1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 459

EXERCISE

1. a. Use SOLVESYS or your computer software to solve the prey-predator
   initial value problem

dx

dt = 2x - .5xy; x(O) = 2

dy

dt = -2y + xy; y(O) = 3 .5

on the interval [O, 4]. Assume that the unit of time is measured in years

and the unit of population is measured in thousands. What is the period,

T? What is the minimum and maximum populat ion of the prey, x, and the

predators, y? What is the average prey and predator population?

b. Solve the indiscriminate, constant-effort harvesting, prey-predator initial

value problem

dx

dt = 2x - .5xy - .3x; x(O) = 2

dy

dt = -2y + xy - .3y; y(O) = 3.5

on the interval [O, 4]. What is the period, T? How does this period compare

with the period for part a? What is the minimum and maximum population

for the prey and predators? How do these values compare with the correspond-

ing answers in part a? What is the average prey and predator population?

How do these values compare with the corresponding answers in part a?

Modified Prey-Predator Models
In this section, we will consider several modifications to the Volterra-Lotka
prey-predator model.
Internal Prey Competition Model First, suppose that in the absence
of a predator the prey population grows so rapidly that internal competition
within the prey population for important resources such as food and living
space becomes a factor. This internal competition can be modelled by chang-
ing the assumption of Malthusian population growth for the prey (assump-
tion 6 of the previous section) to logistic population growth. The resulting
prey-predator model with internal prey competition is

dx 2

dt = rx - Cx - H xy

(1)

dy

dt = -sy + Qxy

where r, H , C, s, and Q are positive constants.