A predator–prey model with disease in the predator species only

Abstract

In this paper we propose a predator–prey model with logistic growth in the prey population that includes an SIS parasitic infection in the predator population, with the assumption that the predator has an alternative source of food. For simplicity we initially work with a model involving the fractions of the predator which are susceptible and those infected and then translate the results back to the model with absolute numbers. Important thresholds $R_0^1$, $R_0^2$, $R_0^3$ and $R_0^4$ are identified and their implications have been explained. Our theoretical study indicates that the absence of prey may be beneficial for predator when a transmissible disease runs among the predator population. One important conclusion is that infection in the predator species may save the prey from extinction even if $R_0^2$, the basic reproduction number for the prey to be able to invade the predator-only equilibrium, is less than one. Therefore infection in the predator species may be taken as biological control. Finally, analytical results have been supported by numerical simulations with the help of experimental data.

Introduction

Mathematical and statistical models can help us to identify key parameters which determine the rich dynamics of an ecological or epidemiological system. In the development of quantitative theory for interaction of predator and prey, mathematical ecology is also an important factor along with the experimental ecology, see [1] and references therein. Transmissible disease in an ecological situation is fast becoming a major field of study in its own right. The first mathematical description of contagious diseases has been formulated by Kermack and McKendric [2]. Both theoretical and experimental investigations in these two fields namely ecology and epidemiology progressed independently along the years, until the late eighties and early nineties. The ecological literature has increasingly emphasized the importance of parasites in shaping the dynamics of both plant and animal communities, [3], [4], [5].

Some models merging features of the two phenomena, i.e. the demographics of interacting species and an epidemic evolution in such a composite environment, were then considered, see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references therein.

Most of the above models are based on the assumption that the infected prey is more vulnerable to predation except for those developed by Venturino [11], and Haque and Venturino [15]. In these two papers the authors considered the case when disease is spreading among the predator population. The model considered here also studies the situation where an epidemic runs among the predator population but differs from previous models; it includes the recovery rate and also takes into account also the stability of the positive interior equilibrium point. In addition, we assume that the predator has a logistic growth rate since it has sufficient resources for alternative foods; and it is argued that alternative food sources may have an important role in promoting the persistence of predator–prey systems. van Baalen et al. [18] have observed that the switching of the predator species has a significant contribution to the persistence of species in the predator–prey system. They also have analyzed the conditions for stability as well as long-term behavior of the system under bounded population fluctuations. This leads to specific hypotheses about which types of alternative food (in terms of nutritional quality, availability, and handling time) promote persistence and by which mechanism.