General population growth models with Allee effects in a random environment

Abstract

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.

In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.

We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.

We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.

We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.