Lotka Volterra Competition System Research Articles

Dynamics of a delayed Lotka–Volterra competition model with directed dispersal

We consider a diffusive Lotka–Volterra competition system with stage structure, where the intrinsic growth rates and the carrying capacities of the species are assumed spatially heterogeneous. Here, we also assume each of the competing populations chooses its dispersal strategy as the tendency to have a distribution proportional to a certain positive prescribed function. We give the effects of dispersal strategy, delay, the intrinsic growth rates and the competition parameters on the global dynamics of the delayed reaction diffusion model. Our result shows that competitive exclusion occurs when one of the diffusion strategies is proportional to the carrying capacity, while the other is not; while both populations can coexist if the competition favors the latter species. Finally, we point out that the method is also applied to the global dynamics of other kinds of delayed competition models.

Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process.

The competitive relationship is one of the important studies in population ecology. In this paper, we investigate the dynamical behaviors of a two-species Lotka-Volterra competition system in which intrinsic rates of increase are governed by the Ornstein-Uhlenbeck process. First, we prove the existence and uniqueness of the global solution of the model. Second, the extinction of populations is discussed. Moreover, a sufficient condition for the existence of the stationary distribution in the system is obtained, and, further, the formulas for the mean and the covariance of the probability density function of the corresponding linearized system near the equilibrium point are obtained. Finally, numerical simulations are applied to verify the theoretical results.

Lotka-Volterra diffusion-advection competition system with dynamical resources

Competition systems describing the competition between species for resources have been widely studied in the literature and wealthy results have been developed. Most of them (if not all) have essentially assumed that the resources are spatially varying without temporal dynamics. This is an idealized assumption since the most ecological environments and/or biospecies are dynamically changing. Hence the effect of temporal dynamics of resource ought to be taken into account to predict/interpret the competition outcomes more precisely. This constitutes the main motivation of this work and we consider a Lotka-Volterra reaction-diffusion-advection competition system with a dynamical resource whose dynamics is determined by an evolution equation, where the competing species have biased movement (advection) up the resource gradient. We first establish the global existence of classical solutions via Moser iteration and global stability of spatially homogeneous steady states for the constant resource growth rate by method of Lyapunov functionals. When the resource growth rate is spatially varying, we use numerical simulations to demonstrate the possible competition outcomes and find that the asymptotic dynamics of the competition system with dynamical resources is quite different from the case that resources have no dynamics. Moreover, we numerically observe that the advective strategy and/or the relative strength of advection sensory responses are key factors determining the competition outcomes and the asymptotic profiles of the solution.

Analysis of a stochastic Lotka–Volterra competitive system with infinite delays and Ornstein–Uhlenbeck process

In this paper, we construct and analyze a stochastic Lotka–Volterra competitive model with the Ornstein–Uhlenbeck process and infinite delays. First, we verify the existence and uniqueness of the global solution of the system with any initial value. Then, we investigate the pth moment boundedness, asymptotic pathwise estimation, and asymptotic behavior of the solutions of the stochastic system in turn. In addition, we develop sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system by establishing a series of suitable Lyapunov functions. Finally, by solving the corresponding six-dimensional Fokker–Planck equation, we obtain the accurate expression of the local density function of the linear system corresponding to the stochastic system.

Dynamical Behavior of a Lotka-Volterra Competitive System from River Ecology

Dynamical Behavior of a Lotka-Volterra Competitive System from River Ecology

Travelling wave of stochastic Lotka-Volterra competitive system

<p style='text-indent:20px;'>This work is devoted to the travelling wave solutions and asymptotic wave speed for a two-species stochastic competitive system driven by a multiplicative Itô type white noise. By the introduction of sup-solutions and sub-solutions, the existence and the approximations of the travelling wave solutions are proven under some conditions on the parameters. Moreover, combining with the stochastic Feynman-Kac formula, sub-solution, sup-solution, monotone iteration and equilibria of the kinetic system corresponding to the transformed stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

Multiplicity on Limit Cycles of 3D Lotka-Volterra Competitive Systems

Multiplicity on Limit Cycles of 3D Lotka-Volterra Competitive Systems

Existence of Two Limit Cycles in Zeeman’s Class 30 for 3D Lotka-Volterra Competitive System

Existence of Two Limit Cycles in Zeeman’s Class 30 for 3D Lotka-Volterra Competitive System

Spreading speed of a lattice time-periodic Lotka–Volterra competition system with bistable nonlinearity

We study the uniqueness and sign of bistable wave speed for a lattice time-periodic Lotka–Volterra competition system of bistable type. The sign of wave speed reflects the propagation direction of traveling wave which arises as a challenging problem in the field of traveling wave theory. Actually, even for the case of homogeneous environment, there is no common method that can be used to deal with it. First, we show that the speed of the bistable wave of the model is unique. Second, by making use of upper–lower solution method and comparison principle, a set of explicit conditions is built up such that the sign of the wave speed can be justified. From the viewpoint of biology, these conditions reveal which of the species will tend to extinction and which will survive when the environment changes periodically in time.

Propagation dynamics of forced pulsating waves of a time periodic Lotka-Volterra competition system in a shifting habitat

Existence, uniqueness and stability of forced pulsating waves as well as gap formation of reaction diffusion models in a shifting habitat are challenging problems in the field of propagation dynamics. In this paper, we design to tackle these problems for a time periodic Lotka-Volterra competition system. By using alternatively-coupling upper-lower solution method, we establish the existence of forced pulsating traveling wave, as long as the shifting speed falls in a finite interval where the end points are obtained from KPP-Fisher speeds. The asymptotic behaviors of the forced pulsating traveling wave are derived and this helps us show that this wave is unique. Moreover in the Lyapunov sense, the stability of the forced pulsating wave is also studied. Finally, when the forced speed is beyond the finite interval, gap formation is investigated theoretically and illustrated further by numerical simulations.

