point (jco, Ao) is a fixed point. In that case we have that

p (x 0,Ao) = 0 .

Stability o f the fixed point (jco, Ao) is im portant, and so is the need to determ ine the

dynam ics o f the system locally about this fixed point as A, the sensitive param eter, is

varied.

The flow o f the system described by (3.1) is said to be qualitatively the same as the

flow o f its linearization if, by the H artm an-G robm an theorem , the eigenvalues o f the Ja-

cobian o f the linearized system are hyperbolic, that is, the eigenvalues o f the linearization

has no eigenvalues on the im aginary axis.

Definition 4.1.2. A fixed point ( jc, A) = (0 ,0 ) o f a one-param eter fam ily o f one-dim ensional

vector fields is said to undergo bifurcation at A = 0 if the flow for A near zero and jc near

zero is not qualitatively the same as the flow near jc = 0 and A = 0 .

In the situation where these eigenvalues are non-hyperbolic, qualitative resem blance

o f the flow near the fixed point is not guaranteed. Norm al form analysis procedure deter

m ines the contributing terms that play m ajor roles in determ ining system behavior near

the fixed point.



Generically, the one-param eter fam ily o f vector fields defined as

— = p ( x ,X ) = Xx — x3 + 0 (x:4) at

(4.8)

which provides the pitchfork bifurcation shown in Fig. 4.1 can further be perturbed to

Figure 4 .1 : G eneric diagram o f the pitchfork bifurcation resulting from equation (4.8) ignoring higher order term s. The dotted line is the unstable equilibrium .

give the form

Equation (4.9) is a degenerate case o f a fc-codimension bifurcation where k = 2. Theoreti

cally, w hat this says is that the system (3.1) is not structurally stable and is o f codim ension

The study o f behavior o f fixed points is ideally a m uch sim pler approach tow ards anal

ysis o f structural stability o f (3.1). If it were easy to determ ine the equilibrium solutions

o f the system (2.4a)-(2.4c), we w ould sim ply look for those fixed points which are non-

hyperbolic. However, we w ent around this problem because o f the m any non-linear and

rational term s in system (2.4a)-(2.4c).

We wish to determ ine the conditions that give rise to a cusp bifurcation [31] through

the use o f the Theory o f Resultants.

A cusp bifurcation occurs when the follow ing conditions are satisfied:

x .2

(4.9)

k = 2 .



1. A fixed point with a zero eigenvalue.

2. A second derivative o f the vector field vanishes in the direction o f the eigenvector

corresponding to the zero eigenvalue.

The sym m etric transform ation (x , y , z ) > (y ,x,z) reduced the three species system into

a one-dim ensional system

x — p ( x , n ) , x £ M, p € M2. (4.10)

We seek conditions which give rise to a cusp bifurcation. We note that the function p ( x , p )

is smooth since it is a polynom ial function in both x and A, hence adm itting continuous

partial derivatives. In that case w e are led into determ ining the resultant o f the system

p{x) = 0

p'[x) = 0

p"(x) = 0 ;

this last system being a m ultivariate resultant. Therefore

pi (u) = - 1 .2jc5 + (0 .Ik + 5)x4 + ( - 0 .4 + \ . 2d + 0 . \ k ) x 3 + (0.2k + 5d + 0.2k d )x 2

+ (—0.4c? - 0.2k)x + OAkd = 0

p 2 („) = - 6 x 4 + (0.4k + 20 )* 3 + ( - 1 .2 + 3.6 d + 0.3 k)x 2 + (0.4k + 1 0 d + 0 A k d ) x

- 0 .4 c /-0 .2 k = 0

p 3 (n) = - 2 4 x 3 + (1 .2k + 60)x2 + ( - 2 .4 - 7.2c/ + 0.6k)x + 0.4k + 1 Od + OAkd = 0.(4.11)

The process to determ ine w hich param eters influence m ost o f the system behavior is done

through sensitivity analysis. The nonlinear term s in the model system im ply that there are

m any param eters w hose variation will lead to com plicated dynam ics. We assign values

to some o f the param eters as a way to non-dim ensionalize the system. Appendix A .l p ro

vided a num erical approach to determ ine w hich param eter was giving different dynam ics

as that param eter was veing varied. We assigned values to some o f the param eters, leav

ing two free param eters. We observed that the use o f num erical techniques to find the

critical values o f these param eters, though still involving the use o f the theory or resul

tants, w ould be em ployed, only this tim e under a polynom ial system with two quadratic

and one linear equation in these sensitive param eters. Such a problem is often referred to

as a codim ension 2 problem.

