Pattern formation in a ratio-dependent predator-prey model with cross diffusion

Article Mathematics, Applied

Coexistence States of a Ratio-Dependent Predator-Prey Model with Nonlinear Diffusion

Nitu Kumari et al.

Summary: This study investigates a two species ratio-dependent food chain model with nonlinear diffusion terms using bifurcation theory and a priori estimates to prove the existence of positive solution set for the model system. The analysis concludes that a ratio-dependent predator-prey model with nonlinear or cross diffusion can coexist in a habitat surrounded by an inhospitable environment represented by the Dirichlet boundary conditions. Additionally, the successful employment of bifurcation theory in commenting on the existence of positive solutions of a model system where interaction is in accordance with a predator dependent functional response is demonstrated.

ACTA APPLICANDAE MATHEMATICAE (2021)

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Article Engineering, Multidisciplinary

Bifurcation and turing instability for a predator-prey model with nonlinear reaction cross-diffusion

Jianzhi Cao et al.

Summary: In this study, a nonlinear reaction cross-diffusion predator-prey system under Neumann boundary condition is considered, with negative diffusion coefficients introducing local accumulation effect of prey. The paper discusses the criteria for local asymptotic stability of the positive homogeneous steady state with or without cross-diffusion, as well as the conditions for diffusion-driven instability. The existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are studied using Lyapunov-Schmidt reduction, and numerical simulations are carried out to clarify the theoretical results, revealing that Turing instability in the model is induced by the negative diffusion coefficients.

APPLIED MATHEMATICAL MODELLING (2021)

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Article Mathematics, Applied