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Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response

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  • Maiti, Atasi Patra
  • Dubey, B.
  • Chakraborty, A.

Abstract

A stage structure prey–predator model that consists of a system of three nonlinear ordinary differential equations in the presence of discrete time delay is proposed and analysed in this paper. The prey population is divided into two categories: immature and mature prey. The predator population depends on mature prey only and that followed by Crowley–Martin type functional response. We analyse positivity, boundedness and existence of equilibrium points. The local and global stability behaviour of the delayed and non-delayed system are also analysed. Considering delay as a bifurcation parameter, the Hopf-bifurcation is also examined for this system. Then we discuss the stability and direction of Hopf-bifurcation using Normal form theory and Centre manifold theory. Numerical simulation is carried out to verify our analytical findings. We observe that, for a set of values of parameters, the bifurcated periodic solution is supercritical, stable with decreasing period and as the time delay increases, interior equilibrium point disappears. Model of this type may be considered to save the immature prey from the predator population and to maintain the prey–predator relation.

Suggested Citation

  • Maiti, Atasi Patra & Dubey, B. & Chakraborty, A., 2019. "Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 162(C), pages 58-84.
    • Handle: RePEc:eee:matcom:v:162:y:2019:i:c:p:58-84
    DOI: 10.1016/j.matcom.2019.01.009
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    References listed on IDEAS

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    1. Ma, Xiangmin & Shao, Yuanfu & Wang, Zhen & Luo, Mengzhuo & Fang, Xianjia & Ju, Zhixiang, 2016. "An impulsive two-stage predator–prey model with stage-structure and square root functional responses," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 91-107.
    2. Khajanchi, Subhas, 2017. "Modeling the dynamics of stage-structure predator-prey system with Monod–Haldane type response function," Applied Mathematics and Computation, Elsevier, vol. 302(C), pages 122-143.
    3. Yongzhen, Pei & Changguo, Li & Lansun, Chen, 2009. "Continuous and impulsive harvesting strategies in a stage-structured predator–prey model with time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(10), pages 2994-3008.
    4. Zhao, Zhong & Wu, Xianbin, 2014. "Nonlinear analysis of a delayed stage-structured predator–prey model with impulsive effect and environment pollution," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1262-1268.
    5. Guangzhao Zeng & Lansun Chen & Lihua Sun & Ying Liu, 2004. "Permanence And The Existence Of The Periodic Solution Of The Non-Autonomous Two-Species Competitive Model With Stage Structure," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 7(03n04), pages 385-393.
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    1. Ghanbari, Behzad & Günerhan, Hatıra & Srivastava, H.M., 2020. "An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Zhang, Shengqiang & Yuan, Sanling & Zhang, Tonghua, 2022. "A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments," Applied Mathematics and Computation, Elsevier, vol. 413(C).

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