The quintic polynom ial (4.5) is reconsidered to determ ine conditions that give rise to

cusp bifurcation. To determ ine these conditions, the system of polynom ial equations



p(x) = 0

p'{x) = 0

p"{x) = o,

where (4.5) is derived from the general quintic equation (4.3) by considering the values

in Table 4.1 is considered. The transform ation

P2 ^ P2~XP3

transform s the polynom ial p 2 in (4.11) into a linear equation in k and d so that the resultant

is then applied to the system o f polynom ial equations

p i ( jc ) = - 1.2x5 + (0. I k + 5)x4 + ( - 0 .4 + 1 .2d + 0 A k ) x 3 + (0.2k + 5d + 0.2kd)x 2

+ (—0.4d - 0 .2k)x + OAkd = 0,

p 2 (x) = — 24jc3 + (1.2 k + 60)x2 + ( - 2 .4 - 1.2d + 0.6k)x + 0.4A: + 10rf + OAkd = 0,

P 3 ( jc ) = 18x4 + ( - 0M - 40)x3 + ( 1.2 + 3.6d + 0.3k )x 2 - OAd - 0.2k.(4.12)

Treating x as a constant and introducing a variable y, we hom ogenize the system to obtain:

p\ = (—1.2x5 + 5x4 — 0.4x3 )y2 + (O.lx4 + 0.1x3 — 0 .2 x ) k y + (1.2x3

+ 5x2 - 0 A x ) d y + (0.2x2 + 0 .4 )kd

P2 = (—24x3 + 60x2 — 2.4x)y2 + (1.2x2 + 0.6x + 0.4)A:y+ (—7 .2 x + \ 0 ) d y (4.13)

+ (0.4)1W

p 3 = (18x4 — 40x3 + 1,2x2 ) y + (—0.8x3 + 0.3x2 - 0 .2 )£ + (3.6x2 - 0 . 4 )d.

The first two equations o f (4.13) are degree two hom ogeneous polynom ials in the vari

ables k , d , y . The last equation o f (4.13) is a linear hom ogeneous equation in k , d and

r-*

The eigenvalue problem p ( X ) x — 0, w here p ( A) = ^ A 'Aj, was extensively studiesi=0

in [33], [19], [18], [15], [2], We are m ainly concerned with regular m atrix polynom ials,

that is, those polynom ials p ( A) such that d etp (A ) is not identically zero for all A 6 C. In

such polynom ials, the eigenvalues are precisely the roots o f the scalar polynom ial defined

by detp (A ).

D efinition 4.1.3. Let p ( A) be an n x n m atrix polynom ial o f degree k > 1. A pencil

L(A) = AX + y with X , Y belonging to a field o f real or com plex num bers, is called a



linearization o f p (X ) if there exist unim odular m atrix polynom ials E ( X ) , F ( A) such that

E ( X ) L ( X ) F ( X ) = ’ p(A) 00 I ( k - \ ) n

The spaces o f pencils provide a convenient platform to look for linearizations o f struc

tured polynom ials, such as those presented by sym m etric or skew -sym m etric coefficient

m atrices or those defining palindrom ic structures, [21], [12] . Unfortunately, a num ber

o f contributions towards analysis o f system s that are not necessarily structured in any

way have mainly been those w hose coefficient m atrices are square. The problem under

scrutiny has resulted in rectangular coefficient m atrices and the num erical approach to de

term ine the needed eigenvalues and eigenvectors requires that these m atrices be square.

This process requires that the rectangular m atrices that define the eigenvalue problem be

m ultiplied by som e random ly generated m atrices o f the same dim ension. W hile this pro

cedure fails to work for m any num erical analysis problem s due to generation o f m any

useless extrem e eigenvalues, this procedure always w orks for the problem under study in

that the dynam ics are localized and infinite eigenvalues or zero eigenvalues are not worth

considering for the practical problem o f predator-prey dynamics.

A nother observation is that while the theory that include °° and 0 as eigenvalues re

quires that A 5 be nonsingular (for °° eigenvalue) and Ao be nonsingular for 0 eigenvalue,

these requirem ents w ould be necessary if we desire to determ ine lower and upper bounds

for the resulting eigenvalues, [13], [20], It should be m entioned that the values o f the

determ ined eigenvalues are in fact the population sizes o f the prey species as well as

the critical param eter values which give rise to the cusp bifurcation. Therefore we are

interested in finite nonzero positive eigenvalues only.

Com panion m atrices are often considered useful form s o f linearizations o f p ( X ) , [21],

[12].

