chapter 6 models for population population models for single species –malthusian growth model...
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Chapter 6 Models for Population
Population models for single species– Malthusian growth model – The logistic model– The logistic model with harvest– Insect outbreak model
Models for interacting populations – Predator-prey models: Lotka-Volterra systems– Competition models
Other models– With age distribution– Delay models
squirrels
Oak trees
References
J.D. Murray, Mathematical Biology, second edition, Springer-Verlag, 1998.F.C. Hoppensteadt & C.S. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag, 1997A.C. Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, 1997.
Population of interaction species
Three main types of interaction:– Predator-prey: growth rate of one decreased & the other increased
– Competition: growth rate of both decreased
– Mutualism or symbiosis: growth rate of both enhanced
Predator-Prey models:– Lotka-Volterra systems: Lotka, 1925 & Volterra, 1926
– Competition models– Mutualism or Symbiois– General Models
Lotka-Volterra system
Assumption:explain the oscillatory levels of certain fish catches in Adriatic
– Prey in absence of any predation grows in Malthusian way– Predation is to reduce the prey’s per capita growth by a term
preoperational to the prey and predator populations– In the absence of prey, the predator’s death rate is constant– The prey’s contribution to the predator’s growth is
proportional to the prey & the size of the predator population• t: time• N(t): prey population• P(t): predator population
( )( ) [ ( )]
( )( ) [ ( ) ]
dN tN t a b P t
dtdP t
P t c N t ddt
Lotka-Volterra system
Non-dimensionalization
Dimensionless system
Equilibrium– u=v=0– u=v=1
( ) ( )( ) ( )
cN t bP t dat u v
d a a
(1 ), ( 1)du dv
u v v ud d
Lotka-Volterra system
In u, v phase plane:Phase trajectories:
( 1)
(1 )
dv v u
du u v
minln , 1 attain at u=v=1u v u v H H H
Lotka-Volterra system
Explanation:– A close trajectory in u,v plane implies periodic solution of u&v– The constant H determined by u(0) & v(0)– u has a turning point when v=1 & v has one when u=1
Lotka-Volterra system
Trajectory plot: Lotka-Volterra Tool
http://www.aw-bc.com/ide/idefiles/media/JavaTools/popltkvl.html
Different examples– Case 1: a=1, b=1, d=1, c=0– Case 2: a=1, b=1, d=1, c=0.05– Case 3: a=1, b=1, d=1, c=0.5– Case 4: a=1, b=1, d=1, c=1– Case 5: a=1, b=1, d=1, c=10
Lotka-Volterra system
Jacobian matrix of the system
Stability: – undetermined: u=v=1
– Unstable: u=v=0
Unrealistic: The solutions are not structurally stable!! Suppose u(0) & v(0) are such that u & v are on trajectory H4. Any small perturbation will move the solution onto another trajectory which does not lie everywhere close to H4
( , )
1
( 1)u v
v uJ
v u
( 1, 1)
0 1,
0u vJ i i
( 0, 0)
1 01, 1
0 1u vJ
Unstable steady state
Lotka-Volterra system
Lotka-Volterra system:– Show that predator-prey interactions result oscillatory behaviors– Unrealistic assumption: prey growth is unbounded in the absence of predation
Realistic predator-prey model
2 2
( , ), ( , ),
( , ) (1 ) ( ) : logistric growth
( , ) (1 ) or ( , ) ( )
[1 ]( ) , ( ) , ( )
aN
dN dPN F N P P G N P
dt dtN
F N P r P R NKh P
G N P k G N P d e R NN
A A N A eR N R N R N
N B N B N
Lotka-Volterra system
Realistic Lotka-Volterra system:
Dimensionless variables
Dimensionless form
1 , 1
, , , , & : positive constants
dN N k P dP h PN r P s
dt K N D dt N
r K k D s h
( ) ( ), ( ) , ( ) , , ,
N t h P t k s Dr t u v a b d
K K h r r K
(1 ) : ( , ), 1 : ( , )du auv dv v
u u f u v bv g u vd u d d u
Lotka-Volterra systme
Steady state populations:– u*=0, v*=0 – u*=1, v*=0– Positive steady state:
Stability of the positive steady state
( *, *) 0, ( *, *) 0f u v g u v
2 1/ 2(1 ) [(1 ) 4 ]* *, *
2
a d a d du v u
2( *, *)
* ** 1
: ( * ) *u v
au auu
A J u d u d
b b
Lotka-Volterra system
Linear stability condition2 2
* *tr 0 * 1 , det 0 1 0
( * ) * ( * )
au a auA u b A
u d u d u d
Competition models
Assumption: two species compete for the same limited food source
The Model:
Nondimensionalization ?Steady state ?Stability ?
