An ecogenetic disease-affected predator–prey model

Ilaria Usville,Ezio Venturino,Cristina Paola Viola

doi:10.1080/23311835.2016.1195716

Abstract

A nonlinear ecoepidemic model of new type is introduced here, in that it contains genetically distinguishable subpopulations. Further, in the system a predator is present, that hunts these two disease-affected genotypes. Under the assumptions of the model, the disease cannot endemically survive in the predator-free environment. The healthy prey can thrive in the absence of the predators, but this is in line with previous results and does not appear to be due to the effects of the epidemics. On the other hand, the disease affects the stability of the purely demographic equilibria.

Highlights

Mathematical ecogenetic models have recently been introduced (Venturino, 2012; Viberti & Venturino, 2014)
The model trajectories can settle toward the disease- and predator-free equilibria, toward the predatorfree one, or the system can achieve coexistence of all the populations
To answer one of the questions raised in the Introduction, the disease can be eradicated, and the predators have a fundamental role in it, as the parameters related to their dynamics appear in the necessary stability condition (Equation 12)

Summary

Introduction

Mathematical ecogenetic models have recently been introduced (Venturino, 2012; Viberti & Venturino, 2014). The basic age description corresponds to a linear, or more generally nonlinear (Gurtin & Levine, 1979), wave equation, with an initial and a boundary condition at age zero, for which standard numerical schemes are available (Smith, 1985) These considerations could be extended to interacting populations (Venturino, 1984, 1987) A structure can be superimposed on diseases: the classical Kermack– McKendrick model (Kermack & McKendrick, 1927) considers the population partitioned among susceptibles, infected, and recovered individuals. Motivated by the fact that different genotypes may experience different responses to external interferences in animals, as mentioned above, and to predators, and to pathogens, we consider here a hypothetical ecogenetic model, similar to some other systems already studied, that in comparison with the current literature (Venturino, 2012; Viberti & Venturino, 2014), makes a step forward, namely it introduces a disease among the genetically distinguishable population, that affects only one genotype. E < 1 denotes the conversion factor of captured prey into new predators, is the disease contact rate and represents its recovery rate

Equilibria

The disease-free equilibrium E2 At E2 one explicit eigenvalue is available

Conclusions