D efin ition 4.1.4. A com panion m atrix [21] C(X) = X X + Y from the linearization o f p (X )

is called the first com panion form if

X = diag(A *,/(*_!)„) and Y =

Ak - 1 A * _ 2- I n 0

Ao0

- I n 0



It is called the second com panion form if

X = diag(A *,/(*_!)„) and Y =

Ak - 1

Ak - 2

Ao

-In

0

0

0

- I n0

/ kI

0=0

In general, for an eigenvalue problem o f the form p ( X ) y / = j ^ A'A, J y/ = 0, the

linearization

L(X) — X

will reduce the £th order eigenvalue problem into a first order equation. In our situa

tion, k — 5 so that the quintic eigenvalue problem is written in block m atrix form and

represented by its corresponding linearization:

Ak 0 . . . 0 ' Ak - l A k- 2 • • • ^ 0

0 In +~In 0 . . . 0

0

. 0 0 In . 0 - I n 0 _

' a 5 0 . . . 0 ' A 4 A 3 ■■■ A o

0 h

••• 0+

- 1 5 0 . . . 0

0 0 h .0 - I s 0 _

L(X ) = X

It should be m entioned that linearization o f this quintic eigenvalue problem is not unique.

Equivalent transform ations o f the block matrices

'A 5 0 . . . 0 ' ' a 4 A 3 ■■■ Ao0 I5 and

- I s 0 . . . 0

0

0 1

0

0 - I s 0

yield the same results. This is consistent with the standard linearization procedure o f

introducing the variables x\ = X k 'x , X2 = X k l x X3 = Xk - 2 , k - 3. ■Xk- 1 = X x ,X k = X.

For writing convenience and easy notation let k = x , d = y, y = z and x — X, and

consider the lexicographical arrangem ent o f the linear and quadratic term s in x . y and

z to be

and

{x2 , xy ,x z ,y 2 , yz , z2}



respectively.

We therefore rewrite the hom ogeneous polynom ials (4.13) as

pi (x.y.z, A) = —0 .2x —0.4y + (0.3x + 3.6y + 1.2 z)A 2 + (—0 .8x — 40z)A 3 + 18zA4,

P2 ( x , y , z , k ) = 0 .4ry + ( -0 .2 x z -0 .4 y z )A + (0.2xy + 5yz)A2

+ (0 . Ixz 4- 1.2 yz — 0.4z2 )A 3 + (0 . \xz + 5z2 )A4 + (—1.2 z2 )A^,

/?3 (x.y,z, A) = 0 .4 x y + 10yz + (0.6xz — l . l y z — 2.4z2)A + ( 1 .2xz + 60z2)A2 + ( - 2 4 z 2)A3.(4.14)

This arrangem ent leads us to solve a quintic eigenvalue problem o f the form

+ Ag, + A2g 2 + A3g 3 + A4g4 + A5g5 1 ^ = 0 ;

with f . g j , i = 1,5 functions o f x .y and z; and

f ( x , y , z ) = { —0.2x,0.4xy,0.4xy + 10yz},

g i (x , y , z ) — {0, - 0 .2 x z - 0 .4 y z ,0 .6 x z - 7 .2 y z - 2 .4 z 2},

g 2 (x,y,z) = {0.3x + 3 .6 y + 1 .2 z ,0 .2 x y + 5yz, 1.2xz + 60z2},t o (4.15)

g 3 (x,y,z) = { - 0 .8 x - 4 0 z ,0 .1 x z + 1 .2 y z - 0 .4 z , - 2 4 z },

g4 (x,y,z) = {18z.0.1xz + 5z2 ,0},

g5(*,:y,z) = {0, — 1 .2z2,0}.

The coefficients in the quadratic system o f equations, com bined with those o f the origi

nal system o f hom ogenized polynom ials will form a 6 x 6 matrix whose determ inant is

the resultant o f the three polynom ials. A ppendix A.3 provides the num erical approach

through a M atlab program to solve the resultant o f three polynom ials. Exact values o f the

critical param eters k and d, as well as the prey species density x w here a cusp bifurcation

occurs are generated.

We let yq = A4 t/r, i//2 = A3 y/, y/3 = A2 t/r, 1//4 = A yf and y/5 = yr the quintic eigen

value problem p ( X ) y / = I ^ ^ 'A j I yt = 0 is transform ed into the equationV«=o )

A 5 ( X y / i ) + A 4 yf\ + A 3 y/ 2 + A 2 y/3 + A 1 \f/ 4 + Ao y/ 5 = 0 . (4.16)

From (4.14) and (4.15) we form block m atrices that define the m atrix polynom ial that

in turn defines the quintic eigenvalue problem . We define block m atrix entries o f (4.16),

using coefficients o f polynom ials (4.14) as shown in Table (4.2). These block m atrices

are not square m atrices. A num erical technique to m ake them square before one could

linearize the quintic eigenvalue problem requires that each o f these block m atrices is


CHAPTER 4. THREE SPECIES SYSTEM

Table 4.2: Form ation o f block m atrix entries for the quintic eigenvalue problem .

