1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 467

As b efore, x is the prey population, y is the predator population, and a, b, c,

d, e and k are positive constants.

EXERCISE

1. Use computer software to solve system (1) with a = c = 12 , b = 1.2,

d = e = 2, and k = 1 on the interval [O, 5] for the following initial conditions:

a. x(O) = 2, y(O) = 1 b. x(O) = 2, y(O) = 1.35 c. x(O) = 2, y(O) = 2

In each case, produce a phase-plane graph of y versus x. What do you notice
about the phase-plane graphs?

Competing Species Models

Two similar species sometimes compet e with one another for the same lim-
ited food supply and living space. Thus, each sp ecies removes from the en-
vironment resources that promote the growth of the other. This situation
usually occurs when two different species of predators are in competition with
one another for the same prey. For this reason the models to be discussed

are sometimes called competitive hunters models. When two predator

sp ecies comp ete with one another, one species nearly always becomes extinct
while t he other, more efficient species, survives. This biological phenomenon
is called the principle of competitive exclusion.

Let us make the following assumptions regarding two competing species
with population sizes x and y.

1. There is sufficient prey to sustain a ny level of the predator populations.


2. The rates of change of the predator populations depend qu adratically
   on x and y.


3. If one predator species is absent, there is no change in its population
   size.


4. In the absence of one predator species, the other predator species will
   grow according to the Malthusian law.

These assumptions lead to the competitive hunters model

(1)

dx

dt =Ax - Bxy

dy

- = Cy-Dxy

dt

where A , B , C , and D are positive constants.