1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 465

Leslie-Gower Prey-Predator Model

In 1960 , P. H. Leslie and J. C. Gower studied the following prey-predator

model

(1)

dx

• =ax -cxy
  dt

dy = dy-ey2

dt x

where x is the prey population, y is the predator population and a, c, d, and

e are positive constants. The first equation of this system is the same as in

the Volterra-Lotka prey-predator model. Thus, in the absence of predators

the prey population is assumed to grow according to the Malthusian law.

EXERCISE

1. Use computer software to solve the Leslie-Gower prey-predator model,
   system (1), with a = 1, c = .1, d = 1, and e = 2.5 on the interval [O, 6]

for initial populations of x(O) = 80 and y(O) = 20. Display a graph of x(t)

and y(t) and display a phase-plane graph of y versus x. Estimate limt, 00 x(t)
and limt,oo y(t).

A Different Uptake Function
Thus far, we have assumed that the rate of change of the prey population,
x, due to predation by the predator population, y , is proportional to the
product xy. However, since the predator population's collective appetite and
food requirement can be satisfied when there is an abundance of prey, when x
is large relative toy, the rate of change in the prey population should approach

a function which is proportional to y alone. It has been suggested that the

uptake function xy of the Volterra-Lotka prey-predator model be replaced
by the uptake function xy/(1 + kx). Thus, the new prey-predator model to
be considered is

(1)

dx

dt

bxy

ax----

1 + kx

dy dxy

dt = -cy + 1 + kx

where a, b, c, d, and k are positive constants.

EXERCISES

1. a. Find the critical point of the system

dx

4

x _ 8xy

dt 1+2x

(2)

dy _

2

8xy

dt - - Y + 1+2x

in the first quadrant.