Abstract
The fear response is an important anti-predator adaptation that can significantly reduce prey's reproduction by inducing many physiological and psychological changes in the prey. Recent studies in behavioral sciences reveal this fact. Other than terrestrial vertebrates, aquatic vertebrates also exhibit fear responses. Many mathematical studies have been done on the mass mortality of pelican birds in the Salton Sea in Southern California and New Mexico in recent years. Still, no one has investigated the scenario incorporating the fear effect. This work investigates how the mass mortality of pelican birds (predator) gets influenced by the fear response in tilapia fish (prey). For novelty, we investigate a modified fractional-order eco-epidemiological model by incorporating fear response in the prey population in the Caputo-fractional derivative sense. The fundamental mathematical requisites like existence, uniqueness, non-negativity and boundedness of the system's solutions are analyzed. Local and global asymptotic stability of the system at all the possible steady states are investigated. Routh-Hurwitz criterion is used to analyze the local stability of the endemic equilibrium. Fractional Lyapunov functions are constructed to determine the global asymptotic stability of the disease-free and endemic equilibrium. Finally, numerical simulations are conducted with the help of some biologically plausible parameter values to compare the theoretical findings. The order $\alpha$ of the fractional derivative is determined using Matignon's theorem, above which the system loses its stability via a Hopf bifurcation. It is observed that an increase in the fear coefficient above a threshold value destabilizes the system. The mortality rate of the infected prey population has a stabilization effect on the system dynamics that helps in the coexistence of all the populations. Moreover, it can be concluded that the fractional-order may help to control the coexistence of all the populations.
Highlights
The conventional notion that predator affects the prey population only through direct killing has been changed to a great extent in recent past [1]
To make the discussions more realistic and novel, we have considered the fear effect in the prey due to predation, since Tilapia (Oreochromis niloticus) under stressful circumstances react by boosting or completely hindering reproduction [7]
We extend the mathematical model proposed by Greenhalgh in [37] by incorporating the fear effect in the susceptible prey in terms of Caputo fractional derivative
Summary
The conventional notion that predator affects the prey population only through direct killing has been changed to a great extent in recent past [1]. Later in 2017, Greenhalgh et al [37] modified their earlier studies in [36] by taking into account that predators feed on both the susceptible and infected preys(tilapia) They presumed that the diseased prey significantly influences the growth rate of susceptible prey and the carrying capacity of the predator is dependent on the total number of prey (tilapia). We extend the mathematical model proposed by Greenhalgh in [37] by incorporating the fear effect in the susceptible prey (tilapia) in terms of Caputo fractional derivative. The memory effect in FDEs that provides data between two distinct integer values motivates us to study the model using a fractional derivative. The paper is organized as follows: In section 2, we describe a modified predator-prey interaction model with fear effect involving Caputo fraction derivatives.
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