The influence of diseases on Lotka-Volterra systems

Abstract

Single species population dynamics have been studied since the past century, in the first researches of Ver-hulst. The first interacting species model was proposed by Volterra and then also studied by Lotka. On the other hand, the classical model that considers epidemics in a population was proposed by Kermack and McKendrick. Although the two fields have been the subject of widespread research in re­cent years, hardly any work has been done to study the effect of a disease on an environment where two competing species are present. Here we analyze multiple modifications of the basic Lotka-Volterra model, to account for a disease spreading among one of the two species. We choose the simplest epidemiological models, the SI and SIS, where only susceptibles and infectives are counted. We analyze two different types of incidences, simple mass action and the standard incidence. The results seem to indicate that either the disease dies out, leaving only “neutral” cycles of the Lotka-Volterra system or one species disappears and all individuals in the other one eventually be­come infected. For some particular choices of the parameters, however, endemic equilibria in which both populations survive seem to arise. © 1993 Rocky Mountain Mathematics Consortium.