Spatio-temporal Bazykin’s model with space-time nonlocality

Swadesh Pal; Malay Banerjee; Vitaly Volpert; Swadesh Pal; Malay Banerjee; Vitaly Volpert

doi:10.3934/mbe.2020262

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4801-4824. doi: 10.3934/mbe.2020262

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Spatio-temporal Bazykin’s model with space-time nonlocality

1.
Department of Mathematics & Statistics, IIT Kanpur, Kanpur, 208016, India
2.
Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
3.
INRIA, Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
4.
Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia

Received: 06 May 2020 Accepted: 08 July 2020 Published: 10 July 2020

This work deals with a reaction-diffusion model for prey-predator interaction with Bazykin's reaction kinetics and a nonlocal interaction term in prey growth. The kernel of the integral characterizes nonlocal consumption of resources and depends on space and time. Linear stability analysis determines the conditions of the emergence of Turing patterns without and with nonlocal term, while weakly nonlinear analysis allows the derivation of amplitude equations. The bifurcation analysis and numerical simulation carried out in this work reveal the existence of stationary and dynamic patterns appearing due to the loss of stability of the coexistence homogeneous steady-state.
- Bazykin’s model,
- nonlocal interaction,
- Hopf bifurcation,
- Turing instability,
- spatial pattern
Citation: Swadesh Pal, Malay Banerjee, Vitaly Volpert. Spatio-temporal Bazykin’s model with space-time nonlocality[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4801-4824. doi: 10.3934/mbe.2020262

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Abstract

This work deals with a reaction-diffusion model for prey-predator interaction with Bazykin's reaction kinetics and a nonlocal interaction term in prey growth. The kernel of the integral characterizes nonlocal consumption of resources and depends on space and time. Linear stability analysis determines the conditions of the emergence of Turing patterns without and with nonlocal term, while weakly nonlinear analysis allows the derivation of amplitude equations. The bifurcation analysis and numerical simulation carried out in this work reveal the existence of stationary and dynamic patterns appearing due to the loss of stability of the coexistence homogeneous steady-state.

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