thesis 2015
TRANSCRIPT
Dynamics of Ateles Hybridus Populations in Non-Fragmented andFragmented Landscapes: A Discrete Mathematical Model
By
Matthew J. Buhr
A thesis submitted in partial fulfillmentof the requirements for theUniversity Honors Program
Department of MathematicsThe University of South Dakota
Spring 2015Date of Defense: April 30, 2015
The following members of the Honors Thesis Committee appointedto examine the thesis of Matthew J. Buhr
find it satisfactory and recommend that it be accepted.
Jose D. Flores, Ph.DProfessor
Department of MathematicsUniversity of South DakotaDirector of the Committee
Catalin Georgescu, Ph.DAssociate Professor
Department of MathematicsUniversity of South Dakota
Daniel D. Van Peursem, Ph.DProfessor and Chair
Department of MathematicsUniversity of South Dakota
ii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Basic Ideas of Difference Equations . . . . . . . . . . . . . . . . . . . . . . 42.3 Fixed Points and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Single-Patch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 The Model and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Eigenvalue Analysis of the Single-Patch Model . . . . . . . . . . . . . . . 124.1 Eigenvalue λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Eigenvalues λ2 and λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 Case 1: ∆λ > 0→ λ2,3 ∈ R, λ2 6= λ3 . . . . . . . . . . . . . . . . 144.2.2 Case 2: ∆λ = 0→ λ2,3 ∈ R, λ2 = λ3 . . . . . . . . . . . . . . . . 164.2.3 Case 3: ∆λ < 0→ λ2,3 ∈ C, λ2 = λ3 . . . . . . . . . . . . . . . . 18
5 An Integrated Multi-Patch Model . . . . . . . . . . . . . . . . . . . . . . . 205.1 Multiple Patch Model Diagram and Equations . . . . . . . . . . . . . . . . 215.2 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Extra Stability Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Model Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Community Dynamics on a Variable Parameter . . . . . . . . . . . . . . 286.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Case Study 1: A Variable δ0 Parameter . . . . . . . . . . . . . . . . . . . 296.3 Case Study 2: A Variable p Parameter . . . . . . . . . . . . . . . . . . . . 336.4 Case Study 3: Variable sF and δ0 Parameters . . . . . . . . . . . . . . . . 38
7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
iii
List of Figures
1 Figure 1: A stable equilibrium of coexistence when ∆λ > 0 . . . . . . . . . 15
2 Figure 2: A stable equilibrium of extinction when ∆λ > 0 . . . . . . . . . . 16
3 Figure 6: ∆λ < 0: Oscillatory behavior (Short term) . . . . . . . . . . . . . 20
4 Figure 7: ∆λ < 0: Oscillatory behavior (Long term) . . . . . . . . . . . . . 20
5 Figure 8: A multi-patch stable solution (Extinction) . . . . . . . . . . . . . 25
6 Figure 9: A multi-patch unstable solution . . . . . . . . . . . . . . . . . . 26
7 Figure 10: Dynamics with a low value of δ0 (Short Term) . . . . . . . . . . 29
8 Figure 11: Dynamics with a low value of δ0 (Long Term) . . . . . . . . . . 30
9 Figure 12: Dynamics with a moderate value of δ0 (Short Term) . . . . . . 31
10 Figure 13: Dynamics with a moderate value of δ0 (Long Term) . . . . . . . 32
11 Figure 14: Dynamics with a high value of δ0 (Short Term) . . . . . . . . . 32
12 Figure 15: Dynamics with a high value of δ0 (Long Term) . . . . . . . . . . 33
13 Figure 16: Dynamics with a low value of p (Short Term) . . . . . . . . . . 34
14 Figure 17: Dynamics with a low value of p (Long Term) . . . . . . . . . . . 34
15 Figure 18: Dynamics with a moderate value of p (Short Term) . . . . . . . 36
16 Figure 19: Dynamics with a moderate value of p (Long Term) . . . . . . . 36
17 Figure 20: Dynamics with a high value of p (Short Term) . . . . . . . . . . 37
18 Figure 21: Dynamics with a high value of p (Long Term) . . . . . . . . . . 37
19 Figure 22: Dynamics of an equilibrium with low δ0 (Short Term) . . . . . . 40
20 Figure 23: Dynamics of an equilibrium with low δ0 (Long Term) . . . . . . 40
21 Figure 24: Dynamics of an equilibrium with moderate δ0 (Short Term) . . 41
22 Figure 25: Dynamics of an equilibrium with moderate δ0 (Long Term) . . . 41
23 Figure 26: Dynamics of an equilibrium with high δ0 (Short Term) . . . . . 42
24 Figure 27: Dynamics of an equilibrium with high δ0 (Long Term) . . . . . 42
iv
1 Introduction
Mathematical modeling is a branch of mathematics studying the behavior of systems and
maps in a current state using past events. We want to know how to generate mathematical
representations or models, how to validate them, how to use them, and how and when their
use is limited. Since the modeling of devices and phenomena is essential to both engineering
and science, engineers and scientists have very practical reasons for doing mathematical
modeling. In addition, engineers, scientists, and mathematicians want to experience the
sheer joy of formulating and solving mathematical problems.
Definition 1.0.1. A Mathematical Model is a representation in mathematical terms
of the behavior of real devices and objects [3].
In this study, we create a mathematical model to estimate the dynamics of Ateles
Hybridus, also known as the Brown Spider Monkey, in a non-fragmented and fragmented
landscape. The Brown Spider Monkeys (of several species) live in the tropical rain forests
of Central and South America and occur as far north as Mexico. They have long, lanky
arms and prehensile (gripping) tails that enable them to move gracefully from branch
to branch and tree to tree. These nimble monkeys spend most of their time aloft, and
maintain a powerful grip on branches even though they have no thumbs [4].
Ateles Hybridus are a social species and gather in groups of up to two or three dozen
animals. At night, the groups split up into smaller sleeping parties of a half dozen or fewer.
Foraging also occurs in smaller groups, and is usually most intense early in the day. Spider
monkeys find food in the treetops and feast on nuts, fruits, leaves, bird eggs, and spiders.
They can be noisy animals and often communicate with many calls, screeches, barks, and
other sounds.
Typically, females give birth to only a single baby every one to five years. The var-
iegated spider monkey gives birth to single young, after a gestation of 225 days. Baby
spider monkeys tend to cling to their mother’s belly for around the first four months of
life, after which they climb to her back, eventually developing enough independence to
travel on their own. Young monkeys depend completely on their mothers for about ten
1
weeks, but after that time they begin to explore on their own and play amongst themselves.
Mothers continue to care for their young for the first year of their lives, and often move
about with their offspring clinging to their backs. Indigenous peoples often hunt spider
monkeys for food, and the animals are usually agitated by human contact. Logging and
deforestation continue to shrink the space that spider monkeys are able to call home. The
variegated spider monkey has a complex social system, living in multi-male, multi-female
fission-fusion communities. These groups break up into smaller subgroups to forage, and
have a home range of around 260 to 390 hectares. A wide variety of calls are used, in-
cluding ‘ts chookis’, whoops and wails to locate other subgroups. When two subgroups
reunite there is an excited greeting display, which involves vocalizing, chasing, hugging
with tails entwined, and sniffing of the sternal glands. Like other spider monkeys, this
species is characterised by a slow reproductive rate, with females typically giving birth to
single offspring only once every one to five years.
Ateles Hybridus have undergone endangerment situations for several years. Our goal
is to model the dynamics of Ateles Hybridus given their population structure and lifestyle.
We first develop a single-patch model to model the dynamics of Ateles Hybridus popula-
tions in a single patch. Then, we consider a forced migration parameter of young females
at the time of their sexual maturity and add new parameters into our single-patch model to
account for differences in patch quality, given by hostility and by size. We take various pa-
rameters into account, including survival probabilities of every cohort of Ateles Hybridus,
the birth gender probability, and the rate of reproduction. We aim to develop solutions
to the endangerment issue, and provide feedback based on our mathematical model and
testing.
2
2 Preliminaries
2.1 Difference Equations
The quantities that are involved in mechanics, such as displacement, velocity, and accel-
eration, are typically related to time by smooth functions defined on an entire interval.
Problems in mechanics lead to differential equations. By way of contrast, the mathemat-
ical models to be studied in biology involve quantities whose values are known only at
certain specified times, equally spaced. Such quantities are expressed as functions of time
via sequences. The assumptions in the models can then be expressed by saying that the
former are continuous whereas the latter are discrete. Typically, population models with
a small total population is best modeled with the use of difference equations.
Say there exists a population where no deaths are observed. Thus, the change in
population is affected only by new births each month. This behavior can be modeled by
the equation
individuals
this time-period
=
individuals
last time-period
+
new individuals
this time-period
.
