Deep in the redwood forests in California duskyfooted rats p

Deep in the redwood forests in California, dusky-footed rats provide up to 80% of the diet for the spotted owl, the main predator of the wood rat. So, let O(t) and R(t) stand for the owl population and the rat population at time t in months respectively. Consider the linear system

dR/dt = .1R p · O

dO/dt = .4R .5O

The term p · O in the system measures the deaths of rats due to predation by owls. (In fact, 1000p is the average number of rats eaten by one owl in one month.) Determine the evolution of this system when the predation parameter p = 0.104.

Solution

dR/dt = .1R p · O -------- 1

dO/dt = .4R .5O   -------- 2

Integrating both the equations w.r.t t:

R = .1Rt - p.O.t   -------- 3

and O = .4R - .5Ot -------- 4

So R = .1Rt - 104.O Put this in equation 4

O = .4(.1Rt - 104.O) - .5Ot

O = 0.04Rt - 41.6O - 0.5 O.t

Thus owl population is: 42.6 O = 0.04 R.t - 0.5 O.t or the change in owl population is: derivarive wrt t

42.6 dO/dt = 0.04 R - 0.5 O