1550078481-Ordinary_Differential_Equations__Roberts_

316 Ordinary Differential Equations

system is a simple example of a linear system with variable coefficients (a

linear system in which not all of the coefficient functions are constant).

(6)

y; = - 3y1 + (x + l/x)y2 + sinx.

Here a 11 (x) = x^2 , a 12 (x) =-ex, b 1 (x) = 25, a21(x) = -3, a22(x) = x + 1/x,

and b2(x) = sinx.

DEFINITIONS Homogeneous, Nonhomogeneous, and Nonlinear

Systems of First-Order Equations

If all of the functions bi ( x) are identically zero, then the linear system ( 5)

is said to be homogeneous.

If any bi(x) is not identically equal to zero, then the linear system (5) is

said to be nonhomogeneous.

All systems of first-order differential equations not having the form of

system (5) are called nonlinear systems.

As an example of a nonlinear system of differential equations, we introduce

the Volterra prey-predator model, one of the fundamental models of mathe-

matical ecology. Let y 1 ( x) represent the prey population at time x and let

Y2 ( x) represent the predator population at time x. In the absence of predators,

the population y 1 is assumed to grow according to the Malthusian population

model: dyif dx = ay 1 where a > 0 is the growth rate. The loss of population

due to predation is assumed to be proportional to the number of encounters

between prey and predator- that is , proportional to the product y 1 y 2. Thus,

the rate of change of the prey population becomes dyif dx = ay1 -by 1 Y2 where

b > 0 is a constant which represents the proportion of encounters between the

prey and predators which result in death to the prey. Without the prey to feed

upon, it is assumed that the predators would die off according to the Malthu-

sian population model: dy2/ dx = -cy2 where c > 0 is the death rate. Due

to predation, the predator population is assumed to grow at a rate which is
proportional to the number of encounters between prey and predator. Hence,
the rate of change in predator population becomes dy 2 /dx = - cy 2 + dy 1 y 2

where d > 0 is a constant which represents the proportion of encounters be-

tween prey and predator which is beneficial to the predator. Therefore, the
Volterra prey-predator model is