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Minimal Model of Plankton Systems Revisited with Spatial Diffusion and Maturation Delay

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Abstract

This study revisits the minimal model for a plankton ecosystem proposed by Scheffer with spatial diffusion of plankton and the delay of the maturation period of herbivorous zooplankton. It deepens our understanding of effects of the nutrients and the predation of fish upon zooplankton on the dynamical patterns of the plankton system and also presents new phenomena induced by the delay with spatial diffusion. When the nutrient level is sufficient low, the zooplankton population collapses and the phytoplankton population reaches its carrying capacity. Mathematically, the global stability of the boundary equilibrium is proved. As the nutrient level increases, the system switches to coexistent equilibria or oscillations depending on the maturation period of zooplankton and the predation rate of fish on herbivorous zooplankton. Under an eutrophic condition, there is a unique coexistent homogeneous equilibrium, and the equilibrium density of phytoplankton increases, while the equilibrium density of herbivorous zooplankton decreases as the fish predation rate on herbivorous zooplankton is increasing. The study shows that the system will never collapses under the eutrophic condition unless the fish predation rate approaches infinite. The study also finds a functional bifurcation relation between the delay parameter of the maturation period of herbivorous zooplankton and the fish predation rate on herbivorous zooplankton that, above a critical value of the fish predation rate, the system stays at the coexistent equilibrium, and below that value, the system switches its dynamical patterns among stable and unstable equilibria and oscillations. The oscillations emerge from Hopf bifurcations, and a detailed mathematical analysis about the Hopf bifurcations is carried out to give relevant ecological predications.

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Acknowledgments

The authors greatly appreciate the anonymous referees careful reading and valuable comments. J.W. is partially supported by National Natural Science Foundation of China (No. 11371111), Research Fund for the Doctoral Program of Higher Education of China (No. 20122302110044). J.Z. is partially supported by Natural Science Foundation of Heilongjiang Province of China (No. A201422), Natural Science Foundation of Shandong Province of China (No. ZR2015AQ005) and Program for Young Teachers Scientific Research in Qiqihar University (No. 2012k-M28). J.P.T. is partially supported by National Science Foundation of US (DMS-1446139) and National Natural Science Foundation of China (No. 11371048).

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, People’s Republic of China

    Jiantao Zhao & Junjie Wei

  2. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88001, USA

    Jianjun Paul Tian

  3. School of Science, Qiqihar University, Qiqihar, 161006, Heilongjiang, People’s Republic of China

    Jiantao Zhao

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  1. Jiantao Zhao
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  2. Jianjun Paul Tian
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  3. Junjie Wei
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Correspondence to Junjie Wei.

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Zhao, J., Tian, J.P. & Wei, J. Minimal Model of Plankton Systems Revisited with Spatial Diffusion and Maturation Delay. Bull Math Biol 78, 381–412 (2016). https://doi.org/10.1007/s11538-016-0147-3

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  • DOI: https://doi.org/10.1007/s11538-016-0147-3

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