Stochastic Competition Model Research Articles

A geometric method for asymptotic properties of the stochastic Lotka–Volterra model

In this paper, we use a geometric method to analyze the asymptotic properties of the stochastic Lokta–Volterra model, which mainly include two aspects: uniformly ultimate boundedness and almost sure permanence. The geometric method demonstrates that there exists a bounded region that lies in the interior of the first quadrant such that solutions of the stochastic model starting from the exterior of the region will almost surely go into the interior of it in finite time, meanwhile, solutions which start from the interior of the region will almost surely not leave the interior of it in any finite time. Firstly, we show that the stochastic competition, predator-prey and mutualism model are uniformly ultimately bounded and almost surely permanent by the geometric structures of invariant sets. Secondly we give numerical simulations to illustrate the geometric method as well as the conclusions.

Dynamics of a stochastic three-species competitive model with Lévy jumps

This paper is concerned with a three-species competitive model with both white noises and Lévy noises. We first carry out the almost complete parameters analysis for the model and establish the critical value between persistence in the mean and extinction for each species. The sufficient criteria for stability in distribution of solutions are obtained. Finally, numerical simulations are carried out to verify the theoretical results.

Stationary distribution and ergodicity of a stochastic hybrid competition model with Lévy jumps

Taking white noises, Markovian switching and Lévy jumps into account, an n−species Lotka–Volterra competitive model with random perturbations is proposed and studied. Sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the model are established. These sufficient conditions are sharp in some cases. Some interesting results are revealed: the white noises and Lévy jumps could make the stationary distribution disappear as well as appear; the Markovian switching could make the stationary distribution appear. Some numerical simulations are also introduced to illustrate the main results.

Dynamics of a stochastic delay competitive model with harvesting and Markovian switching

This work is concerned with a two-species stochastic state-switching competitive population model with distributed delays and harvesting. First, necessary and sufficient criteria for the existence of a unique ergodic stationary distribution of the system are established. Then necessary and sufficient criteria for the existence of the optimal harvesting policy are given, and the explicit expression of the optimal harvesting policy is obtained. Finally, some effects of the state-switching on the persistence, extinction and optimal harvesting strategy of the system are discussed with the help of several simulations.

Analysis of a Stochastic Lotka-Volterra Competitive Model with Infinite Delay and Impulsive Perturbations

This paper considers a stochastic Lotka-Volterra competitive model with infinite delay and impulsive perturbations. This model is new, more feasible and more accordance with the actual. The aim is to analyze what happens under the impulsive perturbations. With space $C_{g}$ as phase space, sufficient conditions for permanence in time average are established as well as extinction, stability in time average and global attractivity of each population. Numerical simulations are also exhibited to illustrate the validity of the results in this paper. In addition, a knowledge is given to illustrate that the statement in [21] is incorrect by choosing space $C_{g}$ as phase space. Our results demonstrate that impulsive perturbations which may represent human factor play a key role in protecting the population when environmental noise and interaction rates are disadvantageous to population survival.

Dynamical behavior for a stochastic two-species competitive model

Abstract This paper deals with a stochastic two-species competitive model. Some very verifiable criteria on the global stability of the positive equilibrium of the deterministic system are established. An example with its computer simulations is given to illustrate our main theoretical findings.

Dynamics of a stochastic delay competition model with imprecise parameters

Dynamics of a stochastic delay competition model with imprecise parameters

Mapping of the stochastic Lotka-Volterra model to models of population genetics and game theory.

The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection, and the Moran model with frequency-dependent selection (equivalently a game-theoretic formulation of the Moran model). We demonstrate how these mappings can be used to calculate extinction probabilities and the times until a species' extinction in the SLVC model.

