462 Ordinary Differential Equations
where b, d, and e are positive constants-constants of proportionality for the
processes of maturation, death, and predation. We assume the death rate
for the predator population is proportional to the number of predators, so
dy 3 /dt includes the term -fy 3. In other words, we assume the Malthusian
law holds for the predators. We also assume the rate of increase in the number
of predators due to predation can be modelled as gy2y3. Thus, the equation
for the rate of change of the predator population is
(5c)
dy3
dt = -fy3 + 9Y2Y3
where f and g are positive constants- constants of proportionality for the
processes of death and predation. Consequently, a three component system
of differential equations for modelling a prey-predator system in which
the young prey are protected is
dy1
dt = ay2 - by1 - cy1
(5)
where a, b, c, d, e, f , and g are positive constants.
EXERCISES
- Find and classify all critical points of the prey-predator model with internal
prey competition, system (1). Which critical points are in the first quadrant?
- Use SOLVESYS or your computer software to solve the prey-predator
model with internal prey competition, system (1), for the initial conditions
x(O) = 3, y(O) = 2 on the interval [O, 5] for r = 2, H = .5, s = 2, Q = 1, and
(i) C = 1.5 and (ii) C = .5.
For cases (i) and (ii):
a. Display a graph of x(t) and y(t) on [O, 5].
b. Display a phase-plane graph of y versus x.
Answer the following questions for cases (i) and (ii):
Is the solution periodic?
What happens to the prey population, x (t), as t increases?
What happens to the predator population, y(t), as t increases?
Where are the critical points (x, y) with x 2'. 0 and y 2'. O?
What do you think happens for any initial condition (xo, Yo) where x 0 > 0
and Yo> O?
