The Logistic Growth SDE - Welcome | Department of
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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 {X i ; t I} is a stochastic
process.
Markov Property: P( X (t ) B | U ([to , s]) P( X (t ) B | X (s)) with
probability 1.
Terminology
Diffusion Process: A Markov process with continuous sample
paths such that its probability density function satisfies for any
0 and x (, )
1
t
(i ) lim
t 0
yx
1
t
(ii ) lim
t 0
1
t
(iii ) lim
t 0
p( y, t t; x, t )dy 0
( y x) p( y, t t; x, t )dy a( x, t )
yx
( y x) 2 p( y, t t ; x, t )dy B( x, t )
y x
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.
Terminology
Wiener process: a stochastic process where W(t) depends
continuously on t, W (t ) (, ) and the following hold:
(i )For 0 t1 t2 , W (t2 ) W (t1 ) is normally distributed with mean
zero and variance t2 t1
(ii )For 0 t1 t2 , the increments W (t2 ) W (t1 ) and
W (t1 ) W (t0 ) are independent
(iii ) Prob{W (0) 0} 1
Ito’s Integral
Stochastic dynamics yields differential equations of the form
X (t ) f ( X (t ), t )dt g ( X (t ), t ) (t )
where (t ) is Gaussian white noise.
(1)
The goal is to transform (1) into an integral equation and solve
for X(t).
t
t
0
0
X (t ) X (0) f ( X (s), s)ds g (s, X (s))s ds (2)
Ito’s Integral
The second integral in (2) is undefined. It can be shown that the
Wiener Process is the derivative of the white noise term.
t
W (t ) (s)ds or dW (t ) (t )dt
(3)
0
Using (3) in (2)
t
t
0
0
X t X (0) f ( X (s), s)ds g ( X (s), s)dW (s) (4)
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
defined
g ( X (t ), t ) W (t )
(5).
Take
We want to define the integral:
t
X (t ) W ( s )dW ( s )
t0
(6).
Ito’s Integral
To examine the behavior of (6) we start by assuming it to be
Riemann-Stieltjes integral and integrating. This yields
W 2 (t ) W 2 (t0 )
t W (s)ds
2
0
t
(7)
The partial sums are defined as
n
Sn W ( i )(W (ti ) W (ti 1 ))
(8)
i 1
they converge with finer partitions and arbitrary choice of the
intermediate points i .
Ito’s Integral
The approximation sums converge in mean square. They can be
written as
n
n
i 1
i 1
E (Wti Wti1 )2 ( i ti 1 ).
Convergence depends on the choice of the intermediate point.
Choose i ti 1 .
Ito’s Integral
Equation (6) then becomes
W 2 (t ) W 2 (t0 ) t t0
.
t W (s)dW (s)
2
2
0
t
By convention an Ito SDE is written as
dX (t ) f ( X (t ), t )dt g ( X (t ), t )dW (t )
and satisfies the integral equation
t
t
0
0
X (t ) X (0) f ( X (s), s)ds g ( X (s), s)dW (s).
Ito’s Formula
Suppose X (t ) is a solution to the following Ito SDE:
dX (t ) f ( X (t ), t ) dt g ( X (t ), t ) dW (t ).
If F ( x, t ) is a real-valued function defined for x R
and t [a,b], with continuous partial derivatives,
F
,
t
F
2 F
, and
, then
2
x
x
dF ( X (t ), t ) f ( X (t ), t ) dt g ( X (t ), t ) dW (t ) where
F ( x, t )
F ( x, t )
g 2 ( x, t ) F 2 ( x, t )
f ( x, t )
f ( x, t )
t
x
2
x 2
F ( x, t )
g ( x, t ) g ( x, t )
x
Example: Exponential Growth
Consider the SDE dX (t ) rX (t )dt cX (t )dW (t ) ,
exponential growth with environmental variation, where c and r
are positive constants.
Let F ( x, t ) ln( x). Applying Ito’s formula and integrating from 0
to t, and solving for X(t) yields:
c2
X (t ) X (0) exp([r ])t cW (t )).
2
Example: Logistic Growth
Consider the SDE
X (t )
dX (t ) rX (t ) 1
dt cX (t )dW (t ),
K
logistic growth with environmental variation, where c, K and r
are positive constants.
Let F ( x, t ) 1x . Applying Ito’s formula and integrating from 0 to
t, and solving for X(t) yields:
c2
exp([ r ])t cW (t ))
2
X (t )
1
X (0)
t
r
K
exp([r
0
c2
2
]) s cW ( s)) ds
.
Example: Bimodal Equations
Consider the SDE
X 2 (t )
dX (t ) rX (t ) 1
dt cX (t )dW (t )
K
logistic growth with environmental variation, where c, K and r
are positive constants.
Let F ( x, t ) x12 . Then applying Ito’s formula and integrating from
0 to t, and solving for X(t) yields:
c2
exp([r ])t cW (t ))
2
X 2 (t )
1
X (0)
t
r
K
exp([r
0
c2
2
]) s cW ( s))ds
.
Graphs: The Basic Equations
Graphs: dX (t ) X (t )dt ( x, t )dW (t )
Graphs: dX (t ) rX (t )(1 X (t ) K )dt ( x, t )dW (t )
Graphs: dX (t ) rX (t )(1
X 2 (t )
K
)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
Stochastic Differential Equations
Linda J.S. Allen
Ludwig Arnold
Introduction to Stochastic Differential Equations
Thomas Gard