The Predator-Prey Equations
An application of the nonlinear system of differential equations in
mathematical biology /ecology: to model the predator-prey relationship of a
Suppose in a closed eco-system (i.e. no migration is allowed into or out of
the system) there are only 2 types of animals: the predator and the prey.
They form a simple food-chain where the predator species hunts the prey
species, while the prey grazes vegetation. The size of the 2 populations can
be described by a simple system of 2 nonlinear first order differential
equations (a.k.a. the Lotka-Volterra equations, which originated in the study
of fish populations of the Mediterranean during and immediately after WW
Let x(t) denotes the population of the prey species, and y(t) denotes
the population of the predator species. Then
x′ = ax − αxy
y′ = −cy + γxy
a, c, α, and γ are positive constants.
Note that in the absence of the predators (when y = 0), the prey population
would grow exponentially. If the preys are absence (when x = 0), the
predator population would decay exponentially to zero due to starvation. This system has two critical points. One is the origin, and the other is in the
0 = x′ = ax − αxy = x(a − αy) → x = 0 or a − αy = 0
0 = y′ = −cy + γxy = y(−c + γx→ y = 0 or −c + γx = 0
Therefore, the critical points are γ α 0).and
The Jacobian matrix is
a −α y − x
γ y −c + x
At (0, 0), the linearized system has coefficient matrix
A = 0 −c
The eigenvalues are a and −c. Hence, it is an unstable saddle point.
At ( ,a ) , the linearized system has coefficient matrix
0 − αc
A = aγ
The eigenvalues areac i. It is a stable center. (Previously, we have
learned that the purely imaginary eigenvalues case in a nonlinear system is
ambiguous, with several possible behaviors. But in this example it really is
a center. See its phase portrait on the next page.) The phase portrait of one such system of Lotka-Volterra equations is shown
here: When x(t) and y(t) are plotted individually versus t, we see that the periodic
variation of the predator population y(t) lags slightly behind the prey
population x(t). An example is shown below.
Note: The higher curve is that of x(t), the lower curve that of y(t).
Notice that y lags x slightly. Variations of the basic Lotka-Volterra equations
One obvious shortcoming of the basic predator-preysystem is that the
population of the prey species would grow unbounde