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Dynamics comparison between non-spatial and spatial systems of the plankton–fish interaction model

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Abstract

This paper considers a plankton–fish interaction model by comparing the dynamics between non-spatial and spatial systems. In the model, both the fish and phytoplankton populations are assumed to be growing logistically, the fish population is nonlinearly harvested, and the interaction between phytoplankton and zooplankton is described by the Crowley–Martin functional response. Stability analyses for the plankton–fish interaction model have been carried out for both non-spatial and spatial systems. The theoretical results are then supported by numerical simulations. In the case of non-spatial system, single-parameter bifurcation diagrams are used, while in the spatial system numerical simulations for one- and two-dimensional cases are performed. In the case of a non-spatial system, the conditions for the existence of a positive equilibrium point, as well as their local and global stability analyses, have been obtained. In the spatial system, the stability and existence of Hopf bifurcation have been discussed, and different conditions for Turing pattern information have been established through diffusion-driven instability analysis. Taken together, these results show that spatial heterogeneity, the mortality rate of phytoplankton, and the constant harvesting of the fish population play important roles in the dynamical behavior of the marine system.

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Acknowledgements

The authors express their gratitude to the associate editor and the reviewers whose comments and suggestions have helped to improve the manuscript. The authors thank Dr. Vikas Rai for helpful discussion during revision. This work has been supported by the Science & Engineering Research Board (SERB), DST, Govt. of India, under grant No. MTR/2017/000301 to the corresponding author.

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  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Dhanbad, Jharkhand, 826004, India

    Sarita Kumari & Ranjit Kumar Upadhyay

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  1. Sarita Kumari
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  2. Ranjit Kumar Upadhyay
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Correspondence to Ranjit Kumar Upadhyay.

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Kumari, S., Upadhyay, R.K. Dynamics comparison between non-spatial and spatial systems of the plankton–fish interaction model. Nonlinear Dyn 99, 2479–2503 (2020). https://doi.org/10.1007/s11071-019-05415-6

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  • DOI: https://doi.org/10.1007/s11071-019-05415-6

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