If, for example, the population takes two time-periods to become reproductive and then
produce only one offspring, then the last term on the right hand side of the previous
equation, provided the current month is at least the third month, is given by the equation
new individuals this
time-period
=
individuals
two time-periods ago
.
If we let the current month be the k-th time-period (k ≥ 3), then the last month would
be the (k − 1)st month and the one before that would be the (k − 2)th month. Then, the
above equation would be expressed as an equation of variable Ni, where Ni represents the
number at time i.
Nk = Nk−1 +Nk−2, k = 3, 4, 5, . . .
3
This is the well known Fibonacci equation and a prime example of a difference equation.
Since we consider an initial population of Ateles Hybridus of under 1,000 inhabitants, a
discrete time-scale is most sufficient.
2.2 Basic Ideas of Difference Equations
The idea of a difference equation can now be formulated in a general way, applicable to a
wide variety of biological problems. Difference equations arise in problems like the previous
example.
Definition 2.2.1. Let a rule express a recursive sequence, where members of a sequence
are in terms of previous members of a sequence. If the rule defines the kth member of the
sequence in terms of the (k-1)st member (and possibly also the number k itself), then it is
said to be a first-order difference equation [1].
Once a value is specified for y1, the difference equation then determines the rest of the
sequence uniquely. The value given for y1 is called an initial condition and the sequence
obtained is called a solution of the difference equation.
Definition 2.2.2. An Initial Condition of a system is a set of starting-point values
belonging to or imposed upon the variables in an equation that has one or more arbitrary
constants. [1].
In our model of Ateles Hybridus, we use biological data [5] to best give realistic
initial conditions for our patch populations to be tested under various survival and birth
probabilities. In this way, we do not have an unbalance in our population that would be
deemed unrealistic in real life.
Definition 2.2.3. Let a rule express a member of a sequence in terms of previous members
of a sequence. If the rule defines the kth member of the sequence in terms of the (k-2)th
member (and possibly also the (k-1)st member or the number k itself), then it is said to be
a second-order difference equation [1].
4
A unique solution for second order difference equations is determined once the initial
values of both y1 and y2 are specified. Difference equations of third and higher orders
may be defined in a similar way. This process of repeatedly substituting old values back
into the difference equation to produce new ones is known as iteration. It is clear that
this process will eventually produce yk for any prescribed value of k. For some difference
equations it is possible to find a simple formula giving the solution yk as a function of k.
Such a formula is said to provide a ‘closed-form’ solution of the difference equation and
enables values for large times, such as y100, to be calculated directly, without the need to
calculate all the preceding members of the sequence.
2.3 Fixed Points and Stability
In the applications of difference equations to biological systems, a solution represents some
quantity measured at equal intervals of time.
Definition 2.3.1. A solution in which the measured values do not change with time is
called a constant or steady-state solution [1].
Definition 2.3.2. An orbit is a collection of points related by the evolution function of
the dynamical system. The orbit is a subset of the phase space and the set of all orbits is
a partition of the phase space, that is, different orbits do not intersect in the phase space
[1].
Although a solution chosen at random is unlikely to be automatically in a steady-state,
it may approach a steady-state solution over a long period of time.
Definition 2.3.3. Let f : I → I. A fixed point is a point x such that f(x) = x [1].
Obviously, the orbit of a fixed point is the constant sequence x0, x0, x0, . . . . Fixed points
have the advantage of a simple graphical interpretation, which often provides information
about fixed points even in cases where we cannot solve equations explicitly. A number k
is a fixed point of a function f if and only if the point (k, f(k)) is a point of intersection of
the graphs of y = f(x) and y = x [3].
5
Theorem 2.3.4. If x0 is some fixed point for a function f , then we say that x0 is a source
and is unstable if |f ′(x0)| > 1. On the other hand, x0 is a sink and is asymptotically stable
if |f ′(x0)| < 1. If |f ′(x0)| = 1, this test is inconclusive and other tests must be used. We
note that if |f ′(x0)| = 1, x0 is called non-hyperbolic.
Proof. See [3].
Definition 2.3.5. A scalar λ is called an Eigenvalue of an n × n Matrix A if there
is a nontrivial solution x of the equation A~x = λ~x. Such an x is called an eigenvector
corresponding to the eigenvalue λ.
Theorem 2.3.6. Eigenvalue Stability Theorem. If all roots of the characteristic
equation at an equilibrium point satisfy |λ| < 1, then all solutions of the system with
initial values sufficiently close to an equilibrium will approach the equilibrium point as
t→∞ and the equilibrium point is known as a stable equilibrium point [3].
Theorem 2.3.7. Eigenvalue Instability Theorem. If all roots of the characteristic
equation at an equilibrium point satisfy |λ| ≥ 1, then all solutions of the system with initial
values sufficiently close to an equilibrium will approach the equilibrium as t → −∞ and
the equilibrium point is known as an unstable equilibrium point [3].
In a discrete-time system, the Jury Criterion [2] can be used to determine its stability.
A system is stable if and only if all roots of the characteristic polynomial
Char(λ) = |A− λI| = (λ) = a0λn + a1λ
n−1 + · · ·+ an−1λ+ an (2.3.1)
are inside the unit circle. To use the Jury Criterion, we can begin by multiplying our
polynomial a(λ) by −1 if necessary to make a0 positive. Then, form the array
6
a0 a1 . . . an−1 an
an an−1 ... a1 a0
b0 b1 . . . bn−1 .
bn−1 bn−2 . . . b0 .
c0 c1 . . . . .
cn−2 cn−3 . . . . .
(2.3.2)
where the third row entries are based on second-order determinants divided by a0 of the
first two rows, starting with the first and last columns, then the first and second-to-last
columns, such as
b0 = a0 −ana0an
b1 = a1 −ana0an−1
bk = ak −ana0an−k
(2.3.3)
and the fourth row is made by reversing the third row, and the fifth row is given by
ck = bk −bn−1b0
bn−1−k. (2.3.4)
If all the terms in the first columns of the odd rows are positive, then the polynomial aλ
is such that the system is stable.
7
3 Single-Patch Model
We implement a discrete model to study the population dynamics of Ateles Hybridus in
a single patch. Data [5] suggest that for a population level of under 1,000 inhabitants, a
discrete model is most suitable. Different patches resemble a landscape which has been
fragmented over the past few years. A population is divided into categories by sex: male
and female. Furthermore, the population is broken down so that the female population
is broken into subgroups: adult females and young females, to account for an age of
reproductive ability. Additionally, females are the dispersing sex in spider monkeys. In
our population, a young female acquires its reproductive ability around their seventh year,
at which point they disperse from their group or “family” in search of another group where
they will spend their reproductive life. This activity will require the adult females to select
a target patch other than their original one, and successfully cover the distance between
their current patch and their selected one. An additional hostility factor includes a target
patch that is close to its carrying capacity in which the female could have a considerable
amount of trouble staying alive, hence having to make a second decision. Because of the
given variables in female dispersal throughout the patches in question, we consider three
ecological processes. These are the natural per-capita birth and death rate, the average
time for females to reach reproductive ability, and eventually, a forced migration process
at the time of female adulthood.
3.1 State Variables
A patch is composed of a single group of individuals divided into male and female coun-
terparts, where females are further divided into two subgroups, which are those who have
reached reproductive ability, and those who have not. We assume that the time to reach
reproductive ability is, on average, seven years of age. Each one of these groups is repre-
sented by the variables M,Y, F , where M = Males, Y = Young (Unreproductive) Females
and F = Females. Parameters for the model are estimated from previous studies and
published data [5]. We assume that new individuals are the result of births at a per-capita
8
birth rate r in years. Out of these new individuals a proportion p are male at birth. Thus,
(1− p) represents the proportion of the population which are female at birth.
3.2 The Model and Parameters
We begin our discussion with a single-patch model to model general (linear) behavior of
Ateles Hybridus. The dynamics of the population of Ateles Hybridus in a single patch is
given by the discrete system of equations
Mn+1 = pbFn + (sM)Mn
Yn+1 = (1− p)bFn + (sY − δ0)Yn
Fn+1 = (sF )Fn + Ynδ0
. (3.2.1)
Parameter Symbol Parameter Definition
Mn, Mn+1 Population of males at stages n and n+ 1, respectively
Yn, Yn+1 Population of young females at stages n and n+ 1, respectively
Fn, Fn+1 Population of adult females at stages n and n+ 1, respectively
p Probability of births being male
b Average number of births per female per stage
sM Male survival Probability per stage
sY Young female survival Probability per stage
sF Female survival Probability per stage
δ0 Percentage of current young females reaching sexual maturity
The model given by 3.2.1 is written in the matrix form
Mn+1
Yn+1
Fn+1
=
sM 0 pb
0 sY − δ0 (1− p)b
0 δ0 sF
Mn
Yn
Fn
, (3.2.2)
9
where the projection matrix J , defined by Xn+1 = JXn is given by
J =
sM 0 pb
0 sY − δ0 (1− p)b
0 δ0 sF
. (3.2.3)
3.3 Equilibria
Equilibria in a linear system occur when the population of Ateles Hybridus at a given
stage is the same as the population of Ateles Hybridus at the immediate next stage.