/ , = [ -0 .2 , -0 .4 ,0 ]

h = [0 ,0 .4 ,0 ,0 ,0 ,0 ]

/ 3 = [0 ,0 .4 ,0 ,0 ,1 0 ,0 ]

= [Aoi, Ao2 ,Ao3];

A01 = oneBlock{3, dout, 1, / I }

Aq2 — oneB lock{3,dout, 2 , /2 }

A03 = o n eB lo ck { 3 ,d o u t,2 ,/3 }

g\ = [0 , 0 , 0 ]

g 2 = [ 0 ,0 ,- 0 .2 ,0 ,- 0 .4 ,0 ]

g 3 = [0 ,0 ,0 .6 ,0 , - 7 .2 , -2 .4 ]

M = [^ 10, ^ 11, ^ 12];

A 10 = oneB lock{3,dout, 1 , g l }

A n = oneB lock{3,dout, 2 ,g2}

A 12 = oneB lock{3 ,dou t,2 ,g3}

hi = [0 .3,3.6,1.2]

h 2 = [0 , 0 .2 ,0 , 0 ,5 ,0]

/i3 = [0 ,0 ,1 .2 ,0 ,0 ,6 0 ]

A2 = [^21 , ^ 22 , ^ 23];

A21 = oneB lock{3,dout, 1,/?1}

A 2 2 = oneB lock{3 ,dout,2 ,/i2}

A23 = oneB lock{3,dout,2 ,/?3}

ii = [ - 0 .8 ,0 ,- 4 0 ]

h = [0 ,0 ,0 .1 ,0 ,1 .2 , —0.4]

I3 = [0 ,0 ,0 ,0 ,0 ,-2 4 ]

A 3 = [^31 , ^ 32 , ^ 33];

A31 = oneBlock{3, dout, 1, i \ }

A32 = oneB lock{3 ,dou t,2 ,/2}

A33 = oneBlock{3, dout, 2, /3}

j \ = [0,0,18]

j i — [0 , 0 , 0 . 1 , 0 , 0 ,5]

h = [0 , 0 , 0 , 0 , 0 , 0 ]

A4 — [A41 ,A42,A43];

A41 = oneB lock{3,dout, 1, j 1}

A42 = oneB lock{3,dout, 2, y'2}

A43 = oneB lock{3,dout,2 ,y '3}

h = [0 , 0 , 0 ]

k2 = [0 , 0 , 0 , 0 , 0 , - 1 .2 ]

k3 = [0 , 0 , 0 , 0 , 0 , 0 ]

^ 5 = [^51 , ^ 52 , ^ 53]-

A51 = oneB lock{3,dout, 1 , k \ }

A52 = oneB lock{3 ,dou t,2 ,£2}

A53 = oneB lock{3,dout,2 ,£3}



m ultiplied by the transpose o f a random ly generated m atrix of the same dim ension. The

technique in solving the quintic eigenvalue problem , as said earlier, requires linearization

o f the system through com panion m atrices so that we would eventually solve the generic

eigenvalue problem

A x = Xx.

The m atrix A ideally should be square,even though this problem is actually one o f solving

n polynom ials in n + 1 variables X , x \ , . . . , x n. To solve this system, the system o f po ly

nom ials is augm ented with an equation o f a linear com bination o f the variables jc,- using

some random ly chosen coefficients, [18]. Problem s m ay arise where the system m ay have

n isolated solutions instead o f n + 1 solutions. Corrections are often m ade by considering

the solution o f the system

q\ = (A + <?n)(xi + e \ 2)

q 2 = (X + e2 l )(x2 + e22)

q n = (A + €n \) (xn -r s n2)

q n + 1 = c \ x \ H H c nx n + Cn+ 1 •

It is observed that the resulting eigenvalues and eigenvectors that are obtained must

be selected carefully as som e are extrem e cases which do not respond to the biological

system requirem ents. The approach that was em ployed for this contains a section for

checking the eigenvalues through solution identity o f the corresponding polynom ials, (see

A ppendix A.3). A m ore detailed explanation about solutions o f hom ogeneous system s

through resultant theory, w here the Bezout resultant approach is discussed, can be found

in [33] and [18].

Results o f the M atlab code that num erically solved the quintic eigenvalue problem

using block m atrices defined in Table (4.2) are given in Table (4.3). By considering the

first set o f values from table 4.3, as well as recalling the param eter values that provided for

the quintic polynom ial (4.3), we observe that the corresponding coexistence equilibrium

under the sym m etric transform ation

is num erically determ ined to be

(x,y,z) = (0 .1561 ,0 .1561,2 .0000).



Table 4.3: Critical population density at critical param eter values.