Asymptotic behaviour of a class of resource competition biology species system by the fractional Brownian motion

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with \(n\) species and \(k\) resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result. doi:10.1017/S1446181117000098

Permanence of a stochastic delay competition model with Levy jumps

Permanence of a stochastic delay competition model with Levy jumps

Nonlocal birth-death competitive dynamics with volume exclusion

A stochastic birth-death competition model for particles with excluded volume is proposed. The particles move, reproduce, and die on a regular lattice. While the death rate is constant, the birth rate is spatially nonlocal and implements inter-particle competition by a dependence on the number of particles within a finite distance. The finite volume of particles is accounted for by fixing an upper value to the number of particles that can occupy a lattice node, compromising births and movements. We derive closed macroscopic equations for the density of particles and spatial correlation at two adjacent sites. Under different conditions, the description is further reduced to a single equation for the particle density that contains three terms: diffusion, a linear death, and a highly nonlinear and nonlocal birth term. Steady-state homogeneous solutions, their stability which reveals spatial pattern formation, and the dynamics of time-dependent homogeneous solutions are discussed and compared, in the one-dimensional case, with numerical simulations of the particle system.

ASYMPTOTIC BEHAVIOUR OF A CLASS OF RESOURCE COMPETITION BIOLOGY SPECIES SYSTEM BY THE FRACTIONAL BROWNIAN MOTION

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

Optimal harvesting policy of a stochastic two-species competitive model with Lévy noise in a polluted environment

As well known that the sudden environmental shocks and toxicant can affect the population dynamics of fish species, a mechanistic understanding of how sudden environmental change and toxicant influence the optimal harvesting policy requires development. This paper presents the optimal harvesting of a stochastic two-species competitive model with Lévy noise in a polluted environment, where the Lévy noise is used to describe the sudden climate change. Due to the discontinuity of the Lévy noise, the classical optimal harvesting methods based on the explicit solution of the corresponding Fokker–Planck equation are invalid. The object of this paper is to fill up this gap and establish the optimal harvesting policy. By using of aggregation and ergodic methods, the approximation of the optimal harvesting effort and maximum expectation of sustainable yields are obtained. Numerical simulations are carried out to support these theoretical results. Our analysis shows that the Lévy noise and the mean stress measure of toxicant in organism may affect the optimal harvesting policy significantly.

Optimal harvesting of a stochastic delay competitive model

In this paper an $n$-species stochastic delay competitive model with harvesting is proposed. Some dynamical properties of the model are considered. We first establish sufficient conditions for persistence in the mean of the species. Then asymptotic stability in distribution of the harvesting model is studied. Next the optimal harvesting effort and the maximum harvesting yield are given by using the ergodic approach. Finally the analytical results are illustrated through simulation figures using MATLAB followed by discussions and conclusions.

The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment

Lévy noise, accounting for sudden catastrophic shocks, exists widely in the natural world. However, the effect of Lévy noise on the survival of a population is not well understood. In this paper, we develop and investigate a stochastic competitive model of two species with Lévy noise in an impulsive polluted environment. The threshold between weak persistence in the mean and extinction is established for each species. Sufficient conditions for strong persistence in the mean, stability in the mean and stochastic permanence are obtained. These results show that Lévy noise can change the persistence and extinction of each species significantly. Numerical simulations is carried out to illustrate the theoretical results.

Survival and Stationary Distribution Analysis of a Stochastic Competitive Model of Three Species in a Polluted Environment.

In this paper, we develop and study a stochastic model for the competition of three species with a generalized dose-response function in a polluted environment. We first carry out the survival analysis and obtain sufficient conditions for the extinction, non-persistence, weak persistence in the mean, strong persistence in the mean and stochastic permanence. The threshold between weak persistence in the mean and extinction is established for each species. Then, using Hasminskii's methods and a Lyapunov function, we derive sufficient conditions for the existence of stationary distribution for each population. Numerical simulations are carried out to support our theoretical results, and some biological significance is presented.

