1550078481-Ordinary_Differential_Equations__Roberts_

464 Ordinary Differential Equations

(i) a= 2, b = c = d = e = f = g = h = 1

(ii) a= 2, b = .5, c = d = e = f = g = h = 1

In each case, display a single graph showing YI ( t) , Y2 ( t) and y3 ( t) on the

interval [O, 6]. What happens to YI(t), y2(t), and y3(t) as t increases?

8. a. Find the critical points of the prey-predator model in which the young
   prey are protected, system (5).
   b. Use computer software to solve system (5) for the initial conditions

YI(O) = 1, Y2(0) = 1, y3(0) = 1 on the interval [O, 10] for the following values

of the constants.

(i) a= 2, b = c = d = .5, e = f = g = 1

(ii) a= 2, b = c = d = e = f = g = 1

In each case, display a single graph showing YI ( t), Y2 ( t), and y3 ( t) on the

interval [O, 10]. What happens to YI(t), y2(t), and y3(t) as t increases?

Leslie's Prey-Predator Model

In 1948, P. H. Leslie proposed the following prey-predator model

(1)

dx

- = ax - bx^2 - cxy

dt

dy = dy-ey2.

dt x

The first equation in system (1) for the rate of change of the prey popula-
tion, x, is the same as the Volterra-Lotka prey-predator model with internal
prey competition. That is, in the absence of predation the prey population
is assumed to grow according to the logistic law model and has maximum
population size b/a. The second equation in system (1) for the rate of change
of the predator population, y, resembles the logistic law equation except the
second term has been modified to take into account the density of the prey.
If y/x = d/e, the predator population is at its equilibrium value. If there

are many prey per predator, y/x is small (y/x < d/e) and, therefore, the

predator population grows nearly exponentially. And if there are few prey

per predator, y/x is large (y/x > d/e) and the predator population decreases.

EXERCISE

1. a. Find the critical point of Les lie's prey-predator model, system (1) in the
   first quadrant and determine if it is stable or unstable.
   b. Use computer software to solve system (1) with a = 2, b = e = .5,
   c = d = 1 on the interval [O, 5] for the initial conditions x(O) = 1, y(O) = 1.
   Display a single graph showing x(t) and y(t). Display a phase-plane graph of
   y versus x. Estimate limt, 00 x(t) and limt, 00 y(t).