X k d

0.1561

0.1377

0 .1 0 0 2

0.0905

-0 .0 5 0 1

-0 .0 4 5 2

This critical equilibrium is the cusp equilibrium . The Jacobian m atrix at the coexistence

equilibrium is

J n h i J\ 3 \ / 8.8826 0.1335 1050.7264

h \ h i J 1 3 = 0.1335 8.8826 1050.7264

h \ Jyi J 3 3 J \ 0 .1894 0.1894 -6 4 .6 2 9 8

The eigenvalues A/, i = 1,2,3, o f the Jacobian m atrix evaluated at this equilibrium are

A, = 14.0729, A2 = 8.7491, A3 = -6 9 .6 8 6 7 ,

which shows stability in the direction o f the top predator species and instability in the

direction o f the two sw itching prey system.

We obtained critical values for x and p defined in Table 4.3 for the system (4.10),

where p = (k .d) . By a param eter-dependent shift o f the coordinate E, = x + 8 ( p ) and

expanding p ( x ) ( = p(E, — 8 ( p) ) in Taylor series, and taking into account that p ( 0 ,0 ) =

p ' ( 0 , 0 ) = p " ( 0 , 0 ) = 0 but p"'(x) ^ 0 , we obtain the corresponding normal form for the

cusp bifurcation that assum es the generic equation

f\ = j 3 i + / 3 2 t ? - t 7 3-

A plot o f the equilibrium m anifold

M = {(i7,j3i,02) : 0 i + 021] — T73 = 0}

has its projection onto the ( 0 i ,0 2)-plane as shown in Fig. 4.2.

4.2 Symmetry-Breaking Bifurcation

W hile analysis o f the system under sym m etry assum es that individuals in the same species

are identical, obviously they are not. We however continue to study sym m etry by con

sidering that alm ost sym m etric system s are regarded as small perturbations o f sym m etric

systems.



0f t

Figure 4.2\ O ne-dim ensional cusp projection o f the equilibrium m anifold. A nondegenerate fold bifurcation occurs upon crossing either branch (7j and Ti) at any point other than the origin.

Under sym m etry

In the case o f sym m etric eigenvectors, we have a turning point type o f bifurcation, whereas

in the antisym m etric case we obtain a pitchfork bifurcation. We also observe that if the

full system is stable about a fixed point, the same dynam ics are observed in the sym m etric

system, that is, there is no loss o f stability. H owever this is not the case in the antisym

m etric system, w here change o f stability is observed through a pitchfork bifurcation.

any any eigenvector o f the three species system is either symmetric

or antisym m etric, represented by


Chapter 5

Discussions and Conclusions

5.1 General observations

M ost published m odels o f predator-prey system s that exhibit com plicated dynam ics such

as chaos involve term s that result in oscillations. Indeed an ideal predator-prey system

should show cyclical behavior owing to the fact that when prey density is low, predator

density, if solely dependent on this prey for food, w ould also decrease due to scarcity

o f food. The result is that a critical population size for the predator species is reached,

beyond which extinction occurs. D uring this period o f low predation, prey density rises

again, w hich in turn increases predator density. In reality the predator population does not

depend solely on one food source. This has led to m odel systems exhibiting either lim it

cycles or stable foci.

We briefly explain the assum ptions we considered for the model. M uch o f this re

search was centered on understanding the specific system interaction that exhibits the

switching behavior in the special tw o-species system. Instead o f approaching the m odel

system through the standard cyclical behavior o f predator-prey dynam ics, the interact

ing term s rule out the possibility o f lim it cycles. The effect o f the interaction between

the corresponding two species was expressed through the response a,y, i , j = 1 , 2 , / ^ j .

The predation intensity was m easured by the absolute value o f this coefficient Ofy. The

positive or negative sign o f a ,7 reflected a positive or negative contribution to the density-

dependent factor in the per capita growth rate o f the species. This way, one w ould be able

to determ ine which species is the predator and which is the prey.

The resulting equations could have been re-scaled by introducing new variables for

which the carrying capacity o f the environm ent disappeared, but in an effort to reflect

ecological reality, w e chose to keep the original m odel. Indeed the carrying capacity be

came the sensitive param eter we chose to vary while other param eters rem ained constant.

The work by B arradas et. al. showed m ore variations o f param eters. They tried to tie

the resulting phase portrait diagram s to environm ental changes. In our study, only the

carrying capacity param eter becam e the sensitive param eter which varied. It is difficult

to vary two or m ore param eters sim ultaneously given the nonlinearities involved in the

model. We instead chose to approach the sim plifications by considering the system under

59


CHAPTER 5. DISCUSSIONS AND CONCLUSIONS 60

symmetry. The results o f this approach lead to the dynam ics o f a pitchfork bifurcation

outcome.

