Stochastic Lotka-Volterra System Research Articles

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Stationary distribution of stochastic population dynamics with infinite delay

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C∞-smooth density.

Structure-Preserving Numerical Methods for Stochastic Poisson Systems

We propose a class of numerical integration methods for stochastic Poisson systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An a-generating function approach with \alpha\in [0,1] is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These integrators are proved to preserve both the Poisson structure and the Casimir functions of the SPSs. Applications to a three-dimensional stochastic rigid body system and a three-dimensional stochastic Lotka-Volterra system show efficiency of the proposed methods.

The asymptotic stability of numerical analysis for stochastic age-dependent cooperative Lotka-Volterra system.

In this study, we explore a stochastic age-dependent cooperative Lotka-Volterra (LV) system with an environmental noise. By applying the theory of M-matrix, we prove the existence and uniqueness of the global solution for the system. Since the stochastic age-dependent cooperative LV system cannot be solved explicitly, we then construct an Euler-Maruyama (EM) numerical solution to approach the exact solution of the system. The convergence rate and the pth-moment boundedness of the scheme have also been obtained. Additionally, numerical experiments have been conducted to verify our theoretical results.

Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection

Motivated by the remarkable works of Busse and his collaborators in the 1980s on turbulent convection in a rotating layer, we explore the long-run behavior of stochastic Lotka-Volterra (LV) systems both in pull-back trajectories and in stationary measures. A decomposition formula is established to describe the relationship between the solutions of stochastic and deterministic LV systems and the stochastic logistic equation. By virtue of this formula, it can be verified that every pull-back omega limit set is an omega limit set of the deterministic LV system multiplied by the random equilibrium of the stochastic logistic equation. The formula is also used to derive the existence of a stationary measure, its support and ergodicity. We prove the tightness of stationary measures and that their weak limits are invariant with respect to the corresponding deterministic system and supported on the Birkhoff center.The developed theory is successfully utilized to completely classify three dimensional competitive stochastic LV systems into 37 classes. The expected occupation measures weakly converge to a strongly mixing measure and all stationary measures are obtained for each class except class 27 c). Among them there are two classes possessing a continuum of random closed orbits and strongly mixing measures supported on the cone surfaces, which weakly converge to the Haar measures of periodic orbits as the noise intensity vanishes. The class 27 c) is an exception, almost every pull-back trajectory cyclically oscillates around the boundary of the stochastic carrying simplex characterized by three unstable stationary solutions. The limit of the expected occupation measures is neither unique nor ergodic. These are consistent with symptoms of turbulence.

Lévy flight movements prevent extinctions and maximize population abundances in fragile Lotka–Volterra systems

Multiple-scale mobility is ubiquitous in nature and has become instrumental for understanding and modeling animal foraging behavior. However, the impact of individual movements on the long-term stability of populations remains largely unexplored. We analyze deterministic and stochastic Lotka-Volterra systems, where mobile predators consume scarce resources (prey) confined in patches. In fragile systems (that is, those unfavorable to species coexistence), the predator species has a maximized abundance and is resilient to degraded prey conditions when individual mobility is multiple scaled. Within the Lévy flight model, highly superdiffusive foragers rarely encounter prey patches and go extinct, whereas normally diffusing foragers tend to proliferate within patches, causing extinctions by overexploitation. Lévy flights of intermediate index allow a sustainable balance between patch exploitation and regeneration over wide ranges of demographic rates. Our analytical and simulated results can explain field observations and suggest that scale-free random movements are an important mechanism by which entire populations adapt to scarcity in fragmented ecosystems.

Global dissipativity of stochastic Lotka–Volterra system with feedback controls

In this paper, a class of stochastic Lotka–Volterra system with feedback controls is considered. The purpose is to establish some criteria to ensure the system is globally dissipative in the mean square. By constructing suitable Lyapunov functions as well as combining with Jensen inequality and It[Formula: see text] formula, the sufficient conditions are established and they are expressed in terms of the feasibility to a couple linear matrix inequalities (LMIs). Finally, the main results are illustrated by examples.

