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

Abstract

SUMMARYA predator-prey model with transmissible disease in the prey species is proposed and analysed. The essential mathematical features are analysed with the help of equilibrium, local and global stability analyses and bifurcation theory. We find four possible equilibria. One is where the populations are extinct. Another is where the disease and predator populations are extinct and we find conditions for global stability of this. A third is where both types of prey exist but no predators. The fourth has all three types of individuals present and we find conditions for limit cycles to arise by Hopf bifurcation. Experimental data simulation and brief discussion conclude the paper.

Ecoepidemiology: a More Comprehensive View of Population Interactions

“…In [57] a Leslie-Gower model with the predators selectively feeding only on infected prey is presented:…”

Stability and Hopf Bifurcation of a Delay Eco‐epidemiological Model With Nonlinear Incidence Rate

“…Mathematical biology, namely predator-prey systems and models for transmissible diseases are major fields of study in their own right. But little attention has been paid so far to merge these two important areas of research (Haque M. and Venturino E., [14]; Haque M. and Venturino E., [15]; Greenhalgh D. and Haque M., [11]; Liu M.X., Jin Z. and Haque M., [26]; Venturino E. [33]; Xiao and Chen, [35,36]; Hethcote, Wang et al, [18]). In order to study the influence of disease on an environment, two or more interacting species should be presented.…”

“…(A 1 ) In the absence of infection and predation, the prey population density grows logistically with carrying capacity K > 0 and an intrinsic birth rate constant (r > 0) (Greenhalgh D. and Haque M., [11]; Haque M. and Venturino E., [14])…”

“…In the presence of disease, the total prey population N (t) is divided into two distinct classes, namely, susceptible population S(t) and infected population I(t) (Greenhalgh D. and Haque M., [11]; Haque M. and Venturino E., [14]). Therefore, at any time t, the total density of prey population is N (t) = S(t) + I(t).…”

“…However, the infective population I(t) still contributes with S(t) to population growth toward the carrying capacity (Greenhalgh D. and Haque M., [11]; Haque M. and Venturino E., [14]). …”