Equilibria occur when
Mn+1 = Mn
Yn+1 = Yn
Fn+1 = Fn
We assume that continuation of the population is dependent on all three members of the
population. Thus, either there will be a stable extinction equilibrium, or a tri-coexistence
equilibrium where all three state variables corresponding to the population are alive at
a time. Since all three state variables corresponding to the population depend on one
another, an equilibrium (M,Y, F ) = (M 6= 0, 0, 0), (M,Y, F ) = (0, Y 6= 0, 0), (M,Y, F ) =
(0, 0, F 6= 0), (M,Y, F ) = (0, Y 6= 0, F 6= 0), (M,Y, F ) = (M 6= 0, 0, F 6= 0), (M,Y, F ) =
(M 6= 0, Y 6= 0, 0) cannot exist. Either the coexistence equilibrium or the extinction
equilibrium will be stable at one time; not both. To justify those criteria, we create simple
population assumptions used in our model:
• It is assumed males are living and available to fertilize females at a given time,
• The system is partially decoupled, as the above assumption means as long as M > 0,
then females can reproduce,
• Consider a submatrix of J , Jsub, where
10
Jsub =
sY − δ0 b(1− p)
δ0 sF
= 0.
We show that since Xn+1 = JXn, then solving (J − I)Xn = 0 allows us to solve for
equilibrium points. If det(J − I) 6= 0, then Xn = 0. If δ0 =(1− sF )(1− sY )
(1− p)b+ sF − 16= 0,
or (x, y, z) 6= (0, 0, 0), then there exists a (0, 0, 0) equilibrium point.
11
4 Eigenvalue Analysis of the Single-Patch Model
We analyze the values of the three eigenvalues in our linear system. We then determine
whether the values of the eigenvalues warrant a stable system, given by the Eigenvalue
Stability Theorem and the Eigenvalue Instability Theorem.
4.1 Eigenvalue λ1
We determine the eigenvalues of our system by considering the matrix J (Equation 3.2.3)
and using the formula
det|J − λI3| = det
sM − λ 0 pb
0 sY − δ0 − λ b(1− p)
0 δ0 sF − λ
,
= |sM − λ| ∗ det
sY − δ0 − λ b(1− p)
δ0 sF − λ
,= 0. (4.1.1)
Since the above equation also provides the fact that λ1−sM = 0, we conclude that λ1 = sM .
Since a 3× 3 matrix will contain at most three eigenvalues λ1, λ2, λ3. The remaining two
eigenvalues λ2 and λ3 are solved by extracting a submatrix Jsub from the original projection
matrix J, given by
Jsub =
sY − δ0 b(1− p)
δ0 sF
, (4.1.2)
and finding its own characteristic equation, using the Trace of Jsub, Tr(Jsub) = sY +sF−δ0,
and the determinant of Jsub, det(Jsub) = sF (sY − δ0)− δ0(1− p)b. Here, we get
Char(A− λI) = det|J − λI3| = (sM − λ)p2(λ) (4.1.3)
12
4.2 Eigenvalues λ2 and λ3
The remaining two eigenvalues of our system are determined by considering our matrix
Jsub and using the formula
p2(λ) = det|Jsub − λI| = det
sY − δ0 − λ (1− p)b
δ0 sF − λ
= 0. (4.2.1)
If we expand formula 4.2.1, we obtain the following equation for two eigenvalues λ2 and
λ3, Char(λ), which is a quadratic equation on the variable λ, which is
Char(λ) = λ2 + (δ0 − sF − sY )λ+ (sY sF − δ0sF − δ0b+ δ0bp) = 0. (4.2.2)
Applying the quadratic formula on equation 4.2.2, we find explicit solutions to our eigen-
values λ2 and λ3 to determine the stability and behavior of the population model at
equilibrium points.
The formula for the eigenvalues λ2 and λ3 is given by
λ2,3 =−(δ0 − sF − sY )±
√(δ0 − sF − sY )2 − 4(sY sF − δ0sF − δ0b+ δ0bp)
2. (4.2.3)
The discriminant ∆λ from equation 4.2.3 is given by
∆λ = (δ0 − sF − sY )2 − 4(sY sF − δ0sF − δ0b+ δ0bp). (4.2.4)
Equation 4.2.3 plays a large importance in our model. The values of each of the eigenvalues
will determine the stability of the equilibrium points, and the future of the population
dynamics of Ateles Hybridus. There are three possibilities for eigenvalues λ2 and λ3,
given by
• λ2 6= λ3, λ2,3 ∈ R, ∆λ > 0,
• λ2 = λ3, λ2,3 ∈ R, ∆λ = 0,
13
• λ2 = λ3, λ2,3 ∈ C, ∆λ < 0.
The nature of the eigenvalues λ2,3 depends on ∆λ. The value of ∆λ will determine the final
values of the eigenvalues, and ultimately the overall dynamics of the model. Specifically,
the behavior of solutions over time will differ depending on the overall value of ∆λ. We
analyze values of ∆λ on a case-by-case basis.
4.2.1 Case 1: ∆λ > 0→ λ2,3 ∈ R, λ2 6= λ3
We show that conditions for the eigenvalues λ2 and λ3 to be real and distinct. Consider
∆λ > 0, then
∆λ = (δ0 − sF − sY )2 − 4(sY sF − δ0sF − δ0b+ δ0bp) > 0.
The quantity ∆λ > 0 shows that all respective solutions are defined in R. The equation
reordered with respect to δ0 gives
∆λ = δ20 + 2(sF − sY + 2b(1− p))δ0 + (sF − sY )2 > 0, (4.2.5)
which is a quadratic equation with respect to δ0. The discriminant ∆δ of this equation is
given by
∆δ = (2sF − 2sY + 4b(1− p))2 − 4(sF − sY )2. (4.2.6)
The discriminant ∆δ < 0 by definition since ∆λ > 0. Thus, we have
∆δ =(2sF − 2sY + 4b(1− p))2 − 4(sF − sY )2,
(sF − sY + 2b(1− p))2 − (sF − sY )2 < 0.
(4.2.7)
Biologically, δ0 is defined to be a positive value, 0 < δ0 ≤ 1.
Theorem 4.2.1. Eigenvalues λ2 and λ3 are real and distinct when sF < sY . Then, Ateles
Hybridus populations exhibit non-oscillatory behavior, and either tend toward a survival
or extinction equilibrium point, or are unstable.
14
0 10 20 30 40 50 60 70 80100
200
300
400
500
600
700
800
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 1: Case 1: A stable equilibrium of coexistence when sF = .5, sY = 1, sM = .8, δ0 =.5, p = .5, b = 1 with initial conditions (M,Y, F ) = (400, 100, 400). In this case, ∆λ > 0and sF < sY .
Proof. From Equation 4.2.6, applying the rule for difference of squares on (sF−sY +2b(1−
p))2 − (sF − sY )2 < 0, we have
((sF − sY + 2b(1− p)) + (sF − sY ))((sF − sY + 2b(1− p))− (sF − sY )) < 0,
(sF − sY + b(1− p))(2b(1− p)) < 0
(4.2.8)
Biologically, it is given that 2b(1− p) > 0, thus we know that 2(sF − sY ) + 2b(1− p) < 0,
or 0 < b(1− p) < sY − sF . Since b(1− p) is a positive term, it follows that sY − sF > 0,
or sY > sF .
An example of this phenomenon is given in Figure 1. A related example where solutions
tend toward extinction is given in Figure 2.
Remark 4.2.2. Stability for this case occurs when |λ2,3| < 1 after the condition sY > sF
is given. Knowing that this case has the condition sY > sF , we later use this information
to develop strong conservation strategies.
15
0 5 10 15 20 25 300
100
200
300
400
500
600
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 2: The short term behavior of a stable equilibrium of extinction when sF = .49, sY =1, sM = .8, δ0 = .5, p = .5, b = 1 with initial conditions (M,Y, F ) = (400, 100, 400). In thiscase, ∆λ > 0 and sF < sY .
