Influence of Allee effect on the spatiotemporal behavior of a diffusive predator-prey model with Crowley-Martin type response function.

The present paper is dealt with a predator-prey model in which the growth of the prey population is influenced by the Allee effect while the predator species are contended with the prey population following the Crowley-Martin type response function. The proposed model is comprehensively analyzed in terms of stability and manifestation of bifurcation of the system. The system unveils the bi-stability together with the existence of a separatrix. In view of the eminence of spatial ecology, the dynamical complexity emanating from the induction of the Allee effect in prey species of a Crowley-Martin reaction-diffusion predator-prey model is also investigated profoundly. The results of numerical simulations reveal that the present system dynamics is motivated by both the Allee effect and diffusion-controlled pattern formation growth to hot spots, stripe-hot spot mixtures, stripes, labyrinthine, stripe-cold spot mixtures, and cold spots replication. The theoretical consequences of the spatiotemporal model under study are validated through numerical simulations.

Dynamics of a three species ratio-dependent food chain model with intra-specific competition within the top predator.

In this paper, a ratio-dependent food chain model has been considered. The total population has been divided into three classes, namely prey, predator and top-predator population. We have also incorporated intra-specific competition of predators in the model. We have studied the boundedness, dissipativeness and permanence of the solutions of the system and analyzed the existence of various equilibrium points and stability of the system at those equilibrium points. The system exhibits Bogdanov-Takens bifurcation, saddle-node bifurcation, Hopf bifurcation for suitable choice of the relevant parameters. The results of extensive numerical simulation are provided to support the validity of the theoretical findings. The ecological implications of our analytical and numerical findings are discussed.

Study of a tri-trophic prey-dependent food chain model of interacting populations.

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf-Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.