THE LOTKA-VOLTERRA MODEL
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:

• Sardines have food in non limited quantities
• Sharks are the only obstacle to the growing of the population of sardines
• The population of sharks is limited by the number of sardines they can eat

From those hypothesis, you can establish two equations:

• As sardines have enough food, you can consider that in the absence of sharks, x(t) can be described with an exponentially growth function.
•
x'(t)=ax(t)    with a>0
• In the same idea, we can say that in the absence of sardines, the population of sharks exponentially decay.
•
y'(t)=-by(t)    with b>0
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: • a: natural growth rate of sardines in the absence of sharks
• c: natural death rate of sharks in the absence of sardines
• b: death rate per encounetr of sardines due to predation
• d: efficiency of turning predated sardines into sharks
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).

• For the first critical point, we obtain the following matrix and eigen values:  As one of the eigen value is real and positive, we can conclude that the first critical point is unstable.

• For the second critical point, we have the following matrix and eigen values:  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.