4.2.2 Case 2: ∆λ = 0→ λ2,3 ∈ R, λ2 = λ3
Consider ∆λ = 0 for equation 4.2.4, then there are two eigenvalues λ2 and λ3 in R that
are equal. The determinant is given by
∆λ = (δ0 − sF − sY )2 − 4(sY sF − δ0sF − δ0b+ δ0bp) = 0. (4.2.9)
The quantity ∆λ = 0 shows that all solutions will be defined in R. The equation with
respect to δ0 is given by
δ20 + 2(sF − sY + 2b(1− p))δ0 + (sF − sY )2 = 0. (4.2.10)
From this equation, the discriminant ∆δ ≥ 0 by definition since ∆λ = 0. We have
∆δ =(2sF − 2sY + 4b(1− p))2 − 4(sF − sY )2,
(sF − sY + 2b(1− p))2 − (sF − sY )2 ≥ 0.
16
Theorem 4.2.3. Eigenvalues λ2 and λ3 are real and equal when b(1− p) ≥ sY − sF .
Proof. Applying the rule for difference of squares on (sF−sY +2b(1−p))2−(sF−sY )2 ≥ 0,
we have
((sF − sY + 2b(1− p)) + (sF − sY ))((sF − sY + 2b(1− p))− (sF − sY )) ≥ 0,
(sF − sY + sF − sY + 2b(1− p))(2b(1− p)) ≥ 0.
(4.2.11)
We are given that 2b(1 − p) > 0 biologically, thus we know that sF − sY + b(1 − p) ≥ 0,
or b(1− p) > sY − sF .
Theorem 4.2.4. Eigenvalues when ∆λ = 0 are less than one in magnitude when |δ0−sF−sY |2
<
1, giving rise to a stable system.
Using Equation 4.2.3, stability occurs when |λ2,3| < 1. (We have previously satisfied
condition |λ1| < 1 by default.) We have the derivation
|δ0 − sF − sY |2
< 1 (4.2.12)
Stability of the system is determined by the value of the three eigenvalues. If |λ2,3| < 1,
then the system has a stable equilibrium point. When ∆λ = 0, then λ2,3 =−(δ0 − sF − sY )
2.
Thus, when equation 4.2.12 is satisfied, the system is stable. We observe that with the
biological limitations 0 < δ0 ≤ 1, 0 ≤ sF ≤ 1, 0 < sY ≤ 1, the inequality always holds
and the system is always stable. If δ0 = 0, the system could theoretically be unstable(|δ0 − sF − sY |
2< 1
)but then the term sF would be undefined since no females would be
in the population (no young females would survive to adulthood). Additionally, if sY = 0,
the system could again theoretically be unstable,
(|δ0 − sF − sY |
2< 1
)then then the
term sF would be undefined again, since no young females would transition to adulthood.
For a complete biologically-defined system, the system must always be stable in this case.
17
4.2.3 Case 3: ∆λ < 0→ λ2,3 ∈ C, λ2 = λ3
When we consider ∆λ < 0 for equation 4.2.4. The eigenvalues are defined on C. Eigenval-
ues are complex in this case and the discriminant ∆λ is given as
∆λ = (δ0 − sF − sY )2 − 4(sY sF − δ0sF − δ0b+ δ0bp) < 0 (4.2.13)
The equation 4.2.13 in terms of δ0 is written as
δ20 + 2(sF − sY + 2b(1− p))δ0 + (sF − sY )2 < 0. (4.2.14)
Biologically, young females must transition into adult females to give the existence of adult
females, thus δ0 > 0. The only option in this case is that there are two real roots for δ0.
Thus, ∆λ < 0 between the two real roots, therefore we must have ∆δ > 0 Applying the
quadratic formula on 4.2.14 with respect to δ0, two roots of δ0 can be found explicitly
δ01,2 = −(sF − sY + 2b(1− p))±√
(sF − sY + 2b(1− p))2 − (sF − sY )2. (4.2.15)
We must have
∆δ =(2sF − 2sY + 4b(1− p))2 − 4(sF − sY )2,
(sF − sY + 2b(1− p))2 − (sF − sY )2 > 0.
We remark that in Equation 4.2.15 another biological condition 0 ≤ p < 1 is imposed. If
in the case that p = 1, then ∆δ = 0, which would be a contradiction since ∆δ > 0. We
then expand the difference of squares:
([sF − sY + 2b(1− p)] + [sF − sY ])([sF − sY + 2b(1− p)]− [sF − sY ]) > 0,
[2sF − 2sY + 2b(1− p)][2b(1− p)] > 0,
[sF − sY + b(1− p)][b(1− p)] > 0.
18
Since b(1 − p) > 0 biologically, this implies sF − sY + b(1 − p) > 0. Then we have the
following result.
Theorem 4.2.5. The Model 3.2.1 does not have periodic solutions and hence does not
have cycles.
Proof. We are given that complex solutions theoretically will arise when ∆λ < 0 which we
saw led us to Equation 4.2.15 whence we proved the inequality sF − sY + b(1 − p) > 0
exists in this case. Since b > 0 and (1 − p) > 0, then since sF − sY + b(1 − p) > 0,
sF −sY +2b(1−p) > 0 as well. Solutions δ01,2 that arise from equation 4.2.15 are negative.
Biologically, the limitation 0 < δ0 ≤ 1 exists. Therefore, no suitable values for δ0 exist
that would make λ2,3 complex. Hence, the model 3.2.1 does not have periodic solutions
and hence does not have cycles.
Remark 4.2.6. Oscillations of Ateles Hybridus populations are possible, but they do not
exist in cycles. Instead, when we take the equation
J(~x) = λ(~x) (4.2.16)
and combine it with the fact that
Xn+1 = JXn (4.2.17)
a combined equation over n iterations is given by
Jn(~x) = λn(~x). (4.2.18)
Thus, when λ < 0, an oscillatory behavior of the trajectory is observed. Examples are
given by Figures 3 and 4.
19
0 10 20 30 40 50 60 70 80100
150
200
250
300
350
400
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 3: Dynamics of Ateles Hybridus show short-term oscillatory behavior when sF =.37, sY = .7, sM = .5, δ0 = 1, p = .1, b = 1. In this case, b(1− p) > sY − sF .
0 20 40 60 80 100 120 140 160 180 200100
150
200
250
300
350
400
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 4: Dynamics of Ateles Hybridus show long-term oscillatory behavior when sF =.37, sY = .7, sM = .5, δ0 = 1, p = .1, b = 1. In this case, b(1− p) > sY − sF .
5 An Integrated Multi-Patch Model
We have analyzed the dynamics of a single-patch system. We also analyzed effects of
certain parameters by setting them to random numbers, using other set parameter values,
20
and running simulations with multiple initial conditions. With this information, we better
understand the dynamics of a single patch. Biologists studying the endangerment of Ateles
Hybridus have observed that young females migrate to a different patch at the time they
reach reproductive capabilities and become part of the female cohort (and hence no longer
part of the young female cohort). We use conclusions given to us by the single-patch model
analysis and apply it to an integrated multi-patch model using the assumption of forced
migration of young females to a new patch, and we create new parameters to account for
differences in patch quality, and determine the dynamics of Ateles Hybridus in a multi-
patch setting. It has been reported for these animals that they tend to segregate in very
conservative female to male ratios when they are in ideal ecological conditions [5]. It is also
important to consider the direction of the path of young females to their target patch at the
time they reach reproductive maturity. We consider that a female which is migrating to its
target patch would move straight to the patch instead of a random pathway. Additionally,
in our multi-patch estimation model, we consider that no females are allowed to stay at
the patch they were born; all females must migrate to a different patch at the time they
reach reproductive maturity. We assume that home patches that have lesser quality will
translate as higher chances of a female reaching their target patch of a higher quality, and
vice versa.
5.1 Multiple Patch Model Diagram and Equations
Given our results from our single-patch model, we can integrate our findings to estimate
behavior in a modified multiple-patch model. Our multi-patch model is similar to the
model for a single-patch, with some changes in definitions of parameters, and extra scalar
parameters added. Mortality rates are different for each group but are equal between
patches. In our multi-patch model, we consider forced migration of a young female to a
new patch at the time they acquire reproductive maturity. We create the restriction on
our parameters Hi and Zi such that −∞ < Hi < ∞ and 0 < Zi. Since Hi represents a
hostility parameter of a given patch, high values of Hi (Hi > 0) correspond to a high level
21
Parameter Symbol Parameter Definition
Mn,i, Mn+1,i Population of males in patch i at stages n and n+ 1
Yn,i, Yn+1,i Population of young females in patch i at stages n and n+ 1
Fn,i, Fn+1,i Population of adult females in patch i at stages n and n+ 1
Hi Hostility parameter of a given patch i
δ0,i Female maturation probability in a patch i
Zi Size parameter of a given patch i
Table 1: Table of Parameter Symbols and Definitions Used in Multiple-Patch Model
of hostility, and thus, a low level of attractiveness of the patch to a migrating monkey.
