ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Pattern Formation Scenario through Turing Instability in Interacting Reaction-Diffusion Systems with Both Refuge and Nonlinear Harvesting

Journal of Applied Nonlinear Dynamics 9(1) (2020) 1--21 | DOI:10.5890/JAND.2020.03.001

Lakshmi Narayan Guin$^{1}$, Esita Das$^{1}$, Muniyagounder Sambath$^{2}$

$^{1}$ Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India

$^{2}$ Department of Mathematics, Periyar University, Salem-636011, Tamil Nadu, India

Abstract

Of concern in the present theoretical study is to carry out the complex dynamics of a reaction-diffusion predator-prey model incorporating constant proportion of prey refuge and nonlinear prey harvesting with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the feasible interior equilibrium point for nonspatial model was established. The conditions of diffusion-driven instability were obtained and the Turing space in the parameters space was given as well. Consequently, we present the evolutionary procedure that occupies organism distribution and their interaction of spatially distributed species with diffusion and locate that the model dynamics reveals a diffusion-controlled formation growth to hole patterns or labyrinthine patterns or hole-stripe patterns replication over the whole spatial domain. The analytical results are then authenticated with the help of numerical simulations. Our results points out that the diffusion has an immense impact on the prey refuge as well as prey harvesting and extend well the findings of spatiotemporal dynamics in the reaction-diffusion model.

Acknowledgments

The first two authors gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)). We would like to express thank the anonymous referee and the editor for supportive remarks and ideas.

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