We also observed that even when the dynam ics involve the top predator and one prey

species, through response functions o f the H olling Type III where there rem ains the choice

factor, the overall result is that we have only stable foci in the event there is the existence

o f the coexistence equilibrium solution.

The standard approach to obtaining equilibria and determ ining the nature o f stabil

ity could not be pursued due to nonlinearities in the system. Considering the system

under sym m etry w e required the use o f the theory o f resultants to determ ine conditions

necessary for turning points. In this case, sym m etry m eant trying to provide the sam e en

vironm ental conditions for the three species. W hile this m ay not be realistic biologically,

a perturbation o f the original system under sym m etry lead to sym m etry-breaking analy

sis. The change o f stability through sym m etry-breaking again gave rise to characteristic

sym m etry-breaking bifurcation o f the pitchfork type. Conditions for a cusp bifurcation

further sim plified the analysis, but we observed that w ithout com putational num erical

analysis, a cusp bifurcation could only be observed through simulations.

It is im portant to find conditions giving rise to cusp bifurcation. The dynam ics of

populations leading to a cusp bifurcation need careful param eter variations because the

existence o f small perturbations near the cusp peek could result in some catastrophic jum p

between stable branches. Ecologically, this could lead to catastrophic regim es where one

species m ay go extinct.

O ur results show that these bifurcation diagram s allowed the visualization o f different

types o f interactions and the dynam ics o f variability o f certain param eters. Specifically,

an increase in the environm ental carrying capacity £,, led to different equilibria and their

stability conditions. We noted that the increase in carrying capacity o f one species resulted

in a direct decrease in the abundance o f the other species.

The results o f this m odel show that in the presence o f the third top predator, the system

stabilizes and the three species is observed to persist. The conditions for coexistence in the

species are a requirem ent for persistence. The top predator is understood to play a m ajor

ecological role in controlling population abundances in a given ecosystem . The fact that

under very low density o f one prey species, the two species system continues to persist,

m eans that the top predator acts as a regulatory m echanism for continued existence o f the

three species, albeit under set environm ental conditions.



5.1.1 Extensions

The dynam ics o f the three species system was considered as a w hole only under condi

tions o f sym m etry and hence perturbations leading to sym m etry-breaking. Suggestions

for analysis o f this system included considering one o f the species as a linear m ultiple of

the other. On the analysis o f the corresponding two species system, we realized that the

top predator w ould continue to m ake food choices, despite the absence o f the other food

source. We note the deficiency in this regard but in reality, alternative food sources are

usually provided in the environm ent. The dogfish-lobster-snail relationship is one where

the dogfish does not solely depend on lobsters and snails, but also feed on other m arine

life. Despite this observation one m ight w ish to change the m odel system o f the corre

sponding two species system to accom m odate the fact that the top predator no longer has a

choice in the absence o f low abundance o f one prey system due to predation. A nother pos

sibility w ould be to change the density-dependent carrying capacity o f the environm ent

for the predator population in favor o f a response function of the H olling Type II which

w ould benefit the growth rate o f the predator. These changes w ould perhaps provide a

model that is m ore biologically realistic.

We wished that through sensitivity analysis o f param eters that influence the dynam ics

o f the system, we would identified values that gave rise to interesting dynam ics. It is true

that given the m any nonlinearities in the m odel system, a num ber o f param eters could

have been chosen for sensitivity analysis w hose change produced som e drastic system

behavior. We therefore wish to further this study by doing more sensitivity analysis o f the

param eters involved.

The approach we chose o f analyzing the system under full sym m etry [38] is ju s t one

approach to analyze the system, especially given lim ited com puter resources. However

the algebraic com putations involved through the use o f Resultants Theory necessitated

com putational num erical analysis code to produce num erical results needed for a cusp

bifurcation, (see Appendix A.3). The com putational techniques em ployed are not yet

com pletely understood. We believe that through the use o f DSTool or AUTO, software

packages designed to solve dynam ical system s with non-linear term s like this m odel, one

can obtain param etrizations where interesting dynam ics such as chaos is obtained.

The sw itching o f predator-prey role is a rare phenom enon in ecological system s. This

switching provides an intrinsic m echanism for which persistence o f species could be ex

plained, for such a switch excludes the predator as sole resource beneficiary. These inter

actions occur in the presence o f o ther species w hich benefit from the resulting outcom es.



That benefit either leads to extinction o f one or m ore species while certain conditions or

to continued existence o f all species. We believe that m ore work is required for systems

exhibiting such properties.