On the other hand, low values of Hi, (Hi < 0), correspond to a low level of hostility
in a given patch i, and thus, a high level of attractiveness of the patch to a migrating
monkey. The size parameter Zi is combined with the hostility parameter to account for
the size of the patch to create an ideal proportion to create the overall term e−HiZi . Based
on biological data, if a patch is larger, monkeys do not find as much hostility in a given
patch i, thus, larger patches would require higher hostility levels to maintain the same level
of attractiveness or to a migrating monkey. We also declare some immediate conclusions.
• In the case where Zi = 0, this indicates a patch with zero-size and hence is non-
existent or undefined.
• In the case where Hi = 0, this indicates a patch with zero hostility compared to
the emigration patch of a migrating monkey, which we would conclude follows the
dynamics of the single-patch model.
• In the case where Hi � 0, the chance that any monkey would migrate to this patch
is near zero.
• In the case where Hi � 0, the chance that any monkey would migrate to this patch
is very high.
22
• The parameter Zi is assumed not to be high as to keep with the dynamics of a
multi-patch model.
The dynamics of the population of Ateles Hybridus in a multi-patch model is given by the
following discrete system of equations. We note that the total population of adult females
in the ith patch is the result of adding all the young females coming from other patches
reaching the ith patch and finally being accepted.
Mn+1,i = pbFn,j 6=i + (sM)Mn,j 6=i
Yn+1,i = (1− p)bFn,j 6=i + (sY − δ0,i)Yn,j 6=i
Fn+1,i = (sF )Fn,j 6=i + e−HiZi δ0,iYn,j 6=i
. (5.1.1)
The model is written in a matrix form by
Mn+1,i
Yn+1,i
Fn+1,i
=
sM 0 p ∗ b
0 sY − δ0,i (1− p) ∗ b
0 e−HiZi δ0,i sF
Mn,j 6=i
Yn,j 6=i
Fn,j 6=i
.
The Projection Matrix is written as
J =
sM 0 pb
0 sY − δ0,i (1− p)b
0 e−HiZi δ0,i sF
. (5.1.2)
5.2 Equilibria
The multiple-patch case is very similar to the single-patch case. Even though a term e−HiZi is
included in the multi-patch model, the model is linear. Equilibria in a linear system occur
when the population of Ateles Hybridus at a given stage is the same as the population
of Ateles Hybridus at the immediate next stage. Migration in our multi-patch model is
assumed to be a forced migration, but equilibrium values in each patch still depend on
the population levels in consecutive stages in that patch only, even if new monkeys are
23
migrating from other patches, and monkeys from a patch i will emigrate to a different
patch at adulthood.
Equilibria occur in a patch i when
Mn+1,i = Mn,i
Yn+1,i = Yn,i
Fn+1,i = Fn,i
• It is assumed that continuation of the population is dependent on all three mem-
bers of the population. Thus, either there will be an extinction equilibrium that is
automatically stable, or a tri-coexistence equilibrium where all three cohorts of the
population are alive at a time.
• Since all three cohorts in the population depend on one another, an equilibrium
(M 6= 0, 0, 0), (0, Y 6= 0, 0), (0, 0, F 6= 0), (0, Y 6= 0, F 6= 0), (M 6= 0, 0, F 6= 0), (M 6=
0, Y 6= 0, 0) cannot exist.
• Either the coexistence equilibrium or the extinction equilibrium will be stable at one
time; not both.
Additionally, the population assumptions are used in our model:
• It is assumed males are living and available to fertilize females at a given time.
• The system is partially decoupled, the above assumption implies as long as M > 0,
then females can reproduce.
• Consider a submatrix of J , Jsub, where
Jsub =
sY − δ0,i b(1− p)
e−HiZi δ0,i sF
= 0
24
5.3 Extra Stability Cases
Stability cases for Ateles Hybridus were analysed intensely in a single-patch model which
can be efficiently transferred and integrated into the multi-patch model. The big difference
between the single-patch model and the multi-patch model is the patch hostility-size term
e−HiZi . If the hostility Hi is zero compared to patches of emigration, then the patch has no
additional effect on migrating monkeys, and we can take the case as if it is the same patch.
If the size Zi is zero in a selected patch i, then the patch does not exist and is undefined,
as shown in the model. We note that it is not realistic to have a large value of Zi. If the
value of Zi hypothetically is large, then the idea of having a multi-patch model is invalid.
Thus, it is assumed Zi is low, to keep in relation with other parameters in the model. If
the hostility Hi of a patch is large, the dynamics of Ateles Hybridus is given in Figure
5 which leads to extinction. Since no young females want to migrate to a patch with a
high hostility, the patch will begin to die out, and the hostility will lower over time. On
0 200 400 600 800 1000 1200 1400 1600 1800 20000
50
100
150
200
250
300
350
400
450
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 5: Extinction-bound Ateles Hybridus in a patch with H � 0
the other hand, if the hostility Hi of a patch is small, the dynamics of Ateles Hybridus is
given in Figure 6 which leads to a temporary blow-up of the population over time. Since
25
all young females want to migrate to a patch with a low hostility compared to their current
patch, the patch will begin to blow up, and the hostility will raise over time. We find that
0 200 400 600 800 1000 1200 1400 1600 1800 20000
2000
4000
6000
8000
10000
12000
14000
Stage
Pop
ulat
ion
at s
tage
n
Spider Monkey Population Dynamics for Various Stages
MalesYoung FemalesFemales
Figure 6: Dynamics of Ateles Hybridus in a patch with H � 0
there will reach a time when the hostility level will naturally raise at a certain point when
the migration level to this patch is large enough. At this time, the hostility level will raise
high enough so that the migration rate will lower. Over an infinite amount of time, the
hostility level in a patch i will level out to a certain point to where all patches will be
uniform, at which point our model will follow the dynamics of a single-patch model.
5.4 Model Simulations
We now investigate with the model the effect of habitat fragmentation in population dy-
namics of Ateles Hybridus. Fragmentation is understood as a biological disturbance in
which an originally continuous landscape is degraded into a series of weakly connected
patches of forest. It has the obvious immediate effect that the remaining group of in-
dividuals experience a reduction in available area and resources, but more importantly,
a separation effect that is particularly challenging for mobile species. Thus, the newly
26
formed fragments can be characterized by the number of patches left, the area available
for the species and the degree of connectivity between the patches. Connectivity is the
result of a number of connections and the connection quality. For example, patches can
be all connected to each other or there might be only a few of them connected (number
of connections). In addition, the connections present can have different quality, measured
in terms of land cover in the degraded habitat between the patches affecting the ability of
subjects to reach new groups (connection quality).
27
6 Community Dynamics on a Variable Parameter
6.1 Overview
We analyze and run simulations of Ateles Hybridus populations with random initial con-
ditions and a variable parameter, so that we can visualize a parameter’s effect on the
overall population dynamics of Ateles Hybridus over time. This technique is used to de-
termine how free a parameter is to be chosen in accordance with other parameters. In some
cases, different behaviors are observed in the dynamics of the model given varying initial
conditions. The system will maintain the same overall stability as those are dependent
on the eigenvalues only, but we see a difference in initial conditions sometimes translates
to extinction of certain cohorts over a quicker amount of time. Additionally, we calculate
the mean value over all of our iterations at each stage and run an average curve given
by a red line in simulations which broadcast the average dynamic given in relation to the
parameters that were chosen previously and the parameter that is chosen at variable levels.
We give three cases, which were seen to be the most important, realistic, and necessary
parameters to analyze at variable levels. We analyze a case where only δ0 is chosen at
variable levels, a case where p is chosen at variable levels, and a case where both δ0 and
sF are chosen at variable levels. The goal is to understand what one would expect if a
parameter is not known or if more information is needed. There are some questions we
want to answer.
• How do the population dynamics differ across a variable female birth percentage p?
• How do population dynamics differ across a variable female maturation rate δ0,i?
• How do population dynamics differ across a variable survival rate sF in combination
with δ0,i?
When the dynamics of the model are such that populations oscillate at each stage, their
populations may go into extinction or blow up to ∞ immediately, and the exhibited os-
cillatory behavior may be dangerous to the survival of Ateles Hybridus in certain severe
28
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ=0.15 (Short Term)
Figure 7: Short-term behavior of Ateles Hybridus with a low value of δ0 across randominitial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5 and sY > sF .
cases, due to high possibilities of extinction.
