Effect of population size in a predator–prey model

We consider a hybrid version of the basic predator–prey differential equation model: a pure jump stochastic model for the prey variable x coupled with a differential equation model for the predator variable y. This hybrid model is derived from the classical birth and death process. The model contains a parameter ω which represents the number of individuals for one unit of prey: x = 1 corresponds to ω individual prey. It is shown by the mean of simulations and explained by a mathematical analysis based on a result from the singular perturbation theory – the so-called theory of Canards – that qualitative properties of the model like persistence or extinction are dramatically sensitive to ω. For instance, in our example, if ω = 107 we have extinction and if ω = 108 we have persistence. This means that we must be very cautious when we use continuous variables in place of discrete ones in dynamic population modeling even when we use stochastic differential equations in place of deterministic ones.

► We consider a stochastic version of the basic predator–prey differential equation model.
► The model contains a parameter omega which represents the number of individuals for one unit of prey.
► It is shown that qualitative properties of the model like persistence or extinction are dramatically sensitive to omega.
► For instance, in our example, if ω = 107 we have extinction and if ω = 108 we have persistence.
► This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population.