Appendix A

Computational Systems Analysis

A.l Matlab Code for Sensitivity of One Parameter

•/, INITIAL CONDITIONS

’/, speciel specie2p . xO = [3 3] ;

[t x] = odel5s(Spred, p.tt, p.xO, p .options,AO, p);

figure(l), elf, plot(t,x(:,1),’r ’), legend(’x ’),

figure(2), elf, plot(t,x(:,2),’r ’), legend(’y ’),m m m m n m m n m nfunction xp = pred(t,x,AO,p)

xp = x;

f = (5*x(l) - x(l)*x(l))/(l + 0.5*x(l));

g = (5*x(2) - x(2)*x(2))/(1 + 0.5*x(2));

m m m m m m m m m m m m m m m m m m m m m m n

xp(1) = p.R0*x(l)-p.R0*x(l)*x(l)/A0 + g*x(l)*x(2)/A0;

xp(2) = p.R0*x(2)-p.R0*x(2)*x(2)/A0 + f*x(l)*x(2)/A0;

A.2 Matlab Code for Time Series

function threespecies ’/, Paper scaling

63


APPENDIX A. COMPUTATIONAL SYSTEM S ANALYSIS

p.dt = 1; p.tf = 1500.0; p.tt = 0:p.dt:p.tf;

*/.p.A0=.55;

’/, options for ode solverp.options = odeset('abstol’,le-8,’reltol', le-8, ’maxstep’,p.dt);

'/. INITIAL CONDITIONS

'/, preyl prey2 predatorp.xO = [0.1265 0.1265 2*0.1265]; X under symmetry

x = p.xO; p .limit=[100:1:500]; for bl=2:.l:5 [t x] = ode23(@pprey, p.tt, p.x0,p.options,bl, p);

yl=x(p.limit,1): y2=x(p.limit,2) y3=x(p.limit,3)°/.ymaxl=max(yl) ;’/,ymax2=max(y2) ; ymin3=min(y3); ymax3=max(y3); plot(bl,ymax3,’r ’); hold onplot(bl,ymin3,’*g’) hold on'/.axis ([0 1 0 2]) ; legend(’umax’,’umin’)'/.legend (’ umax ’)

end

"/.figure (1), elf, plot (t ,x(: , 1), ’r ’), legend(’A ’),

function xp = pprey(t,x,bl,p)

xp = x;

fl = ( (3*x(2)-x(2)*x(2))/(l+0.2*x(2)*x(2))); f2 = ( (3*x(l)-x(l)*x(l))/(l+0.2*x(l)*x(l))); f3 = x(3)/(l+0.2*x(l)+0.2*x(2))-l;

xp(l) =



0.3*x(l)-(0.3*x(l)*x(l))/bl+(f2*x(l)*x(l))/bl-0.2*x(l)*x(l)*x(3)/(l+0.2*x(l)*x(l));

xp(2) =0.3*x(2)-(0.3*x(2)*x(l))/bl+(fl*x(2)*x(2))/bl-0.2*x(2)*x(2)*x(3) /(1+0.2*x(2)*x(2));

xp(3) = x(3)*f3 ;

global Alin

dout=3; M=12; N=10;

’/, k=l we have a linear polynomial, and for "/. k=2,3 we have quadratic polynomials

'/iP_l (x, y , z , \lambda) =-0. 2x-0. 4y+ (0. 3x+3. 6y+l. 2z)\lambda‘2+(-0. 8x-40z)\lambda~3 ’/, +18z\lambda~4

*/.p_2 (x, y , z , \lambda) = 0 . 4xy+ (-0. 2xz-0. 4yz) \lambda+ (0. 2xy+5yz) \lambda~2’/, + (0. lxz+1. 2yz-0. 4z~2) \lambda~3"/, + (0. lxz+5z~2) \lambda”4+(-1. 2z~2)\lambda~5

A.3 Matlab Code for Quintic Eigenvalue Problem

y."/.p_3 (x, y , z , \lambda) 0.4xy+10yz+(0.6xz-7.2yz-2.4z‘2)\lambda

+(1.2xz+60z~2)\lambda~2+(-24z~2)\lambda~3•/.*/."/, set the polynomials fl = [-0.2,-0.4,0]; f2

gl = [0,0,0]; g2 = [0,0,-0.2,0,-0.4,0] ; g3 = [0,0,0.6,0,-7.2,-2.4] ;

hi = [0.3,3.6,1.2]; h2 = [0,0.2,0,0,5,0]; h3 = [0,0,1.2,0,0,60]

il = [-0.8,0,-40]; i2 = [0,0,0.1,0,1.2,-0.4]; i3 = [0,0,0,0,0,-24];

jl = [0,0,18]; j2 = [0,0,0.1,0,0,5] ; j3 = [0,0,0,0,0,0] ;

kl = [0,0,0]; k2 = [0,0,0,0,0,-1.2] ; k3 = [0,0,0,0,0,0] ;

"/, We will now make the rectangular resultant matrices °/, A_k, k=l,5, such that we want'/t(\lambda"5 A5+\lambda~4 A4+\lambda~3 A3+\lambda~2 A2