Developing a new stochastic competitive model regarding inventory and price

Within the competition in today’s business environment, the design of supply chains becomes more complex than before. This paper deals with the retailer’s location problem when customers choose their vendors, and inventory costs have been considered for retailers. In a competitive location problem, price and location of facilities affect demands of customers; consequently, simultaneous optimization of the location and inventory system is needed. To prepare a realistic model, demand and lead time have been assumed as stochastic parameters, and queuing theory has been used to develop a comprehensive mathematical model. Due to complexity of the problem, a branch and bound algorithm has been developed, and its performance has been validated in several numerical examples, which indicated effectiveness of the algorithm. Also, a real case has been prepared to demonstrate performance of the model for real world.

Models of genetic drift as limiting forms of the Lotka-Volterra competition model.

The relationship between the Moran model and stochastic Lotka-Volterra competition (SLVC) model is explored via time scale separation arguments. For neutral systems the two are found to be equivalent at long times. For systems with selective pressure, their behavior differs. It is argued that the SLVC is preferable to the Moran model since in the SLVC population size is regulated by competition, rather than arbitrarily fixed as in the Moran model. As a consequence, ambiguities found in the Moran model associated with the introduction of more complex processes, such as selection, are avoided.

Two-strain competition in quasineutral stochastic disease dynamics.

We develop a perturbation method for studying quasineutral competition in a broad class of stochastic competition models and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic susceptible-infected-susceptible (SIS) model. Here we extend previous results due to Parsons and Quince [Theor. Popul. Biol. 72, 468 (2007)], Parsons etal. [Theor. Popul. Biol. 74, 302 (2008)], and Lin, Kim, and Doering [J. Stat. Phys. 148, 646 (2012)]. The second model, a two-strain generalization of the stochastic susceptible-infected-recovered (SIR) model with population turnover, has not been studied previously. In each of the two models, when the basic reproduction numbers of the two strains are identical, a system with an infinite population size approaches a point on the deterministic coexistence line (CL): a straight line of fixed points in the phase space of subpopulation sizes. Shot noise drives one of the strain populations to fixation, and the other to extinction, on a time scale proportional to the total population size. Our perturbation method explicitly tracks the dynamics of the probability distribution of the subpopulations in the vicinity of the CL. We argue that, whereas the slow strain has a competitive advantage for mathematically "typical" initial conditions, it is the fast strain that is more likely to win in the important situation when a few infectives of both strains are introduced into a susceptible population.

Uncovering overlapping cluster structures via stochastic competitive learning

In this paper, we present a method for determining overlapping cluster structures in a networked environment. The technique is built upon a definition of a stochastic competitive model, which permits one to describe the behavior of the model for different types of networks. In the referred model, several particles navigate in the network and compete with each other with the purpose of occupying as many vertices as possible. While visiting new vertices in the network, the particles mark the vertices with a unique domination signature in an attempt to repel intruder particles from possibly capturing them in future steps. Such an indication is soft, in a way that, according to the network’s topology, other particles can eventually nullify this domination and become the new owners of those vertices. With this in mind, we show that the particles’ domination levels provide a natural and intuitive indicator of whether or not a specific vertex or cluster structure presents overlapping characteristics. Additionally, as opposed to the majority of the techniques designed for uncovering overlapping structures, we demonstrate that this task can be performed in an efficient manner with the particle competition model, since the underlying stochastic process is able to generate useful dynamical information with no additional computational cost. As a result, the model’s computational complexity is maintained low. Thus, this method turns out to be a good alternative for finding overlapping cluster structures in real-world data. Moreover, a convergence analysis of the competitive dynamical system is discussed. We conclude that the model usually does not converge to a fixed point, but instead it is bounded by a time-dependent finite region. Furthermore, an upper bound of this region is estimated. Computer simulations reveal that this overlapping index works well in synthetic and real-world data sets. Finally, in order to show the robustness of the model, an application on handwritten digits and letters recognition is provided, where the model is able to achieve high clustering accuracy and is capable of finding the overlapping structures that match our intuition.