Limits of Difference Equations

Date: 05/14/2000 at 13:13:53
From: Christopher Moh
Subject: Difference Equations

Hello,

I would like to know how to find the limit of F and R in the following 
system of equations. This system of equations is similar to that of 
the Lotka-Volterra ecological model.

Assume F(x) and R(x) are functions of Foxes (predators) and Rabbits 
(prey), respectively. Assuming that both can reproduce naturally and 
that rabbits can be converted to foxes at a certain rate (by being 
eaten), we get the following difference equations:

     R(x+1) = a * R(x) - b * F(x)
     F(x+1) = c * F(x) + d * R(x)

where a, b, c and d are constants and F(x) and R(x) are the number of 
foxes and rabbits at time period x respectively.

Let us assume, for the purposes of this problem, F(1) = R(1) = 1000.

The question I want to ask is, can I find lim R(x) and lim F(x) as x 
tends towards infinity?  If so, how?

Thanks.

Date: 05/14/2000 at 18:38:20
From: Doctor Anthony
Subject: Re: Difference Equations

I shall use 't' rather than x to give the time intervals.

     R(t+1) = a.R(t) - b.F(t)   .......[1]
     F(t+1) = c.F(t) + d.R(t)   .......[2]

[1] x c plus [2] x b gives:

     c.R(t+1) + b.F(t+1) = (ac+bd)R(t)

and so

     b.F(t+1) = (ac+bd)R(t) - c.R(t+1)

and then

       b.F(t) = (ac+bd).R(t-1) - c.R(t)

substitute into [1]

       R(t+1) = a.R(t) - (ac+bd).R(t-1) + c.R(t)

       R(t+1) = (a+c).R(t) - (ac+bd).R(t-1)

       R(t+1) - (a+c).R(t) + (ac+bd).R(t-1) = 0

So we have a standard homogeneous difference equation which is easily 
solved.

If we solve the quadratic equation:

     x^2 - (a+c)x + (ac+bd) = 0

and get roots p and q then

     R(n) = A.p^n + B.q^n   (A and B are constants)

where n corresponds to the nth interval of time. By the look of this 
result R(n) will tend to infinity if either p or q is greater than 1, 
and if both are less than 1 then R(n) will tend to 0.

Similar calculations can be made for F(t). 
         
- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/