1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 461

law, since they compete with one another for space in which to grow. In the

absence of the plants, the mammals are assumed to die out according to the

Malthusian law. And in the absence of the mammals, the reptiles are also

assumed to die out according to the Malthusian law. Hence, the population

dynamics for these three interacting species is represented by the system of

differential equations

(4)

dy1

dt

where a, b, c, d, e, f, g, and h are positive constants.

Next, suppose that one species preys upon the adults of a second species but
not upon the young of that species. This situation can occur when the young
prey are protected in some manner- perhaps by their coloration, by their
smaller size, by their living quarters, or simply by the physical intervention
of the adults. The model for this prey-predator system with protected young
prey leads to a system of differential equations with three components.

Let Y1 denote the number of young prey, Y2 denote the number of adult
prey, and y3 denote the number of predators. Thus, the total prey population
at time t is y 1 ( t) + Y2 ( t). We assume the birth rate for the prey is proportional
to the number of adult prey- that is, we assume a Malthusian type of growth
for the prey. We assume the number of young prey maturing to adulthood is
proportional to the number of young prey. And we assume the death rate for
the young prey is proportional to the number of young prey. Hence, the rate
of change for the number of young prey is

(5a)

dy1

- = ay2 - by1 - cy1

dt

where a , b, and c are positive constants- constants of proportionality for
the processes of birth, maturation, and death. Next, we assume the adult
prey population increases due to the maturation of the young into adults, so
dy2f dt includes the term by 1 to reflect the maturation process. We assume
the death rate of the adult population due to causes other than predation to
be proportional to the number of adults, so dy2 / dt includes the term -dy 2
to model the death of adults. And we model the predation by including the
term -ey 2 y 3. Thus, the equation for the rate of change of the adult prey
population is

(5b)