A delayed eco-epidemiological model with nonlinear incidence rate and Crowley–Martin functional response for infected prey and predator

Abstract

A prey–predator model with disease in prey population is considered here. The prey population is classified into two categories: susceptible prey and infected prey. The nonlinear incidence rate, at which the susceptible prey becomes infected, is incorporated. Here, predator consumes infected prey only according to Crowely–Martin functional response. Time lag is introduced to convert the susceptible prey to infected one, and the present model becomes delayed eco-epidemic model. The positivity, boundedness and permanence of the solutions of both models are guaranteed here. The existence of equilibrium points is assured. Local as well as global stability analyses of both the models around the equilibrium points are performed and finally summarized in a table, comparing the conditions between two models. The occurrence of bifurcation (saddle-node, transcritical, Hopf and Bogdanov–Takens) in the non-delay model is explored here. The nature and direction of Hopf bifurcation are searched using normal form and centre manifold theorems for delay model. Lastly, the numerical simulation is carried out to verify the analytical findings. An effort is made to search the parameters, by paying special attention to newly introduced parameters, which control the dynamics of both models. Accordingly, stable/unstable zones on $$x\tau _\mathrm{c}$$ -planes, where x is any considered parameter and $$\tau _\mathrm{c}$$ is the value of critical delay time, are proposed. The stranger attractor is searched in the delay model. Bifurcated diagrams of population with respect to newly introduced parameters are exemplified and discussed. When the population (S, I and P) starts oscillating, their time-averaged values are presented to show their variation due to aforesaid parameters. Moreover, pattern of all the curves $$S=S(x)$$ , $$I=I(x)$$ and $$P=P(x)$$ , where x is any aforesaid parameters, are certified using their respective gradient.