The biological control paradox

Abstract

We propose an eco-epidemiological model with disease in the prey population and the disease is not inherited. The disease spreads over the susceptible prey through the Holling type-II incidence rate and the prey becomes infected so that the number of infected prey is expected to be saturated. Here, the infected prey is assumed to recover by their natural immune. The predator consumes both susceptible and infected preys according to the ratio-dependent Michaelis–Menten type functional response. We incorporate two-time delays: incidence delay for the conversion of susceptible prey into infected one and gestation delay for the predator growth. Mathematical requirements (positivity, boundedness, permanence of the solution, existences of equilibrium points (graphically based on nullclines concept) and local as well as global stability around equilibrium points) for our proposed model to behave well are taken care here. Numerical simulation is carried out extensively to validate the analytical restrictions, imposed additional conditions and find the approximate solution of our highly nonlinear differential equations. System dynamicity is performed with special attention to newly introduced parameters. Population variation due to some of the parameters is presented and interpreted eco-epidemiologically. When the population starts oscillating, their time-averaged values are presented to show their variation. In order to make the present model more realistic, consumption of susceptible by predator and conversion from susceptible to predator rates are varied simultaneously by considering a linear relationship among these two rates. We apply the necessary tools of nonlinear dynamics such as bifurcation diagram, phase diagram, spectrum and maximum Lyapunov exponent to affirm system’s complex dynamical behavior. Stable/unstable zones are proposed in xτ1c-plane where x is any considered parameter, and τ1c is the corresponding critical value of the incidence delay. Bifurcation diagrams of population with respect to both the delays are presented, and their relative contribution (incident or/and gestation delay) to the system is explored.