Page 358 - Advanced engineering mathematics

338 CHAPTER 10 Systems of Linear Differential Equations

• Complex eigenvalues with nonzero real part—(0,0) is a spiral source (positive real part) or
spiral sink (negative real part);
• Pure imaginary eigenvalues—(0,0) is a center (periodic solutions).

10.6.2 Predator/Prey and Competing Species Models

We will show how phase portraits are used in the analysis of two types of systems that arise
in important applications. These systems will be 2 × 2, but are nonlinear, hence they are not as
easily solved as linear constant coefficient systems. Nevertheless, the phase portraits will display
the qualitative behavior of solutions of the phenomena being modeled.

A Predator/Prey Model

Begin with a predator/prey model. Suppose an environment includes two species having popu-
lations x(t) and y(t) at time t. One species y(t) consists of predators, whose food is in the prey
(x(t)) population. For example, we could be looking at rabbits and foxes in a wilderness area. Or
we could have birds preying on young sea turtles near an island where the turtles lay their eggs.
For convenience in the discussion, we will use rabbits and foxes as a prototypical predator/prey
setting.
As a simplification, assume that the rabbits have no other natural enemies in the setting, and
that every encounter of a rabbit with a fox results in the fox eating the rabbit.
To model these two populations, suppose that at time t, the rabbit population increases at a
rate proportional to x(t), which is the number of rabbits at this time, but also decreases at a rate
proportional to encounters of rabbits with foxes, which is modeled by a product x(t)y(t) of the
rabbit and fox populations at that time. Then, for some positive constants a and b,

x (t) = ax(t) − bx(t)y(t).

The foxes are assumed to increase at a rate proportional to their encounters with rabbits (hence
proportional to x(t)y(t)) but to decrease at a rate proportional to their own population (because in
the absence of rabbits the foxes have no food and die). Thus, for some positive numbers c and k,

y (t) = cx(t)y(t) − ky(t).
We now have a 2 × 2 system for these populations:

x = ax − bxy

y = cxy − ky.
This is a nonlinear system because of the xy terms.
If the initial rabbit population is x(0) = α> 0, and there are no foxes, then b = 0, and the
at
rabbit population increases exponentially with x(t)=αe . If the initial fox population is y(0)=β
and there are no rabbits, then c = 0 and, with no food, the fox population dies out exponentially
according to the rule y(t) = βe −kt .
Phase portraits reveal an interesting characteristic of the populations in the case that α and β
are both positive. Clearly all trajectories will be in the first octant of the x, y-plane, since popula-
tions must be nonnegative. Figure 10.25 is a typical trajectory of this system. The horizontal and
vertical lines through P : (k/c,a/b) separate the first quadrant into four regions I, II, III, and IV,
and trajectories move about P through these regions. Follow a typical point (x(t), y(t)) around
one trajectory. Suppose a population pair (x(t 0 ), y(t 0 )) is in region I at some time t 0 (so the rab-
bit population at this time is greater than k/c and the fox population less than a/b). Now both
populations may be large. This produces more encounters, hence more rabbit kills. In this region,
x (t)< 0 and y (t)> 0, so the rabbit population is declining and the fox population increasing.

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