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