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Mathematical modelling and analysis of two-component system with Caputo fractional derivative order

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  • Owolabi, Kolade M.

Abstract

A class of generic spatially extended fractional reaction-diffusion systems that modelled predator-prey interactions is considered. The first order time derivative is replaced with the Caputo fractional derivative of order γ ∈ (0, 1). The local analysis where the equilibrium points and their stability behaviours are determined is based on the adoption of qualitative theory for dynamical systems ordinary differential equations. We derived conditions for Hopf bifurcation analytically. Most significantly, existence conditions for a unique stable limit cycle in the phase plane are determined analytically. Our analytical findings are in agreement with the numerical results presented in one and two dimensions. The system of fractional nonlinear reaction-diffusion equations has demonstrated the usefulness of understanding the dynamics of nonlinear phenomena.

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  • Owolabi, Kolade M., 2017. "Mathematical modelling and analysis of two-component system with Caputo fractional derivative order," Chaos, Solitons & Fractals, Elsevier, vol. 103(C), pages 544-554.
    • Handle: RePEc:eee:chsofr:v:103:y:2017:i:c:p:544-554
    DOI: 10.1016/j.chaos.2017.07.013
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    References listed on IDEAS

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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Reyes-Reyes, J. & Adam-Medina, M., 2016. "Modeling diffusive transport with a fractional derivative without singular kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 467-481.
    3. Ghosh, Joydev & Sahoo, Banshidhar & Poria, Swarup, 2017. "Prey-predator dynamics with prey refuge providing additional food to predator," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 110-119.
    4. Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & López-López, M.G. & Alvarado-Martínez, V.M. & Guerrero-Ramírez, G.V., 2016. "Triple pendulum model involving fractional derivatives with different kernels," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 248-261.
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