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.
The center manifold is an invariant manifold that plays a crucial role in the bifurcation analysis of dynamical systems. The center manifold existence theorem assures the local existence of an invariant submanifold of the state space of a dynamical system around a non-hyperbolic equilibrium point. Center manifold theory is essential in the reduction of different bifurcation scenarios to their normal forms. Our study focuses on a predator-prey interactive system with density-dependent growth in predators subject to a contagious disease. The disease is assumed to be horizontally transmitted, and the rate of recovery of the infected predator is assumed to be density-dependent. At the trivial (zero) equilibrium, the center manifold is calculated whose dynamical behaviour is similar to that of the original system. Further, using the center manifolds, the normal form of a Hopf bifurcation point is determined fromwhich the criticality of the system can be deduced. Finally, numerical simulations are performed with biologically plausible parameters to substantiate the analytical findings. Using numerical continuation methods we detect Generalized Hopf and Zero-Hopf bifurcation points. We discuss their normal form coefficients, compute their two-parameter unfoldings and relate these results to the mathematical theory of codimension two bifurcations.
The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity. Hunting cooperation in predators is one of the most remarkable features of the predation process, which benefits the predators by developing fear upon their prey. This study investigates the dynamical behavior of a modified LV-type predator–prey system with Michaelis–Menten-type harvesting of predators where predators adopt cooperation strategy during hunting. The ecologically feasible steady states of the system and their asymptotic stabilities are explored. The local codimension one bifurcations, viz. transcritical, saddle-node and Hopf bifurcations, that emerge in the system are investigated. Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation. A backward Hopf-bifurcation is detected when the harvesting effort is increased, which destabilizes the system by generating periodic solutions. The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient. Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations. Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased. We have also explored codimension two bifurcations viz. the generalized Hopf and the Bogdanov–Takens bifurcation exhibit by the system. To visualize the dynamical behavior of the system, numerical simulations are conducted using an ecologically plausible parameter set. The existence of the bionomic equilibrium of the model is analyzed. Moreover, an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.