Prey-Predators systems 
The model with no fishing
Let us collect some information about pouts and sharks.
The increase in the number of pouts is directly proportional to the number of existing pouts.
This qualitative information about the time evolution of the population of pouts is modeled by the equation:
where \(A\) is a real positive constant.
The decrease in the number of sharks is directly proportional to the number of existing sharks.
Then, the evolution of the population of sharks is described by the equation:
where \(C\) is a real positive constant.
(Click on the figure to change the number of encounters.)
The intervention of the mathematician and physicist Vito Volterra allowed the mathematical formulation of an important relation in the food chain between sharks and pouts. When these two species meet, the sharks eat the pouts: the first ones are predators of the second ones, which, because of this this, are called prey. In this way, these encounters favor the sharks and disfavor the pouts; and naturally, are more likely to happen if the number of elements of each specie is big.
In the Volterra model, the encounters are translated by the product of the number of elements in each specie; this way of counting of the favourable cases to the encounters of the two species has a probabilistic motivation. Then
decrease in the number of pouts \(=B\). number of pouts. number of sharks
increase in the number of sharks \(=D\). number of pouts. number of sharks
If we aggregate the information of increases and decreases with time of each species (adding the increases and subtracting the decreases), we obtain the following expressions:
variation of pouts \(=A\). number of pouts \(–B\). number of pouts. number of sharks
variation of sharks \(=–C\). number of sharks \(+D\). number of pouts. number of sharks
This model, like all models, admits some simplifications but, as we shall see, it represents the dynamics of a prey-predator system well.