1 1 2 2 2 11 1 12 2 2 21
1 1 2 2
1 1 2 2 12 21
1 , 1 ,
, , , , & : positive constants; r's: linear birth rates; K's: carrying capacities
dN N N dN N Nr N b r N b
dt K K dt K K
r K r K b b
Mutualism or Symbiosis
Assumption: The interaction is to the advantage of all, e.g. plant or seed dispersers
Nondimensionalization ?Steady state ?Stability ?
1 1 2 2 2 11 1 12 2 2 21
1 1 2 2
1 1 2 2 12 21
1 , 1 ,
, , , , & : positive constants; r's: linear birth rates; K's: carrying capacities
dN N N dN N Nr N b r N b
dt K K dt K K
r K r K b b
General Models
Kolmogorov equations
Example of three species: Lorenz (1963)
– Steady state ?– Stability ?– A periodic behavior cant arise
1 2( , ,..., ), 1, 2,...,ii i n
dNN F N N N i n
dt
( ), ,
, , : positive constnats
du dv dwa v u u w b u v u v c w
dt dt dta b c
Model with age distribution
Deficiency of ODE models– No age structure & size– Birth rate & death rate depend on age!
Dependence of birth rate & death rate on age
Model with age distribution
Kinetic or mesoscopic model– t: time– a: age, – n(t,a): population density at time t in the age range [a,a+da]– b(a): birth rate of age a– : death rate of age a– In time range [t,t+dt], # of population of age a dies– The birth rate only contribute to n(t,0)– no births of age a>0
( )a( ) ( , )a n t a dt
Model with age distribution
Conservation law for the population
Von Foerster equation (PDE)
( , ) ( ) ( , )
: contribution to the change in n(t,a) from individials getting old
da1: since a is chronological age
n ndn t a dt da a n t a dt
t an
daa
dt
0
( , ) ( , )( ) ( , ), 0, 0;
(0, ) ( ), 0; ( ,0) ( ) ( , ) , 0
n t a n t aa n t a t a
t a
n a f a a n t b a n t a da t
Model with age distribution
Characteristics: on which
Integrate along the characteristic line:– When a>=t
0
0
( ), ,( )1 ( )
( ), .
t a t a t a tda ta t
t t t t a a tdt
( )dn
a ndt
0
0
0
( , ) (0, ) exp ( ) ( ) exp ( ) , .
(0, ) (0, ) ( )
a a
a a t
n t a n a s ds f a t s ds a t
n a n a t f a t
Characteristic lines
Model with age distribution
– When a<t
– Where n(t-a,0) solves
– It is a linear integral equation, can be solved numerically by iteration!!
0
0 0
( , ) ( ,0) exp ( ) ( ,0) exp ( ) ,a a
n t a n t s ds n t a s ds a t
0 0
0 0
( ,0) ( ) ( , ) ( ) ( , ) ( ) ( , )
( ) ( ,0) exp ( ) ( ) ( ) exp ( )
t
t
t a a
t a t
n t b a n t a da b a n t a da b a n t a da
b a n t a s ds da b a f t a s ds da
Model with age distribution
Similarity solution: The age distribution is simply changed by a factor
ODE
Plug into the boundary condition
( , ) ( )tn t a e r a
0
( )[ ( ) ] ( ) ( ) (0) exp ( )
adr aa r a r a r a s ds
da
0 0
0
0 0
(0) ( ) (0) exp ( )
1 ( ) exp ( ) : ( ) ! solution
at t
a
e r b a e r a s ds da
b a a s ds da
Model with age distribution
Population grow
Population decay
Critical threshold S for population growth
S>1 implies growth & S<1 implies decay, S is determined solely b the birth & death!!
0 0 (0) 1
0 0 (0) 1
0 0
(0) ( ) exp ( )a
S b a s ds da
Delay models
Deficiency: birth rate is considered to act instantaneously
In practice: – a time delay to take account of the time to reach maturity– finite gestation period
Delay Model in general
Logistic delay model
( )( ( ), ( )) with T>0 the delay
dN tf N t N t T
dt
( ) ( )( ) 1
dN t N t TrN t
dt K
Delay Models
Oscillatory behaviors, e.g.
Nondimensional form
Steady state: N=1Linearize around N=1
( ) ( ) cos2 2
dN tN t T N t A
dt T T
( )*( *) * *
N tN t t rt T rT
K
( )( )[1 ( )]
dN tN t N t T
dt
( )( ) 1 ( ) ( )
dn tN t n t n t T
dt
Delay models
Look for solutions
Unstable of N=1 since Application in physiology: dynamic diseases
( ) t Tn t ce e
Re( ) 0