# American Institute of Mathematical Sciences

March  2020, 13(3): 823-834. doi: 10.3934/dcdss.2020047

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

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

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

Received  April 2018 Revised  May 2018 Published  March 2019

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.

Citation: Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 823-834. doi: 10.3934/dcdss.2020047
##### References:

show all references

##### References:
Dynamic behaviour of fractional system (16) with $\alpha = 0.50$. Other parameters are as fixed in (17)
Dynamic behaviour of fractional system (16) with α = 0:79. Other parameters are as fixed in (17).
Dynamic behaviour of fractional system (16) with $\alpha = 0.91$. Other parameters are as fixed in (17)
A strange attractor for dynamic system (16) with $\alpha = 0.48$
One-dimensional distribution of time-fractional reaction-diffusion system (19) for $\alpha = 0.11$
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:25
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:45
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:79
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:91
 [1] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Zakia Hammouch, Dumitru Baleanu. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 975-993. doi: 10.3934/dcdss.2020057 [2] Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203 [3] Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317 [4] Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430 [5] M. M. El-Dessoky, Muhammad Altaf Khan. Application of Caputo-Fabrizio derivative to a cancer model with unknown parameters. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3557-3575. doi: 10.3934/dcdss.2020429 [6] Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247 [7] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259 [8] Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure and Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005 [9] Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure and Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481 [10] Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621 [11] Willian Cintra, Carlos Alberto dos Santos, Jiazheng Zhou. Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3913-3931. doi: 10.3934/dcdsb.2021211 [12] Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 2949-2973. doi: 10.3934/dcdss.2020132 [13] Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92 [14] Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 [15] Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148 [16] Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265 [17] Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069 [18] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [19] Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035 [20] Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2239-2255. doi: 10.3934/dcdsb.2021007

2020 Impact Factor: 2.425