6.2 Case Study 1: A Variable δ0 Parameter
In this case, we run simulations of the population dynamics of Ateles Hybridus over time
when the parameter δ0,i is carefully selected across different values with random initial
conditions. All other parameters are carefully and realistically selected and monitored, to
develop a strong understanding of the possible dynamics one expects to see if parameters
were not known, or if a field researcher needs a stronger understanding. We analyze both
short and long-term behavior of Ateles Hybridus under the conditions we give for each
situation. Short term behavior is defined as 15 stages, where long-term behavior is defined
as 40 stages. After analyzing the short term behavior of Ateles Hybridus with a low rate
of δ0, we notice that the final outcome for every cohort in the population shares similar
dynamics, a gentle decrease to extinction. Although in this study the fact that a decrease
in population to extinction is observed, the key is noticing whether a change in δ0 while
29
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 0.15 (Long Term)
Figure 8: Long-term behavior of Ateles Hybridus with a low value of δ0 across randominitial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5
leaving other parameters the same will have a desired effect on the overall population
dynamics over time. The long term behavior confirms that the populations do go to
extinction with the other parameters given. The short term behavior with a low value of
δ0 is given by Figure 7, and the long term behavior of Ateles Hybridus with a low value
of δ0 is demonstrated in Figure 8, where the mean population at each stage is calculated
and represented in the simulation. We compare these dynamics to dynamics with a larger,
moderate, value of δ0, where it is set instead at 0.5 instead of .15. After analyzing both the
short and long term behavior across 15 and 40 stages, respectively, we notice there is no
change to the dynamics of the population. The short term behavior with a moderate value
of δ0 is demonstrated in Figure 9, and the long term behavior of Ateles Hybridus with
a moderate value of δ0 is demonstrated in Figure 10 where the mean population at each
stage is calculated and represented in the simulation. We observe contrasting dynamics
after raising the value of δ0 to 0.85. Instead of an observed convergence to an extinction
equilibrium point, as is the case with a low and moderate value of δ0, the populations of all
30
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ=0.5 (Short Term)
Figure 9: Short-term behavior of Ateles Hybridus with a moderate value of δ0 acrossrandom initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5
cohorts exhibit oscillatory behavior. The oscillatory behavior in the young female cohort
is so strong, the population is sometimes knocked into extinction, depending on given
initial conditions. Oscillatory behavior given by this could pose a threat to the survival
probability of Ateles Hybridus in a patch with these parameters given. The short term
behavior with a high value of δ0 is demonstrated in Figure 11, and the long term behavior
of Ateles Hybridus with a high value of δ0 is demonstrated in Figure 12 where the mean
population at each stage is calculated and represented in the simulation. We observe
that when δ0 is high, dynamics include an initial oscillatory behavior, which corresponds
to an eigenvalue being negative in the system, ∆λ < 0. This shows that varying δ0 with
parameters p, b, sY , sF , sM chosen can induce oscillatory behavior and that δ0 has strength.
When oscillatory behavior is not observed, the eigenvalues of the system are real, as is
discussed by cases 1 and 2 of the eigenvalue analysis, ∆λ > 0 and ∆λ = 0.
31
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 0.5 (Long Term)
Figure 10: Long-term behavior of Ateles Hybridus with a moderate value of δ0 acrossrandom initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 15−20
0
20
40
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: δ=0.85 (Short Term)
Figure 11: Short-term behavior of Ateles Hybridus with a high value of δ0 across randominitial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5
32
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 40−20
0
20
40
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 0.85 (Long Term)
Figure 12: Long-term behavior of Ateles Hybridus with a high value of δ0 across randominitial conditions for (M,Y, F ), where p = 1
2, b = 1, sM = .5, sY = .7, sF = .5
6.3 Case Study 2: A Variable p Parameter
In this case, we run simulations of the population dynamics of Ateles Hybridus over
time when parameter p is selected across different values with random initial conditions,
where all other parameters are realistically selected and monitored, to develop a strong
understanding of possible dynamics one may expect to see if parameters were not known,
or if a field researcher needs a stronger understanding. The parameter p measures the
probability that a new birth is a male. Therefore, we define (1 − p) as the probability
that a new birth is a female. We run simulations with p = .15, p = .5, and p = .85 to
develop a realistic representation of the strength the parameter p has on the population,
and whether there is a bias as to which gender’s appearance in a population has the
most importance in the dynamics. We analyze both short and long-term behavior of
Ateles Hybridus under the conditions we give for each situation. Short term behavior is
defined as 15 stages, where long-term behavior is defined as 40 stages. After analyzing the
short term behavior of Ateles Hybridus with a low rate of p, we observe that for large
33
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: p=0.15 (Short Term)
Figure 13: Short-term behavior of Ateles Hybridus with a low value of p across randominitial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
50
100
150
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
50
100
Stage
Pop
ulat
ion
Females
0 10 20 30 400
100
200
300
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: p = 0.15 (Long Term)
Figure 14: Long-term behavior of Ateles Hybridus with a low value of p across randominitial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
34
values of p, the population tends to the extinction equilibrium more often than when p
is small. The long term behavior confirms that the populations go to extinction when p
is larger and survive when p is smaller. Although the overall dynamics of each individual
selection is not the goal of this study, we observe the changes in the dynamics that are a
consequence of modifying the parameter p alone. The short term behavior with a low value
of p is demonstrated in Figure 13, and the long term behavior of Ateles Hybridus with
a low value of p is demonstrated in Figure 14 where the mean population at each stage
is calculated and represented in the simulation. We compare these dynamics to dynamics
with a larger, moderate, value of p, where it is set instead at 0.5 instead of .15. After
analyzing both the short and long term behavior across 15 and 40 stages, respectively, we
observe that in this interval, the dynamics of the population tend to no longer approach
survival but tend to approach extinction. The short term behavior with a moderate value
of p is demonstrated in Figure 15, and the long term behavior of Ateles Hybridus with a
moderate value of p is demonstrated in Figure 16 where the mean population at each stage
is calculated and represented in the simulation. We observe similar dynamics after raising
the value of p to 0.85. The populations of all cohorts exhibit the same behavior when
p = .85 as when p = .5. The short term behavior with a high value of p is demonstrated
in Figure 17, and the long term behavior of Ateles Hybridus with a high value of p is
demonstrated in Figure 18 where the mean population at each stage is calculated and
represented in the simulation. We observe that oscillatory behavior is not observed in
these simulations given the chosen parameters b, sF , sY , sM with p varied. Therefore, we
conclude that p alone may not have as much strength on changing the oscillatory dynamics
of the population as other dynamics might. The simulations mentioned are conclusions
from eigenvalues being real, as is the case when ∆λ > 0 or ∆λ = 0.
35
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: p=0.5 (Short Term)
Figure 15: Short-term behavior of Ateles Hybridus with a moderate value of p acrossrandom initial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: p = 0.5 (Long Term)
Figure 16: Long-term behavior of Ateles Hybridus with a moderate value of p acrossrandom initial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
36
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Males
0 5 10 150
10
20
30
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: p=0.85 (Short Term)
Figure 17: Short-term behavior of Ateles Hybridus with a high value of p across randominitial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Males
0 10 20 30 400
10
20
30
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: p = 0.85 (Long Term)
Figure 18: Short-term behavior of Ateles Hybridus with a high value of p across randominitial conditions for (M,Y, F ), where δ0 = 1
2, b = 1, sM = .5, sY = .7, sF = .5
37
6.4 Case Study 3: Variable sF and δ0 Parameters
In this study, we run simulations of the population dynamics of Ateles Hybridus over
time when δ0 is chosen at random, as well as one survival parameter, sF , to develop a
strong understanding of possible dynamics one may expect to see if parameters were not
known, or if a field researcher needs a stronger understanding. If δ0 is chosen at random,
this means the proportion of young females that become an adult (and migrate in the
multi-patch model) is randomized in each patch. This in combination with a variation of
the sF parameter is important to see the concluding dynamics of Ateles Hybridus in a
fragmented and non-fragmented landscape. Specifically, we analyze cases where δ0 = .3, .5,
and 1, and we find a complementary value of sF that would warrant equilibrium within
the population. We then analyze the trends in the dynamics to see whether oscillatory
behavior is observed. We analyze both short and long-term behavior of Ateles Hybridus
under the conditions we give for each situation. Short term behavior is defined as 15
stages, where long-term behavior is defined as 40 stages. After analyzing the short term
behavior of Ateles Hybridus with a low value of δ0, we observe that the population has
an equilibrium (with given parameters p, b, sM , sY ) when sF = .75 where initial conditions
are randomized. We do not observe any oscillatory behavior with the given conditions,
and populations tend to approach equilibrium quickly. The short term behavior with
a low value of δ0 is demonstrated in Figure 19, and the long term behavior of Ateles
Hybridus with a low value of δ0 is demonstrated in Figure 20 where the mean population
at each stage is calculated and represented in the simulation. We compare these dynamics
to dynamics with a larger, moderate, value of δ0, where it is set instead at 0.5 instead
of .3. After analyzing both the short and long term behavior across 15 and 40 stages,
respectively, we observe that the population has an equilibrium (with given parameters
p, b, sM , sY ) when sF = .6875 where initial conditions are randomized. This value of sF
is less than the value of sF when the value of δ0, therefore we confirm the an inverse
relationship between sF and δ0. We do not observe any oscillatory behavior with the
given conditions, and populations tend to approach equilibrium quickly. The short term
38
behavior with a moderate value of δ0 is demonstrated in Figure 21, and the long term
behavior of Ateles Hybridus with a moderate value of δ0 is demonstrated in Figure 22
where the mean population at each stage is calculated and represented in the simulation.
