Pattern Formation Study of an Eco-epidemiological Model with Cannibalism and Disease in Predator Population

TL;DR

This study investigates pattern formation in an eco-epidemiological model with cannibalism and disease, analyzing stability, existence of patterns, and effects of movement and parameters through mathematical and numerical methods.

Contribution

It introduces a diffusive eco-epidemiological model with cannibalism and disease, providing stability analysis, existence conditions for patterns, and numerical simulations of pattern formation.

Findings

Stable and unstable patterns identified via Lyapunov spectrum.

Cannibalism and disease influence pattern types and dynamics.

Movement of prey and predators significantly affects pattern formation.

Abstract

Pattern formation analysis of eco-epidemiological models with cannibalism and disease has been less explored in the literature. Therefore, motivated by this, we have proposed a diffusive eco-epidemiological model and performed pattern formation analysis in the model system. Sufficient conditions for local asymptotic stability and global asymptotic stability for the constant positive steady state are obtained by linearization and Lyapunov function technique. A priori estimate for the positive steady state is obtained for the nonexistence of the nonconstant positive solution using Cauchy and Poincar\'e inequality. The existence of the nonconstant positive steady states is studied using Leray-Schauder degree theory. The importance of the diffusive coefficients which are responsible for the appearance of stationary patterns is observed. Pattern formation is done using numerical simulation. Further, the effect of the cannibalism and disease are observed on the dynamics of the proposed model system. The movements of prey and susceptible predator plays a significant role in pattern formation. These movements cause stationary and non-stationary patterns. It is observed that an increment in the movement of the susceptible predator as well as cannibalistic attack rate converts non-Turing patterns to Turing patterns. Lyapunov spectrum is calculated for quantification of stable and unstable dynamics. Non-Turing patterns obtained with parameter set having unstable limit cycle are more interesting and realistic than stationary patterns. Stationary and non-stationary non-Turing patterns are obtained.