Global stability of bistable traveling wavefronts for a three-species Lotka–Volterra competition system with nonlocal dispersal

This paper is devoted to the study of traveling wavefronts for a three-component Lotka–Volterra system with nonlocal dispersal. This system arises in the study of three-species competition model in which there is no competition between two of these three species. It has been shown that this system admits a bistable traveling wavefront. In this paper, we further investigate the stability of bistable traveling wavefronts. By constructing suitable super- and sub-solutions and using a dynamical system approach, we obtain the globally asymptotic stability of the bistable traveling wavefronts.

Linear and nonlinear minimal speed selection of traveling waves for a competitive system with nonlocal dispersal

In this paper, we study the linear and nonlinear selection of the minimal wave speed of traveling waves for a two-species Lotka-Volterra competitive system with nonlocal diffusion. By using the method of upper-lower solutions, we establish some new conditions for linear and nonlinear minimal speed selection. It is shown that the strong competition between two populations will lead to the realization of nonlinear selection, and the minimal speed selection can be realized linearly under appropriate conditions. Two examples are also presented to illustrate the realization of linear and nonlinear selection.

Speed selection of wavefronts for lattice Lotka-Volterra competition system in a time periodic habitat

In this paper, the speed selection for the time periodic traveling wave solutions of a two-species competition lattice model of diffusive Lotka-Volterra type is investigated. By using the upper-lower solution method, an abstract result and several explicit sufficient conditions for linear selection are established. Moreover, a general condition for nonlinear selection is also obtained, which indicates that the minimal speed is nonlinearly selected if the system admits a lower solution with faster decay at the far end. Based on this result, some explicit conditions for nonlinear selection are found by constructing novel lower solutions.

A classification of the dynamics of three-dimensional stochastic ecological systems

The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis’ conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis’ conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka–Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology’s modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.

The dynamics of traveling wavefronts for a nonlocal delay competition system with local vs. nonlocal diffusions

In this paper, we investigate the existence and exponential stability of traveling wavefronts for a two-component nonlocal delay Lotka–Volterra competition system with local vs. nonlocal diffusions. We first obtain the existence of monostable traveling wavefronts connecting two boundary equilibria by Schauder’s fixed point theorem with an explicit construction of a pair of super- and subsolutions. Furthermore, applying the weighted energy method together with the comparison principle, we prove that all the solutions of the corresponding Cauchy problem converge exponentially to the traveling wavefronts provided that the initial perturbations around the traveling wavefronts belong to a certain weighted Sobolev space.

Speed determinacy of the traveling waves for a three species time‐periodic Lotka–Volterra competition system

Some physical environmental conditions changing in time create a great difficulty in studying time‐periodic traveling wave solutions. In this paper, we investigate speed selection of time‐periodic traveling waves for a three species time‐periodic Lotka–Volterra competition system. By transforming the competitive system into a cooperative system, the existence of the traveling wave solutions and speed determinacy of the spread speed are studied via the upper–lower solution method as well as the comparison principle. Through constructing specific types of upper and lower solutions to the system, some sets of sufficient conditions composed of the parameters in the system are established to justify the linear and nonlinear selection of the spread speed. As the positive periodic functions in system reduces to the constant coefficients, parts of our results could cover the previous works, and the other results are novel.

Estimates of Population Size for Traveling Wave Solutions of Spatially Non-local Lotka-Volterra Competition System

Estimates of Population Size for Traveling Wave Solutions of Spatially Non-local Lotka-Volterra Competition System

Speed sign of traveling waves to a competitive Lotka-Volterra recursive system with bistable nonlinearity

<p style='text-indent:20px;'>Traveling wave propagation is a significant phenomenon observed in population biology. Due to the occurrence of nonlocal effect in recursive systems, a deep understanding of the wavefront in the propagation direction (speed sign) is challenging. In this paper we study the sign of wave speed for bistable traveling waves to a two-species competitive system that biologically models the dynamics of two species in competition for a common resource. By a proper choice of the kernel functions, we transfer our model into a coupled localized differential system and shed a new light on how to determine the wave speed sign. Sufficient conditions with symmetry are obtained on the propagation directions of the wavefronts. This symmetry is further verified in the final analysis and numerical simulations are provided to illustrate our theoretical results.</p>

Time-Periodic Traveling Waves for A Periodic Lotka-Volterra Competition System with Nonlocal Dispersal

Time-Periodic Traveling Waves for A Periodic Lotka-Volterra Competition System with Nonlocal Dispersal

Coexistence states of a Lotka Volterra cooperative system with cross diffusion

In this work we study the steady state cross diffusive Lotka Volterra cooperative system. We have incorporated different type of cross diffusive term at each trophic level. Using bifurcation theory, we derive sufficient conditions which ensures existence of positive solutions of the system. The analysis carried out allows us to give an affirmative answer to the question that whether cooperative species under the influence of cross diffusion can co exist or not. We have also compared the results obtained for the competitive Lotka Volterra system with our results and an ecological explanation is provided to understand the difference. • To study a cooperative Lotka Volterra prey predator system with different type of cross diffusion at each trophic level. • To show using bifurcation theory that these positive solution branches tend to have an extension that is globally defined. • To find out through the analysis carried out whether a cross diffusive cooperative system admits coexistence states. • Comparison of the results with the corresponding predator prey system.