Invasive dynamics for a predator-prey system with Allee effect in both populations and a special emphasis on predator mortality

Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey-predator model

Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

Impact of Allee and fear effects in a fractional order prey-predator system with group defense and prey refuge

This paper presents a novel fractional-order model of a prey-predator system that incorporates group defense and prey refuge mechanisms, along with Allee and fear effects. First, we examine the existence, uniqueness, non-negativity, and boundedness of the solution of the system. Second, a comprehensive analysis is conducted on the existence, stability, and coexistence of equilibrium states in the system, which are crucial for comprehending prey-predator system behavior. Our investigation reveals that the coexistence equilibrium undergoes a Hopf bifurcation under five key parameters. Specifically, an increased threshold for the transition between group and individual behavior, influenced by different strengths of the Allee effect, enhances the stability of both populations. This discovery sheds light on the role of group effects in shaping prey-predator interactions and ecosystem stability. Third, system discretization is employed to explore the impact of step size on stimulating stability and to investigate the Neimark-Sacker bifurcation, providing a more comprehensive understanding of system behavior. The role of step size as a constraint on stability is examined, revealing the system's progression from stability to chaos. Consequently, our results offer a more flexible mechanism for adjusting the stability and dynamics of the two species. Finally, numerical simulations are utilized to validate the reasonableness of the research findings.

The (De)Stabilizing effect of juvenile prey cannibalism in a stage-structured model

Cannibalism, or intraspecific predation, is the act of an organism consuming another organism of the same species. In predator-prey relationships, there is experimental evidence to support the existence of cannibalism among juvenile prey. In this work, we propose a stage-structured predator-prey system where cannibalism occurs only in the juvenile prey population. We show that cannibalism has both a stabilizing and destabilizing effect depending on the choice of parameters. We perform stability analysis of the system and also show that the system experiences a supercritical Hopf, saddle-node, Bogdanov-Takens and cusp bifurcation. We perform numerical experiments to further support our theoretical findings. We discuss the ecological implications of our results.