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Stationary Distribution and Periodic Solution of a Stochastic n-Species Gilpin–Ayala Competition System with General Saturation Effect and Nonlinear Perturbations

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Abstract

In this paper, a generalized n-species Gilpin–Ayala competition system with saturation effect and nonlinear perturbations is proposed and examined. We first develop a new mathematical technique called “stochastic \(\epsilon \)-threshold method”, to tackle the nonlinear perturbations and study the competitive coexistence, which includes the existence of ergodic stationary distribution as well as stochastic positive periodic solution. It should be mentioned that the method can be successfully applied to the species coexistence of other biological models. Then, we establish sufficient conditions for exponential extinction of competing species. Finally, several numerical examples are provided to support our theoretical results and analyze the impact of key parameters.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11871473) and the Fundamental Research Funds for the Central Universities (No. 22CX03030A).

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Authors and Affiliations

  1. College of Science, China University of Petroleum (East China), Qingdao, 266580, People’s Republic of China

    Bingtao Han, Daqing Jiang & Baoquan Zhou

  2. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Daqing Jiang

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Appendix A

Appendix A

(I) (Proof of Lemma 2.1): Let \(G(x)=Ax^{\theta }+B-\frac{A\theta }{\theta -\epsilon }\bigl [\frac{B(\theta -\epsilon )}{A\epsilon }\bigr ]^{\frac{\epsilon }{\theta }}x^{\theta -\epsilon }\). By direct calculation, we have

$$\begin{aligned} G'(x)=A\theta x^{\theta -\epsilon -1}\Bigl [x^{\epsilon }-\Bigl (\frac{B(\theta -\epsilon )}{A\epsilon }\Bigr )^{\frac{\epsilon }{\theta }}\Bigr ], \end{aligned}$$

which implies that G(x) has a unique stagnation point \(\overline{x}=(\frac{B(\theta -\epsilon )}{A\epsilon })^{\frac{1}{\theta }}\) on \((0,\infty )\). Thus,

$$\begin{aligned} \inf _{x\in (0,\infty )}G(x)=G(\overline{x})=\frac{B(\theta -\epsilon )}{\epsilon }+B-\frac{B\theta }{\epsilon }=0. \end{aligned}$$

The proof is complete.

(II) (The exponential martingale inequality): Let \(T,\alpha ,\beta \) be any positive numbers and B(t) denote an n-dimensional standard Brownian motion defined on the probability space \( \{\Omega ,\mathscr {F},\{\mathscr {F}_t\}_{t\ge 0},\mathbb {P}\} \). For any \(g=(g_1,\ldots ,g_m)\in L^2(\mathbb {R}_+,\mathbb {R}^{1\times m})\), it then follows

$$\begin{aligned} \mathbb {P}\biggl \{\sup _{0\le t\le T}\biggl [\int _{0}^{t}g(s)dB(s)-\frac{\alpha }{2}\int _{0}^{t}\bigl |g(s)\bigr |^2ds\biggr ]>\beta \biggr \}\le e^{-\alpha \beta }. \end{aligned}$$

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Han, B., Jiang, D. & Zhou, B. Stationary Distribution and Periodic Solution of a Stochastic n-Species Gilpin–Ayala Competition System with General Saturation Effect and Nonlinear Perturbations. Qual. Theory Dyn. Syst. 22, 46 (2023). https://doi.org/10.1007/s12346-023-00743-w

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