Stability and Hopf bifurcation in an eco-epidemiological system with the cost of anti-predator behaviors

Abstract

<abstract><p>The fear effect is a powerful force in prey-predator interaction, eliciting a variety of anti-predator responses which lead to a reduction of prey growth rate. To study the impact of the fear effect on population dynamics of the eco-epidemiological system, we develop a predator-prey interaction model that incorporates infectious disease in predator population as well as the cost of anti-predator behaviors. Detailed mathematical results, including well-posedness of solutions, stability of equilibria and the occurrence of Hopf bifurcation are provided. It turns out that population density diminishes with increasing fear, and the fear effect can either destabilize the stability or induce the occurrence of periodic behavior. The theoretical results here provide a sound foundation for understanding the effect of the anti-predator behaviors on the eco-epidemiological interaction.</p></abstract>

Bifurcation analysis of delayed predator–prey system with fear effect, additional food for predator, prey refuge and predator age structure

“…In the complex and diverse natural world, besides the factors currently considered in this paper, as part of future research, we suggest including non-linear mortality caused by anti-predator activities and stochastic environmental factors into the model to enhance its realism. Additionally, we believe there will be interesting findings when applying the reaction-diffusion model [38][39][40] and infectious disease model [41][42][43]. However, these aspects will be left for further exploration in our future work.…”

The effect of the fear factor on the dynamics of an eco-epidemiological system with standard incidence rate

“…( Wang et al., 2016 ; Wang & Zou, 2017 ). The term accounts for the cost of anti-predator behaviors may take different forms, such as ( Wang et al., 2016 ; Zhang et al., 2023 ), e − kY ( Das & Samanta, 2018 ; Li & Tian, 2023 ), ( Sarkara & Khajanchi, 2020 ). Generally, the fear effect can stabilize or destabilize the considered predator-prey systems, and may cause a significant reduction in the biomass of the population, which are demonstrated in ( Hossain et al., 2020 ; Pal, 2020 ; Sarkara & Khajanchi, 2020 ; Zhang et al., 2019 ).…”

“…In my earlier work, we established the following predator-prey system that incorporates infectious disease in predator population and the cost of anti-predator behaviors ( Zhang et al., 2023 ): where, X , S , I represent the densities of prey, susceptible predator and infected predator at time t , respectively. Here, r is the intrinsic growth rate of the logistically growing prey and K is the carrying capacity of the prey population.…”

“…We summarize the equilibria for system (1.1) as follows: Trivial equilibrium: E 0 = (0, 0, 0); Axial equilibrium: E 1 = ( K , 0, 0); Planar equilibrium: E 2 = ( X 2 , S 2 , 0) exists if ea − bd 1 > 0 and hold, where X 2 and S 2 are defined in Eq (3.1) of ( Zhang et al., 2023 ). If ea − bd 1 > 0 and hold, then system (1.1) has at least one positive equilibrium E 3 = ( X 3 , S 3 , I 3 ), where X 3 , S 3 and I 3 are provided in [23, Theorem 3.1 ].…”

“…Planar equilibrium: E 2 = ( X 2 , S 2 , 0) exists if ea − bd 1 > 0 and hold, where X 2 and S 2 are defined in Eq (3.1) of ( Zhang et al., 2023 ).…”