Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect

Vinoth Seralan; R. Vadivel; Dimplekumar Chalishajar; Nallappan Gunasekaran; Vinoth Seralan; R. Vadivel; Dimplekumar Chalishajar; Nallappan Gunasekaran

doi:10.3934/math.20231165

AIMS Mathematics

2023, Volume 8, Issue 10: 22896-22923. doi: 10.3934/math.20231165

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Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect

1.
Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
2.
Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University, Phuket 83000, Thailand
3.
Department of Applied Mathematics, Virginia Military Institute (VMI), 435 Mallory Hall, Lexington, VA 24450, USA
4.
Eastern Michigan Joint College of Engineering, Beibu Gulf University, Qinzhou 535011, China

Received: 24 May 2023 Revised: 16 June 2023 Accepted: 20 June 2023 Published: 19 July 2023
MSC : 92B05, 34C23, 34C25, 34D20, 39A28

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.
- Allee effect,
- bifurcation,
- discrete system,
- largest Lyapunov exponent,
- local stability,
- predator-prey model
Citation: Vinoth Seralan, R. Vadivel, Dimplekumar Chalishajar, Nallappan Gunasekaran. Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect[J]. AIMS Mathematics, 2023, 8(10): 22896-22923. doi: 10.3934/math.20231165

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Abstract

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.

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