Dynamical behaviour of fractional-order predator-prey system of Holling-type

Kolade M. Owolabi

doi:10.3934/dcdss.2020047

Article Contents

2020, Volume 13, Issue 3: 823-834. Doi: 10.3934/dcdss.2020047

This issue Previous Article Mathematical modeling approach to the fractional Bergman's model Next Article Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations

Dynamical behaviour of fractional-order predator-prey system of Holling-type

Kolade M. Owolabi^,

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

^* Corresponding author: koladematthewowolabi@tdtu.edu.vn (K. M. Owolabi)
^* Corresponding author: koladematthewowolabi@tdtu.edu.vn (K. M. Owolabi)

Received: April 2018

Revised: May 2018

Published: December 2019

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Abstract

In this paper, the local derivative in time is replaced with the Caputo-Fabrizio fractional derivative of order $ \alpha\in(0, 1) $. A two-step fractional version of the Adams-Bashforth method is formulated for the approximation of this derivative. To enhance the correct choice of parameters when numerically simulating the full-system, we examine the stability analysis of the main equation. Two important examples are drawn to explore the dynamic richness of the predator-prey model with Holling type. Simulation results at different instances of $ \alpha $ is in agreement with the theoretical findings.

Keywords:

Mathematics Subject Classification: Primary: 65L05, 65L06, 93C10; Secondary: 34A34, 49M25, 65M70.

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Figure 1. Dynamic behaviour of fractional system (16) with $ \alpha = 0.50 $. Other parameters are as fixed in (17)

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Figure 2. Dynamic behaviour of fractional system (16) with α = 0:79. Other parameters are as fixed in (17).

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Figure 3. Dynamic behaviour of fractional system (16) with $ \alpha = 0.91 $. Other parameters are as fixed in (17)

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Figure 4. A strange attractor for dynamic system (16) with $ \alpha = 0.48 $

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Figure 5. One-dimensional distribution of time-fractional reaction-diffusion system (19) for $ \alpha = 0.11 $

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Figure 6. One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:25

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Figure 7. One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:45

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Figure 8. One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:79

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Figure 9. One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:91

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