We compare these dynamics to dynamics with a larger, high value of δ0, where it is set
instead at 1 instead of .5. After analyzing both the short and long term behavior across 15
and 40 stages, respectively, we observe that the population has an equilibrium (with given
parameters p, b, sM , sY ) when sF = .61 where initial conditions are randomized. This value
of sF is less than the value of sF when the value of δ0, therefore we see that there exists
an inverse relationship between sF and δ0. In this case, oscillatory behavior is observed,
which is especially evident in the young female cohort. Theoretically, for a given set of
initial conditions in a population of Ateles Hybridus, the population of young females
could crash to extinction, which could have everlasting effects on the future dynamics of
the entire population. However, for other initial conditions, this immediate crashing of
the young female cohort is not observed. The short term behavior with a high value of
δ0 is demonstrated in Figure 23, and the long term behavior of Ateles Hybridus with a
high value of δ0 is demonstrated in Figure 24 where the mean population at each stage
is calculated and represented in the simulation. We observe that when the value of δ0 is
high that to warrant an equilibrium solution with the chosen parameters p, b, sM , sY , some
eigenvalue solutions are negative, warranting oscillatory behavior, as is discussed in case 3
of the eigenvalue analysis. In other cases, ∆λ = 0 or ∆λ > 0, as is discussed in Cases 1 and
2 of the eigenvalue analysis. These correspond to when δ0 is low and when it is moderate.
39
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
200
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 0.3, sF =0.75 (Short Term)
Figure 19: Short-term behavior of Ateles Hybridus with a low value of δ0 and moderatevalue of sF across random initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM =
.5, sY = .7
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
200
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: δ = 0.3, sF =0.75 (Long Term)
Figure 20: Long-term behavior of Ateles Hybridus with a low value of δ0 and moderatevalue of sF across random initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM =
.5, sY = .7
40
0 5 10 150
20
40
60
Stage
Pop
ulat
ion
Males
0 5 10 150
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
200
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 0.5 sF =0.6875 (Short Term)
Figure 21: Short-term behavior of Ateles Hybridus with a moderate value of δ0 andmoderate value of sF across random initial conditions for (M,Y, F ), where p = 1
2, b =
1, sM = .5, sY = .7
0 10 20 30 400
20
40
60
Stage
Pop
ulat
ion
Males
0 10 20 30 400
10
20
30
40
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
200
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: δ = 0.5, sF =0.6875 (Long Term)
Figure 22: Long-term behavior of Ateles Hybridus with a moderate value of δ0 andmoderate value of sF across random initial conditions for (M,Y, F ), where p = 1
2, b =
1, sM = .5, sY = .7
41
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Males
0 5 10 15−10
0
10
20
30
Stage
Pop
ulat
ion
Young Females
0 5 10 150
20
40
60
80
Stage
Pop
ulat
ion
Females
0 5 10 150
50
100
150
200
StageP
opul
atio
n
Total
Ateles Hybridus Population Dynamics: δ = 1, sF =0.61 (Short Term)
Figure 23: Short-term behavior of Ateles Hybridus with a high value of δ0 and moderatevalue of sF across random initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM =
.5, sY = .7
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Males
0 10 20 30 40−20
0
20
40
Stage
Pop
ulat
ion
Young Females
0 10 20 30 400
20
40
60
80
Stage
Pop
ulat
ion
Females
0 10 20 30 400
50
100
150
200
Stage
Pop
ulat
ion
Total
Ateles Hybridus Population Dynamics: δ = 1, sF =0.61 (Long Term)
Figure 24: Long-term behavior of Ateles Hybridus with a high value of δ0 and moderatevalue of sF across random initial conditions for (M,Y, F ), where p = 1
2, b = 1, sM =
.5, sY = .7
42
7 Discussion
The original data given [5] and inspiration of the model is based on biological data that
would accurately represent true population dynamics of Ateles Hybridus. The single-
patch model is created as a accurate starter tool that can be used to further estimate
behavior of Ateles Hybridus in a multi-patch model setting. One of the strongest points
noted during the stability analysis is the fact that a minuscule change in parameters can
have a large impact on the final outcome. Additionally, if ∆λ = 0, then solutions are
always stable. We used the inequality to warrant stable solutions when ∆λ = 0, given by
|δ0 − sF − sY |2
< 1 (7.0.1)
We see that the equilibrium corresponding to extinction is stable if and only if it is the
only equilibrium. If it is not the only equilibrium, then surviving populations would tend
toward the tri-coexistence survival equilibrium. We had the benefit of the limitations
on certain parameters to make them biologically realistic. These include the probability
coefficients p, sM , sY , sF , δ0, where 0 ≤ p, sM , sF ,≤ 1 and 0 < sY , δ0 ≤ 1. We were able
to exploit these parameter limitations to infer further results on our single-patch model
that can be integrated into our multi-patch model. For the above example, when we were
determining the value of sY − sF , we know it eventually is equal to a number between
−1 and 1. Therefore, it does not matter what parameter values were chosen, just as long
as their respective limitations were called for. Changing the certain values of sY and sF
does not make much of a difference to our model, as long as sY − sF has the same value.
Spider monkeys are an endangered species, and further research can be done in the area
of further integration of our single-patch model into a multi-patch model that strongly
simulates movement of young females to other patches as they reach their reproductive
stage. Some same equilibrium points may be given, but extra parameters in the model
can understandably give further complexities.
We developed a program that analyzes the importance of parameters p, δ0,i alone, and
43
δ0,i when used in conjunction with sF . After gathering and analyzing simulation data, we
observe that parameters p and δ0,i have a lot to do with the overall dynamics of the popu-
lation when they influence the parameter sF and the need to create a survival equilibrium.
In our case studies 1 and 3, we observe oscillatory dynamics arising when parameter values
were within a certain interval, where outside the interval the oscillatory behavior vanishes.
It is notable that for certain initial conditions of oscillatory behavior, the young female co-
hort dramatically approaches extinction immediately, whereas with other initial conditions
with oscillatory behavior, this is not the case. We see the importance initial conditions
can play on this model. After analyzing a single-patch model and integrating our findings
into a multi-patch model, we understand that the multi-patch model is contingent upon
young females migrating into each patch at each stage. If a hostility coefficient Hi is as
such that females do not want to enter a patch, that patch will die out. Further testing can
be used to see whether extinct patches can be resurrected and reinhabited in the future.
After observing that a minuscule change in sF can have a change in the stability of the
population, we wanted to see what changes would occur in the entire population if we
were able to have random parameter values and initial conditions for every value, as well
for a corresponding set of values at the beginning. We concluded that when everything is
at random, it can be difficult for biologists to create conclusions on the chosen data. Of
course, biologists can obtain extra data from the field by analyzing food sources available,
the psychology of the monkeys in interaction with each other, as well as other animals in
the area. Additionally, biologists can perform climate analysis and habitat analysis to see
if there are any overarching differences between each patch to see if any large changes can
be made.
44
8 Conclusions
After performing a strong analysis on a single-patch model and integrating our findings
into a multi-patch model with additional parameters, we derive strong conclusions that
can be used by biologists to defend against the endangerment issue of Ateles Hybridus.
Additionally, from our community dynamic analysis, we make extra conclusions about
efficient parameter values that biologists could try to reenact in the real life patches to
encourage stability of life in Ateles Hybridus populations.
A system where eigenvalues are negative is best to stay away from. Although systems
where eigenvalues are negative arose many times in our chosen model, the oscillation
factor may make it difficult for biologists to create additional food available for the
populations to take advantage of. Additionally, at the bottom end of each oscillation
leaves the respective cohort in the population vulnerable to extinction, thus bringing
the rest of the patch into extinction. As is seen in our parameter case studies, there
are set initial conditions at which young females would immediately die out of the
population given certain initial conditions, and would need to wait 1 or 2 stages to
be resurrected by new births.
Controlling the female survival percentage is key. Even if it is difficult for biologists
at times to keep a high survival percentage, it is important that it is controlled.
