Dynamic analysis of a plankton–herbivore state-dependent impulsive model with action threshold depending on the density and its changing rate

Wei Li,Yufei Wang,Tonghua Zhang,Huidong Cheng

doi:10.1007/s11071-021-07022-w

Wei Li, Yufei Wang + Show 2 more

Open Access

https://doi.org/10.1007/s11071-021-07022-w

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Abstract

A plankton–herbivore state-dependent impulsive model with nonlinear impulsive functions and action threshold including population density and rate of change is proposed. Since the use of action threshold makes the model have complex phase set and pulse set, we adopt the Poincar $$\acute{\text{ e }}$$ map as a tool to study its complex dynamics. The Poincar $$\acute{\text{ e }}$$ map is defined on the phase set and its properties in different situations are analyzed. Furthermore, the periodic solution of model is discussed, including the existence and stability conditions of the order-1 periodic solution and the existence of the order-k ( $$k\ge 2$$ ) periodic solutions. Compared with the fixed threshold in the existing literature, our results show that the use of action threshold is more practical, which is conducive to the sustainable development of population and makes people obtain more economic benefits. The analysis method used in this paper can study the complex dynamics of the model more comprehensively and deeply.

Highlights

Plankton are at the bottom of the marine food chain
For a herbivore-plankton model with cannibalism, the existence and stability of order-1 periodic solution were discussed in [9], where the model was a system of Impulsive Differential Equations (IDEs), which has more complex and abundant dynamics, but can fully take into account the impact of instantaneous mutations on the state and more deeply reflect the law of things changing
We proposed a herbivore-plankton state-dependent impulsive model with action threshold and nonlinear impulsive function

Summary

Introduction

Plankton are at the bottom of the marine food chain. They are widely distributed and highly reproductive, forming the foundation of the marine food web. The state-dependent feedback control as mentioned above is carried out on the biological population density when it reaches a fixed threshold in [10, 31]. The threshold of the model needs to contains both population density and change rate, i.e. the proportionally dependent action threshold. The adopted state-dependent impulsive model needs to consider both the selection of threshold and impulsive function. Based on the work of [9] and our above analysis, we will propose a plankton-herbivore state-dependent impulsive model with action threshold and nonlinear impulsive function: dp(t) dt. The nonlinear impulsive functions related to the density and fishing rate of the biological population are adopted as follows: α τ 1 + ηp(t) and β σh2(t) h(t) + μ. In the rest of discuss we assume that k < 2p∗ + 2

Definition of the Poincaremap

The natures of Poincaremap

Order-k periodic solution

Numerical simulations and discussion

Conclusion

Conflict of Interest