Analysis of a Stochastic Competitive Model with Saturation Effect and Distributed Delay
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Analysis of a Stochastic Competitive Model with Saturation Effect and Distributed Delay Wenxu Ning1 · Zhijun Liu1
· Lianwen Wang1 · Ronghua Tan1
Received: 25 May 2020 / Revised: 31 August 2020 / Accepted: 17 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This work is concerned with a novel stochastic competitive model with saturation effect and distributed delay, in which two coupling noise sources are incorporated and the interspecific competition delayed terms show saturation effect. A good understanding of exponential extinction, extinction, persistence in the mean and permanence in time average of two species are gained. Also, with the help of Lyapunov function and the global attraction of positive solution, we derive the existence and uniqueness of stationary distribution. Our main results reveal that the coupling noise sources can significantly change the survival results of two species and affect the existence of a unique stationary distribution. Keywords Competitive model · Coupling noise sources · Distributed delay · Survival · Stationary distribution Mathematics Subject Classification (2010) 92D25 · 60H10
1 Introduction To better understand the asymptotic behaviors of the Lotka-Volterra competitive models, various complications have been included in these models which modeled certain types of
Zhijun Liu
[email protected]; zhijun [email protected] Wenxu Ning [email protected] Lianwen Wang [email protected] Ronghua Tan ronghua [email protected] 1
School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, People’s Republic of China
Methodology and Computing in Applied Probability
interactions among different species. Lots of good properties including stability and bifurcation (Ahmad 1993; Chattopadhyay 1996; Song et al. 2004; Li et al. 2019), persistence and extinction (Li and Mao 2009; Liu et al. 2010; Liu and Wang 2011; Bao et al. 2011; Tan et al. 2015; Dong et al. 2015; Zhao et al. 2016; Lu et al. 2019; Lu and Ma 2017; Hu and Liu 2020; Wang et al. 2018), periodic solution and almost periodic solution (Fan et al. 1999; Liu et al. 2008; Zhao 2018; He et al. 2010; Liu and Wang 2014; Xu et al. 2019), optimal control (Liu et al. 2017), stationary distribution and ergodicity (Jiang et al. 2012; Yang et al. 2016; Liu and Zhu 2018; Hu et al. 2020) of such systems have been received wide attention in the last decades. Particularly, Hu and Liu (2020) proposed and studied the following two-species competitive model with saturation effect and coupling noise sources ⎧ y(t) ⎪ ⎨ dx(t) = x(t) r1 − b1 x(t) − c1 dt + a11 x(t)dB1 (t) + a12 x(t)dB2 (t), 1 + y(t) (1) ⎪ ⎩ dy(t) = y(t) r2 − b2 y(t) − c2 x(t) dt + a21 y(t)dB1 (t) + a22 y(t)dB2 (t), 1 + x(t) where the parameters ri , bi and ci (i = 1, 2) are positive, and represent, respectively, the intrinsic growth rates, intraspecific competition rates and interspecific competition rates. The interaction term y/(1 + y) (or x/(1 + x)) shows a saturation value for sufficiently
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