Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton

Ruiqing Shi; Jianing Ren; Cuihong Wang; Ruiqing Shi; Jianing Ren; Cuihong Wang

doi:10.3934/mbe.2020214

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 4: 3836-3868. doi: 10.3934/mbe.2020214

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Research article

Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, China

Received: 18 March 2020 Accepted: 18 May 2020 Published: 25 May 2020

In recent years, some researchers paid their attention to the interaction between toxic phytoplankton and zooplankton. Their studies showed that the mechanism of food selection in zooplankton is still immature and when different algae of the same species (toxic and nontoxic) coexist, some zooplankton may not be able to distinguish between toxic and nontoxic algae, and even show a slight preference for toxic strains. Thus, in this article, a fractional order mathematical model with time delay is constructed to describe the interaction of nutrient-phytoplankton-toxic phytoplankton-zooplankton. The main purpose of this paper is to study the influence of fractional order and time delay on the ecosystem. The sufficient conditions for the existence and local stability of each equilibrium are obtained by using fractional order stability theory. By choosing time delay as the bifurcation parameter, we find that Hopf bifurcation occurs when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to support the analytic results. At last we make some conclusion and point out some possible future work.
- fractional order,
- nutrient-phytoplankton-zooplankton,
- time delay,
- Hopf bifurcation,
- stability
Citation: Ruiqing Shi, Jianing Ren, Cuihong Wang. Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3836-3868. doi: 10.3934/mbe.2020214

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Abstract

In recent years, some researchers paid their attention to the interaction between toxic phytoplankton and zooplankton. Their studies showed that the mechanism of food selection in zooplankton is still immature and when different algae of the same species (toxic and nontoxic) coexist, some zooplankton may not be able to distinguish between toxic and nontoxic algae, and even show a slight preference for toxic strains. Thus, in this article, a fractional order mathematical model with time delay is constructed to describe the interaction of nutrient-phytoplankton-toxic phytoplankton-zooplankton. The main purpose of this paper is to study the influence of fractional order and time delay on the ecosystem. The sufficient conditions for the existence and local stability of each equilibrium are obtained by using fractional order stability theory. By choosing time delay as the bifurcation parameter, we find that Hopf bifurcation occurs when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to support the analytic results. At last we make some conclusion and point out some possible future work.

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