460 Ordinary Differential Equations
Interna l Pre y and Internal Predator Comp e titio n wit h Harvest-
ing Mod e l Now, suppose that in the absence of the other population both
the prey and predator populations obey the logistic law model and that each
population is harvested. The system of differential equations to be investi-
gated then becomes
(2)dx 2- =rx-Cx -Hxy-h 1 x
dt
dy.- = -sy - Py^2 + Qxy - h2y
dt
where r, C, H, h 1 , s, P, Q, and h 2 are all nonnegative constants. Notice that
the Volterra-Lotka prey-predator system, the Volterra-Lotka prey-predator
system with harvesting, and the prey-predator system with internal prey com-
petition are all special cases of this system and may be obtained by setting
various combinations of the constants equal to zero. Consequently, system (2)
is the most general model for prey-predator population dynamics that we have
encountered thus far.
Three Species Models Depending upon the assumptions made, there
are several ways to formulate a system of differential equations to represent
the population dynamics for three interacting species.
F irst of all, suppose a species with population y 1 is prey for two other species
with populations Y2 and y3. Suppose the species with population y 2 is prey
for the species with population y 3. And suppose no other predation occurs.
Further, suppose the population growth for each species, in the absence of
the other species, satisfies the Malthusian law. The system of differential
equations to be studied under these assumptions is
(3)where a, b, c, d, e, f, g, h, and i are positive constants.
M. Braun discussed the following system of d ifferential equations for rep-
resenting the population dynamics for three interacting species which live on
the island of Komodo in Malaysia. One species with population y 1 is a plant.
A second species with population Y2 is a mammal. And the third species
with population y3 is a reptile. The plants are prey for the mammals and the
mammals are prey for the reptiles. No other predation occurs. In the absence
of the mammals, the plants are assumed to grow according to the logistic