An introduction to dynamical systems applied to exploited populations

Abstract

Population dynamics is currently an interdisciplinary subject that aims at the mathematical representation, treatment and modeling of population growth processes, using a variety of applied mathematical techniques and tools. In this work, we present four dynamical models of an exploited population system describing the time evolution of the population. Assuming the natural growth of the population is logistic, we show the effects of predation on the population considering four types of Predatory function; Constant or Quota Yield, Holling Type I, Holling Type II and Holling Type III functional responses. The mathematical analysis of the models shows that under some assumptions, we obtain alternate stable equilibria in the population system using Holling Type II and Type III functions. We observe that using Holling Type III function a desirable situation occurs: the zero population is an unstable equilibrium for all levels of predation, thus, the population can be exploited without risk of extinction.