Description of the model


In order to study the previous phenomenon, Volterra proposed a model.
Given x(t) the number of eatable fishes (we will consider sardines for exemple) and y(t) the number of predators (sharks).

Volterra describes the evolution of those two species with a system of differential equations, based on the following hypothesis:

From those hypothesis, you can establish two equations:

Then you can consider that the product x(t)y(t) is in proportion with the number of meetings between sharks and sardines (to establish this model, Lotka and Volterra argued that consumer and resource population could be treated like particles interacting in an homogeneously mixed gas or liquid, and under these conditions, the rate of encounter between consumers and resources would be proportional to the product of their masses.). Such a meeting is good for the shark and bad for the sardine.
This remark, linked with the previous equations, leads to the following system:

This system is a first order system, which is also autonomous (it doesn't explicitely depends of t).

Critical points

Once, we have this system, it is easy to find its critical points by solving the system x'(t)=0 and y'(t)=0.
We easily obtain two critical points:

Stability of the critical points

To study the stability of the critical points, we can calculate the eigen values of the following matrix:

(if we write the system as x'(t)=f(x,y) and y'(t)=g(x,y))

The study of this matrix results of the linearization of the system near the critical points (to test if a small perturbation is amplified or not by the system).

The two eigen values have a real part which is equal to zero. It is a marginal case in the study of the stability. We can say that this critical point will be a center for the trajectories, whcih will be oscillating around it.