Stochastic Population System Research Articles

Dynamic property of a stochastic cooperative species system with distributed delays and Ornstein–Uhlenbeck process

Scanning the whole writing, we discuss a stochastic cooperative species system with distributed delays under the influences of Ornstein–Uhlenbeck process of mean regression. We successfully obtain the existence and uniqueness of positive solutions for stochastic system at first. Secondly, by studying the Lyapunov function, we present the existence of the stationary distribution of the system. We are relatively familiar with the understanding of the density function of random systems. This paper also gives the expression of the density function of the random system near the unique positive equilibrium. In addition, the asymptotic properties of the p-moment boundedness and solution of the stochastic population system are also studied. In particular, we use numerical simulation to verify the theoretical results in the last section.

Impact of time delay on population model with Allee effect

We investigate a general delayed stochastic single-species population system with Allee effect simulated by coupling between non-Gaussian colored noise and Gaussian white noise. Firstly, the Fokker–Planck function is derived through unified colored noise approximation (UCNA) and small time delay approximation. Then, the expression of stationary probability distribution is obtained. After that, the expression of mean first-passage time is calculated in order to quantify the transition between stable states. Besides, The influence of strong Allee effect and weak Allee effect on the stochastic population system is different. It is not conducive to the survival of the population when the intensity of multiplicative non-Gaussian colored noise is enhanced whether strong Allee effect or weak Allee effect exists. It is beneficial to the stability and survival for the population under strong Allee effect. Further, the phenomenon of resonant activation is firstly discovered in biological system with Allee effect. The increase of time delay will restrain the transition between stable states under strong Allee effect. When weak Allee effect exists, the increase of time delay will reduce stability of the population, even the population may be vulnerable to extinction. The results in our work provide effective guidance for future researches of ecosystem.

A new idea on almost sure permanence and uniform boundedness for a stochastic predator–prey model

The existence and uniqueness of stationary distribution and ergodic properties of a stochastic system are obtained. Especially, different from the existing methods, a new method is introduced to analyze almost sure permanence and uniform boundedness of the stochastic predator–prey model. This new idea is based on geometric structure of invariant set for a stochastic system. More specifically, we obtain our main conclusions by showing the invariant set for the stochastic population system lies in the interior of the first quadrant. It is interesting and surprising that the stochastic population model can guarantee a uniform boundedness almost surely. Some numerical simulations are carried out to support our results.

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

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:

Almost sure permanence of stochastic single species models

A new definition of almost sure permanence of stochastic population systems is proposed in this paper. We consider the stochastic logistic modeldx(t)=x(t)[a(t)−b(t)xθ(t)]dt+σ(t)x(t)dB(t), where B(t) is a standard Brownian motion and θ is a positive constant. Under a simple assumption, a interesting result is obtained as follows:0<liminft→∞x(t)≤limsupt→∞x(t)<∞a.s. The result is analogous to the result in the deterministic case. And some numerical simulations are introduced to support our main results at the end.

Stationary distribution of stochastic population systems under regime switching

This paper is concerned with n-species model of facultative mutualism in random environments. The environment variability in this study is characterized with both white noise and color noise modeled by Markovian switching. We established new sufficient conditions that ensuring that the system model is positive recurrent. We also showed the existence of a unique ergodic stationary distribution. The presented results are demonstrated by numerical simulations.

Stationary distribution of stochastic population systems

In this paper we consider the stochastic differential equation (SDE) population model d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ ( b + A x ( t ) ) d t + σ d B ( t ) ] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004) [2] and Mao (2005) [6]) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.

Existence and Uniqueness for Stochastic Predator-Prey System of Two Species with Age-Structured

In this paper we introduce a stochastic age-dependent predator-prey system. Existence and uniqueness of strong solution for a stochastic population system of two species with age-dependent are proved. The analysis uses Barkholder-Davis-Gundy’s inequality,ItÔ's formula and some special inequalities to achieve our purpose.

On the Pathwise Growth Rates of Large Fluctuations of Stochastic Population Systems with Infinite Delay

In general, time delay and system uncertainty are commonly encountered for all population systems. To examine whether the presence of environmental noise affects infinite delay population systems significantly, this paper perturbs the functional Kolmogorov-type system with infinite delay into the infinite delay stochastic functional differential system By the M-matrix technique, this paper examines the global positive solution and its pathwise estimation for this stochastic functional Kolmogorov-type population system. To illustrate the applications of our theory more clearly, this paper also discusses a Lotka–Volterra system with mixed delays as a special case.

Existence and uniqueness of global positive solutions to the stochastic functional Kolmogorov-type system

In general, any population systems are often subject to various environmental noise. To examine whether the presence of such noise affects these systems significantly, we examine a stochastic functional Kolmogorov-type population system with the general stochastic perturbation. In this paper, we show that the environmental noise may suppress the potential population explosion and guarantee global existence of positive solutions when the noise intensity is strongly dependent on the population size. However, when the noise intensity is weakly dependent on the population size, the stochastic population system behaves similarly to the corresponding deterministic system. To illustrate our idea clearly, we also discuss some stochastic Lotka-Volterra systems as special cases.

Explaining and predicting patterns in stochastic population systems.

Lattice effects in ecological time-series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time-series may require more than one deterministic model, especially when the dynamics are complex.