Prey-Predators systems 
Study of the Volterra model
In what follows \(t\) denotes the time (nonnegative real variable), \(x(t)\) the number of prey at time \(t\) and \(y(t)\) the number of predators at time \(t\). Of course we should consider only integer values of \(x(t)\) and \(y(t)\); however the study of the Volterra model requires that we allow \(x(t)\) and \(y(t)\) to have non-negative real values and define differentiable functions.
The variations over time of the functions \(x\) and \(y\) have their analytical translation in the derivatives \(x^{'} \) and \(y^{'} \) and thus the considerations we have made before are formalised in the following system of differential equations \[\begin{cases} x^{'}= & Ax-Bxy\\ y^{'}= & -Cy+Dxy \end{cases}\] where \(A\), \(B\), \(C\) e \(D\) are positive constants. Solving the system is to determine the function \(t\rightarrow\left(x\left(t\right),y\left(t\right)\right)\) which verifies the two equations.
This non-linear system is solvable: by the Existence and Uniqueness Theorem (details in [4]), for each initial condition \(x(t_{0})\), \(y(t_{0})\) there is a curve \(t\rightarrow\left(x\left(t\right),y\left(t\right)\right)\) which verifies the two equations of the system. However, the proof of this theorem is not constructive and, in particular, we do not know how to determine this solution curve explicitly. We have to make a qualitative analysis with information suggested by numerical integration.
Recall that the aim is to determine the average number of pouts and sharks; we will see that, regardless of the solutions,
average number of sharks \(=\frac{A}{B}\)
If you follow the sequence of the items below you will find a justification of these equalities.
