Exploring the dynamics of a tritrophic food chain model with multiple gestation periods

Ranjit Kumar Upadhyay; Swati Mishra; Yueping Dong; Yasuhiro Takeuchi; Ranjit Kumar Upadhyay; Swati Mishra; Yueping Dong; Yasuhiro Takeuchi

doi:10.3934/mbe.2019234

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4660-4691. doi: 10.3934/mbe.2019234

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Exploring the dynamics of a tritrophic food chain model with multiple gestation periods

1.
Department of Applied Mathematics, Indian Institute of Technology (ISM) Dhanbad, Dhanbad 826004, India
2.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
3.
College of Science and Engineering, Aoyama Gakuin University, Sagamihara 252-5258, Japan

Received: 30 January 2019 Accepted: 29 April 2019 Published: 23 May 2019

This work is mainly focused on the series of dynamical analysis of tritrophic food chain model with Sokol-Howell functional response, incorporating the multiple gestation time delays for more realistic formulation. Basic properties of the proposed model are studied with the help of boundedness, stability analysis, and Hopf-bifurcation theory. By choosing the fixed parameter set and varying the value of time delay, the stability of the model has been studied. There is a critical value for delay parameter. Steady state is stable when the value of delay is less than the critical value and further increase the value of delay beyond the critical value makes the system oscillatory through Hopf-bifurcation. Whereas, another delay parameter has a stabilizing effect on the system dynamics. Chaotic dynamics has been explored in the model with the help of phase portrait and sensitivity on initial condition test. Numerical simulations are performed to validate the effectiveness of the derived theoretical results and to explore the various dynamical structures such as Hopf-bifurcation, periodic solutions and chaotic dynamics.
- food chain,
- Sokol-Howell functional response,
- gestation delay,
- Hopf-bifurcation,
- chaos
Citation: Ranjit Kumar Upadhyay, Swati Mishra, Yueping Dong, Yasuhiro Takeuchi. Exploring the dynamics of a tritrophic food chain model with multiple gestation periods[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4660-4691. doi: 10.3934/mbe.2019234

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Abstract

This work is mainly focused on the series of dynamical analysis of tritrophic food chain model with Sokol-Howell functional response, incorporating the multiple gestation time delays for more realistic formulation. Basic properties of the proposed model are studied with the help of boundedness, stability analysis, and Hopf-bifurcation theory. By choosing the fixed parameter set and varying the value of time delay, the stability of the model has been studied. There is a critical value for delay parameter. Steady state is stable when the value of delay is less than the critical value and further increase the value of delay beyond the critical value makes the system oscillatory through Hopf-bifurcation. Whereas, another delay parameter has a stabilizing effect on the system dynamics. Chaotic dynamics has been explored in the model with the help of phase portrait and sensitivity on initial condition test. Numerical simulations are performed to validate the effectiveness of the derived theoretical results and to explore the various dynamical structures such as Hopf-bifurcation, periodic solutions and chaotic dynamics.

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