Permanence of Stochastic Lotka–Volterra Systems

This paper proposes a new definition of permanence for stochastic population models, which overcomes some limitations and deficiency of the existing ones. Then, we explore the permanence of two-dimensional stochastic Lotka–Volterra systems in a general setting, which models several different interactions between two species such as cooperation, competition, and predation. Sharp sufficient criteria are established with the help of the Lyapunov direct method and some new techniques. This study reveals that the stochastic noises play an essential role in the permanence and characterize the systems being permanent or not.

Kolmogorov-type systems with regime-switching jump diffusion perturbations

Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.

Stochastic optimal control of predator–prey ecosystem by using stochastic maximum principle

A new control strategy for preventing the extinction of predator–prey ecosystem in a random environment is proposed. First, a controlled stochastic Lotka–Volterra system is introduced and its some properties are presented. The stochastic averaging method is applied to transfer the original controlled system to a partially averaged one. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control is determined from the maximum condition and solving the forward-backward stochastic differential equation (FBSDE). For infinite time-interval ergodic control, the adjoint process is stationary and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the prey of optimally controlled system is obtained to show that the controlled ecosystem is more stable than the uncontrolled one. The mean transition time of optimally controlled system is also calculated to show that the proposed control strategy is effective in preventing the possible extinction.

Numerical solutions of stochastic Lotka-Volterra equations via operational matrices

In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.

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:

The Stationary Distribution and Extinction of Generalized Multispecies Stochastic Lotka-Volterra Predator-Prey System

This paper is concerned with the existence of stationary distribution and extinction for multispecies stochastic Lotka-Volterra predator-prey system. The contributions of this paper are as follows. (a) By using Lyapunov methods, the sufficient conditions on existence of stationary distribution and extinction are established. (b) By using the space decomposition technique and the continuity of probability, weaker conditions on extinction of the system are obtained. Finally, a numerical experiment is conducted to validate the theoretical findings.

Analysis on stochastic delay Lotka–Volterra systems driven by Lévy noise

In this paper, two types of stochastic Lotka–Volterra systems (i.e., competition system and predator–prey system) with time delays and Lévy noise are discussed. For each model, we establish sufficient and necessary criteria for stability in time average and extinction of each population. The thresholds between stability in time average and extinction of each model are obtained. Some recent results are improved and extended significantly. These results point out that: (i) time delays are harmless for stability in time average and extinction of our model; (ii) both stability in time average and extinction have close relationships with Lévy noise; (iii) the interaction rates play significant roles in determining the stability in time average and extinction of the populations.

Stochastic Lotka-Volterra systems under regime switching with jumps

A stochastic Lotka-Volterra model with Markovian switching driven by jumps is proposed and investigated. In the model, the white noise, color noise and jumping noise are taken into account at the same time. This model is more feasible and applicable. Firstly, sufficient conditions for stochastic permanence and extinction are presented. Then the moment average in time and the asymptotic pathwise properties are estimated. Our results show that these properties have close relations with the jumps and the stationary probability distribution of the Markov chain. Finally, several numerical simulations are provided to illustrate the effectiveness of the results.

Some New Results on the Lotka-Volterra System with Variable Delay

This paper discusses the stochastic Lotka-Volterra system with time-varying delay. The nonexplosion, the boundedness, and the polynomial pathwise growth of the solution are determined once and for all by the same criterion. Moreover, this criterion is constructed by the parameters of the system itself, without any uncertain one. A two-dimensional stochastic delay Lotka-Volterra model is taken as an example to illustrate the effectiveness of our result.

Dynamics of a stochastic Lotka-Volterra model with regime switching

A stochastic Lotka-Volterra system under regime switching is proposed and investigated. First, sufficient conditions for stochastic permanence and extinction of the solution are established. Then the lower- and upper-growth rates of the positive solution are investigated. In addition, the superior limit of the average in time of the sample path of the solution is estimated. The results show that these properties have close relationships with the stationary probability distribution of the Markov chain. Finally, the main results are illustrated by several examples and figures. Some recent results are extended and improved.

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.

Global Asymptotic Stability of a Neutral Stochastic Lotka-Volterra System

The work addresses a stochastic Lotka-Volterra system with delays of neutral type for which global asymptotic stability criteria are established.

Global asymptotic stability of a general stochastic Lotka–Volterra system with delays

This paper discusses a general stochastic Lotka–Volterra system with delays. Some conditions for the global asymptotic stability are established.