‘/,+\lmabda A1 + AO) to have a'/, nontrivial null vector. These matrices will be '/. 12 x 10

A01 = oneBlock(3,dout,l,fl); A02 = oneBlock(3,dout,2,f2); A03 oneBlock(3,dout,2,f3);

AO = [A01; A02; A03] ;

A10 = oneBlock(3,dout,l,gl); A12 = oneBlock(3,dout,2,g2); A13 oneBlock(3,dout,2,g3);

A1 = [A10;A12;A13];

A21 = oneBlock(3,dout,1,hl); A22 = oneBlock(3,dout,2,h2); A23 oneBlock(3,dout,2,h3);

A2 = [A21;A22;A23];

A31 = oneBlock(3,dout,l,il); A32 = oneBlock(3,dout,2,i2); A33 oneBlock(3,dout,2,i3);

A3 = [A31;A32;A33];

A41 = oneBlock(3,dout,1,jl); A42 = oneBlock(3,dout,2,j2); A43 oneBlock(3,dout,2,j3);

A4 = [A41;A42;A43];

A51 = oneBlock(3,dout,l,kl); A52 = oneBlock(3,dout,2,k2); A53 oneBlock(3,dout,2,k3);

A5 = [A51; A52; A53]

Q=rand(M,N) ; L5 = Q ’*A5 ; L4= Q ’*A4 ; L3=Q’*A3;L2=Q’*A2; L1=Q’*A1; L0=Q’*A0 '/. We'/.get square matrices F and G by multiplying

[M,N] = size(L5) ; I = eye(M); Z= zeros(M,M) ;/.Here I considered the linearization suggested by Mackey et.al.'/, in their paper "Vector Spaces of Linearization for Matrix ‘/.Polynomials" page 1 equation (1.1)F= [ Z,I,Z)Z,Z;Z,Z,I,Z,Z;Z,Z,Z,I,Z;Z,Z,Z,Z,I;L0,L1,L2,L3,L4]; G = [ -I,Z,Z,Z,Z;Z,-I,Z,Z,Z;Z,Z,-I,Z,Z;Z,Z,Z,-I,Z;Z,Z,Z,Z,L5];



[V,E]=eig(F,-G) D= diag(E) ;

pp = isfinite(D) ; D=D(pp); U=V(l:15,pp) ;

'/, In the following loop, type in j=l,2,3,..10 % you will see that some answers are good, others are not.’/, Here we are getting the solution by the technique of '/» normaliing the eigenvector so its last component is one

for k =1:N for j=l:37 ’/,j=input (’ j ’)phi=U(:,j); phi = phi/phi(N) ; lam=D(j); x=phi(6) ; y=phi(9) ; ans= [x,y,lam]checkl= -0.2*x-0.4*y+(0.3*x+3.6*y+l.2)*lam~2

+(-0.8*x-40)*lanr3+18*lam"4 '/, check’/,1st equation

check2= 0. 4*x*y+ (-0. 2*x-0. 4*y) *lcim+ (0. 2*x*y+5*y) *lam*2+(0.l*x+l.2*y-0.4)*lam*3+(0.l*x+5)*lam“4-l.2*lam~5 '/, check 2nd equation

check3= 0.4*x*y+10*y+(-7.2*y+0.6*x-2.4)*lam+(1.2*x+60)*lam"2 + (-24)*lam~3 "/, check 3rd eqn.

endend


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INDEX

asym ptotically stable, 19

B endixson’s Negative Criterion, 16 bifurcation, 26 block m atrix, 53

codim ension 2, 50 coexistence equilibrium , 24 com panion m atrix, 52 com petition, ii conjugacy, 2 0

cusp bifurcation, 49

difference-equation, 5 Driving symmetry, 33 D ulac’s Test, 17

Efficient symmetry, 33 equilibrium solution, 18 excessive asym m etry, 33

full symmetry, 29

H olling Type II, ii H olling Type III, ii hom ogeneous polynom ial, 54 hysteresis, 27

Jacobian, 20

lexicography, 53 Lotka-Volterra, ii

m utualism , ii

non-hyperbolic, 49 nonhyperbolic, 26 normal form, 26

pencil, 51 phase portrait, 19 pitchfork bifurcation, 49 Poincare-Bendixon, 4

71

population floors, 5

quasi-cyclical, 5quintic eigenvalue problem , 53

spiral chaos, 5 stability, 18 structurally stable, 2 0

sw itching, ii sym m etry-breaking, 57

topological equivalence, 2 0

topologically equivalent, 2 0

transcritical bifurcation, 26

unim odal matrix polynom ial, 52


dynamics of a model three species predator-prey …

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