Global asymptotic stability of stochastic Lotka–Volterra systems with infinite delays

Abstract

Over the past few decades, differential equations with time delays have received great attention owing to their extensive applications as models in a variety of scientific areas, such as population dynamics, epidemiology and biology. One of the most interesting topics associated with these delay population systems, especially Lotka–Volterra (or LV for short) systems, is the investigation of the global asymptotic stability (or global attractivity) of the positive solution. There is an extensive literature dealing with the global asymptotic stability of LV systems with delays (see, e.g. Gopalsamy, 1992; Kuang, 1993; Kuang & Smith, 1993; Lu & Takeuchi, 1994; Wang & Zhang, 1995; Bereketoglu & Gyori, 1997; He & Gopalsamy, 1997; Ahmad & Lazer, 2000; Teng & Yu, 2000; Saito, 2002; Teng, 2002; Zhao et al., 2004; Chen, 2005, 2006; Faria, 2009; Hu et al., 2011; Zhang & Teng, 2011 and the references therein), but mainly in the deterministic case. Since population systems are inevitably subject to environmental fluctuations (e.g. variation in intensity of sunlight, temperature, water level, etc.; see, e.g. Renshaw, 1991; May, 2001), it is important and necessary to investigate the global asymptotic stability of stochastic LV models. So far, to the best of our knowledge, there are only a few papers (Jiang et al., 2008; Li & Mao, 2009; Liu & Wang, 2013a,b; Yao & Liu, 2014) which studied the global asymptotic stability of stochastic population systems. Jiang et al. (2008) studied the following stochastic logistic model: