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
yx 
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 )
yx 

( 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 )
(1)
where  (t ) is Gaussian white noise.
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  Wti1 ) 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