Systems where eigenvalue solutions are negative are to be avoided, and we saw from
the community dynamics simulations that when the sF parameter is known and
controlled, the populations were easier to control, and hence would be easier to
manipulate when they are needed. As we also observed in our case study where we
manipulated the parameter p, when p is as such that more females were coming into
the population, the overall dynamics of the model did not tend as much to extinction
as it did when p is as such that males were dominant in the population.
Males need to be alive, but not necessarily in high amounts. If endangerment experts
have limited resources available, they should spend most of those resources keeping
45
the young female and female populations stable. As seen in the majority of our
simulations, the system is partially decoupled with respect to the male cohort. As
long as males are available (M > 0) to fertilize the females, their populations will
not matter as much as keeping the females alive, which are the important ones as
far as the future of the population is concerned.
It is important for patch hostility to be monitored. If a patch becomes too hostile,
young females will not want to migrate there at the time they reach adulthood. If
a patch becomes hostile, endangerment experts can either remove the sources of the
hostility, or find a way to make the patch size larger in order to reduce the effects of
the hostility of the patch. Of course, one must be careful when using this option as
any techniques used to raise the size of the patch may raise the hostility as well.
Force young females to stay alive into adulthood, even if it does not happen in one
stage. We noticed from our model that if δ0,i = 1 or very close to 1, then any system
would produce the strongest oscillations, which could be disastrous if an unforeseen
event were to happen to the patch. Strong oscillations mean there are times when
the population is near zero for a short time in many intervals. When the migration
level is lower per stage per capita, oscillations are not as strong (if they exist) which
will make it easier for the population as a whole to stay alive.
46
References
[1] Adler, R., A. G. Konheim, and M. H. McAndrew. “Topological Entropy.” Transactions
of the American Mathematical Society. 114 (1965): 309-319.
[2] Castillo-Chavez, Carlos, and Fred Brauer. Mathematical Models in Population Biol-
ogy and Epidemiology. New York: Springer, 2001.
[3] Caswell, Hal. Matrix Population Models: Construction, Analysis, and interpretation
of matrix population models in the biological sciences. 1989.
[4] Cordovez, J. M., J. R. Arteaga B, M. Marino, A. G. de Luna and A. Link. “Popu-
lation Dynamics of Spider Monkey (Ateles Hybridus) in a Fragmented Landscape in
Colombia.” Biometrics. (2012): 6-8.
[5] G. Cowlishaw and R. Dunbar. “Primate Conservation Biology.” Chicago University
Press. Chicago. 2000.
[6] Doubleday, W. G., “Harvesting in Matrix Population Models” Biometrics. 31 (1975):
189-200.
[7] F. Michalski and C. A. Peres., “Biological Conservation”. p. 383-396. 2005.
[8] C. A. Peres. “Conservation Biology” 15 (2001). p. 1490-1505.
[9] Y. Shimooka, C. Campbell, A. Di Fiore, A. M. Felton, K. Izawa, A. Link, A.
Nishimura, G. Ramos-Fernandez and R. Wallace. “Demography and group composi-
tion of Ateles” p. 329-348. Cambridge University Press. 2008.
47
9 Appendix A
The following program is used to visualize the final dynamics of our model with certain
parameter inputs. The current input corresponds to an unstable system where the pop-
ulations approach infinity as time approaches infinity. Here, we modified our values of
sM , sY , sF , b, p, and δ0 to determine the behavior and overall result of the population dy-
namics of Ateles Hybridus. We were then able to group our results into cases 1, 2, or 3
based on the behavior that is analyzed in our single patch model. In many cases, small
changes in certain parameter values would turn into large changes in the dynamics of the
model.
function nt=ex2p1(t)
sM=.5;
sF=.49;
p=1/3;
b=.75;
muY=0;
deltaN=.1;
A=[ sM 0 p*b; % enter the matrix
0 1-muY-deltaN (1-p)*b;
0 deltaN sF];
n0 = [400 100 400]’; % enter the initial vector
nt=zeros(3,t); % alocate memory for the vectors
nt(:,1)=n0; % set the initial vector as the first one on the array
for j=2:t % the loop
nt(:,j)=A*nt(:,j-1);
end
plot(nt’);
xlabel(’Stage’)
ylabel(’Population at stage n’)
48
title(’Spider Monkey Population Dynamics for Various Stages’)
legend(’Males’,’Young Females’,’Females’)
end
49
% SimulationOverTime.m - this MATLAB file simulates the
% Ateles-Hybridus difference equation
% M(i+1)=M(i)*sM+p*b*F(i);
% Y(i+1)=(1-p)*b*F(i)+(sY-deltaN)*Y(i);
% F(i+1)=sF*F(i)+deltaN*Y(i);
M0=30; %input(’input initial population M0 of males: ’)
Y0=20; %input(’input initial population Y0 of young females: ’)
F0=15; %input(’input initial population F0 of females: ’)
sM=.8; %Input survival rate of males
sY=.875; %Input survival rate of young females
sF=.125; %Input survival rate of females
p=.5; %Input probability of male birth
b=1; %Input birth rate per female (> 1 allowable)
deltaN=.3;
n=80; %input(’input time period of run: ’)
M=zeros(n+1,1);
Y=zeros(n+1,1);
F=zeros(n+1,1);
t=zeros(n+1,1);
M(1)=M0;
Y(1)=Y0;
F(1)=F0;
for i=1:n
t(i)=i-1;
M(i+1)=M(i)*sM+p*b*F(i);
Y(i+1)=(1-p)*b*F(i)+(sY-deltaN)*Y(i);
F(i+1)=sF*F(i)+deltaN*Y(i);
end
50
t(n+1)=n;
plot(t,M,t,M,’o’)
title(’Male values’),pause
plot(t,Y,t,Y,’*’)
title(’Young Female values’),pause
plot(t,F,t,F,’*’)
title(’Female values’),pause
plot(t,Y,t,M,t,M,’o’,t,Y,’*’)
title(’Male and Young Female values’),pause
plot(M,Y,’o’)
title(’Male vs. Young Female vs. Female’);
51
This is the program that is used to simulate the dynamics of each of the cohorts of the
population of Ateles Hybridus. These are males, young females, and females. We also
included a plot which would graph the total population as well. We created a MATLAB
graph which would generate four subplots displaying each of the cohorts’ dynamics. We
input a value k that would generate the number of stages that would be run in the model,
and input the number of iterations that would be given based on random initial conditions,
and our goal is to see whether random initial conditions had strength within the model.
We did conclude that the parameter p did have strength in the model, and the parameter
δ0 had strength in the model as far as having an influence on the ideal value of sF .
function M1=meanModel2(k)
% Modify J.flores M.Buhr 3/17/2015
% input k=number of simulations of a single state variable
% output the data for the state variable and it graph and the
% graph of the mean.
N=15;
p=.5;
b=1;
mu m=.5;
mu y=.3;
mu f=.5;
delta=.85;
H=1;
A=1;
M1=[];
M2=[];
M3=[];
M4=[];
T end=39;
52
for ii=1:k % number of simulations
M=zeros(1,N);
Y=zeros(1,N);
F=zeros(1,N);
Tot=zeros(1,N);
M(1)=randi(50);
Y(1)=randi(30);
F(1)=randi(70);
Tot(1)=M(1)+Y(1)+F(1);
S=1;
for n=2:T end % number of periods
M(n)=p*b*F(n-1)+(1-mu m)*M(n-1);
Y(n)=(1-p)*b*F(n-1)+(1-mu y-delta)*Y(n-1);
F(n)=(1-mu f)*F(n-1)+Y(n-1)*(delta);
Tot(n)=M(n)+Y(n)+F(n);
end
T=1:T end;
M1=[M1;M]; % change M by Y or Mby F to obtain the data for the other state vari-
ables.
M2=[M2;Y];
M3=[M3;F];
M4=[M4;Tot];
end
subplot(2,2,1)
plot(T,M1)
hold on
plot(T,mean(M1),’LineWidth’,3,’Color’,[1 0 0])
53
xlabel(’Stage’)
ylabel(’Population’)
title([’Males’])
subplot(2,2,2)
plot(T,M2)
hold on
plot(T,mean(M2),’LineWidth’,3,’Color’,[1 0 0])
xlabel(’Stage’)
ylabel(’Population’)
title([’Young Females’])
subplot(2,2,3)
plot(T,M3)
hold on
plot(T,mean(M3),’LineWidth’,3,’Color’,[1 0 0])
xlabel(’Stage’)
ylabel(’Population’)
title([’Females’])
subplot(2,2,4)
plot(T,M4)
hold on
plot(T,mean(M4),’LineWidth’,3,’Color’,[1 0 0])
xlabel(’Stage’)
ylabel(’Population’)
title([’Total’])
text(-45,397,[’Ateles Hybridus Population Dynamics: delta = ’,num2str(delta),’ (Long
Term)’])
end
54