The Logistic Growth SDE

Motivation In population biology the logistic growth model is one of

the simplest models of population dynamics.

To begin studying Stochastic Differential Equations (SDE) we begin by studying the effects of adding a stochastic term to well known deterministic models.

Terminology Stochastic Process: Let I denote an arbitrary nonempty index

set and let Ω,U, P denote a probability space. A family of Rn – valued random variables is a stochastic process.

Markov Property: with probability 1.

; iX t I

( ( ) | ([ , ]) ( ( ) | ( ))oP X t B U t s P X t B X s

Terminology Diffusion Process: A Markov process with continuous sample

paths such that its probability density function satisfies for any

a(x,t) is the infinitesimal mean and is called the drift vector and B(x,t) is the infinitesimal variance and is called the diffusion matrix.

0

0

2

0

1( ) lim ( , ; , ) 0

1( ) lim ( ) ( , ; , ) ( , )

1( ) lim ( ) ( , ; , ) ( , )

y xt

y xt

y xt

i p y t t x t dyt

ii y x p y t t x t dy a x tt

iii y x p y t t x t dy B x tt

0 and ( , )x

Terminology Wiener process: a stochastic process where W(t) depends

continuously on t, and the following hold:( ) ( , )W t

1 2 2 1

2 1

1 2 2 1

1 0

( )For 0 , ( ) ( ) is normally distributed with mean zero and variance

( )For 0 , the increments ( ) ( ) and( ) ( ) are independent

( ) Prob (0) 0 1

i t t W t W tt t

ii t t W t W tW t W t

iii W

Ito’s IntegralStochastic dynamics yields differential equations of the form

where is Gaussian white noise. The goal is to transform (1) into an integral equation and solve

for X(t).

0 0

( ) (0) ( ( ), ) ( , ( )) (2)t t

sX t X f X s s ds g s X s ds

( ) ( ( ), ) ( ( ), ) ( ) (1)X t f X t t dt g X t t t

( )t

Ito’s IntegralThe second integral in (2) is undefined. It can be shown that the

Wiener Process is the derivative of the white noise term.

Using (3) in (2)

0

( ) ( ) or ( ) ( ) (3)t

W t s ds dW t t dt

0 0

(0) ( ( ), ) ( ( ), ) ( ) (4)t t

tX X f X s s ds g X s s dW s

Ito’s Integral The first integral in (4) is the deterministic term and is a

regular integral. The second integral in (4) is the stochastic term and must be

definedTake

We want to define the integral:

( ( ), ) ( ) (5).g X t t W t

0

( ) ( ) ( ) (6).t

t

X t W s dW s

Ito’s IntegralTo examine the behavior of (6) we start by assuming it to be

Riemann-Stieltjes integral and integrating. This yields

The partial sums are defined as

they converge with finer partitions and arbitrary choice of the intermediate points .

0

2 20( ) ( )( ) (7)

2

t

t

W t W tW s ds

11

( )( ( ) ( )) (8)n

n i i ii

S W W t W t

i

Ito’s IntegralThe approximation sums converge in mean square. They can be

written as

Convergence depends on the choice of the intermediate point.

Choose .

1

21

1 1

( ) ( ).i i

n n

t t i ii i

E W W t

1i it

Ito’s IntegralEquation (6) then becomes

By convention an Ito SDE is written as

and satisfies the integral equation

0

2 20 0( ) ( )( ) ( ) .

2 2

t

t

W t W t t tW s dW s

( ) ( ( ), ) ( ( ), ) ( )dX t f X t t dt g X t t dW t

0 0

( ) (0) ( ( ), ) ( ( ), ) ( ).t t

X t X f X s s ds g X s s dW s

Ito’s FormulaSuppose ( ) is a solution to the following Ito SDE:

( ) ( ( ), ) ( ( ), ) ( ).

If ( , ) is a real-valued function defined for

and t [a,b], with continuous partial derivatives, ,

, and

X tdX t f X t t dt g X t t dW t

F x t xFt

Fx

R

2

2

2 2

2

, then

( ( ), ) ( ( ), ) ( ( ), ) ( ) where

( , ) ( , ) ( , ) ( , )( , ) ( , )2

( , )( , ) ( , )

Fx

dF X t t f X t t dt g X t t dW t

F x t F x t g x t F x tf x t f x tt x x

F x tg x t g x tx

Example: Exponential Growth Consider the SDE ,exponential growth with environmental variation, where c and r

are positive constants.

Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

( ) ( ) ( ) ( )dX t rX t dt cX t dW t

( , ) ln( ).F x t x

2

( ) (0)exp([ ]) ( )).2cX t X r t cW t

Example: Logistic Growth Consider the SDE

logistic growth with environmental variation, where c, K and r are positive constants.

Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

( )( ) ( ) 1 ( ) ( ),X tdX t rX t dt cX t dW tK

1( , ) .xF x t

2

2

12

0

exp([ ]) ( ))2( ) .

(0) exp([ ]) ( ))t

crK

cr t cW tX t

X r s cW s ds

Example: Bimodal Equations Consider the SDE

logistic growth with environmental variation, where c, K and r are positive constants.

Let . Then applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:

2 ( )( ) ( ) 1 ( ) ( )X tdX t rX t dt cX t dW tK

21( , )x

F x t

2

2

2

12

0

exp([ ]) ( ))2( ) .

(0) exp([ ]) ( ))t

crK

cr t cW tX t

X r s cW s ds

Graphs: The Basic Equations

Graphs: ( ) ( ) ( , ) ( )dX t X t dt x t dW t

Graphs: ( )( ) ( )(1 ) ( , ) ( )X tKdX t rX t dt x t dW t

Graphs: 2 ( )( ) ( )(1 ) ( , ) ( )X tKdX t rX t dt x t dW t

Considerations Effects of the coefficient of the stochastic

term. How to determine the correct coefficients for a specific problem

Expected Gaussian versus graphed Levy distribution

References An Introduction to Stochastic Process with Applications to

Biology Linda J.S. Allen

Stochastic Differential Equations Ludwig Arnold

Introduction to Stochastic Differential Equations Thomas Gard