Lotka Volterra Competition System Research Articles

Periodic Solutions for the Degenerate Lotka–Volterra Competition System

In this paper we consider periodic solutions for the degenerate Lotka–Volterra competition system. We use the iteration method and energy method to prove the existence of periodic solutions. Furthermore, we give some numerical simulations and explicit solutions to verify our theoretical results. As far as we know, there are no results about the periodic solutions for degenerate Lotka–Volterra competition system before the present work.

Dynamics of a Competitive Lotka–Volterra Systems in $\mathbb{R}^{3}$

We describe the dynamics of the 3-dimensional competitive Lotka–Volterra systems $$ \dot{x}=x(a-x-y-z),\quad \dot{y}=y(b-x-y-z),\quad \dot{z}=z(c-x-y-z), $$ providing the phase portraits for all the values of the parameters $a$ , $b$ and $c$ with $0< a< b< c$ in the positive octant of the Poincaré ball.

Stability switching curves in a Lotka–Volterra competition system with two delays

This study is devoted to consider the dynamic properties of a Lotka–Volterra competition model with two discrete delays. First, it reviews stability conditions of the no-delay model as a benchmark. Second, it examines the delay model and analytically establishes the stability switching curve on which stability is switched to instability and vice versa. Finally, numerical simulations are performed to confirm the theoretical results such as Hopf bifurcations and, further, demonstrate a wide variety of dynamics ranging from simple limit cycle to the existence of multi-stability and the birth of chaotic dynamics.

Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

<p style='text-indent:20px;'>This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue.

Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model

We prove that the critical pulled front of Lotka-Volterra competition systems is nonlinearly asymptotically stable. More precisely, we show that perturbations of the critical front decay algebraically with rate t−3/2 in a weighted L∞ space. Our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate t−3/2 is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.

Long time behavior of stochastic Lotka–Volterra competitive system with general Lévy jumps

Taking general Levy jumps into account, a traditional competition system with stochastic perturbation is proposed and investigated. Sufficient conditions for extinction are discussed as well as weak persistence, nonpersistence in the mean and stochastic permanence. In addition, the critical number between weak persistence in the mean and extinction is obtained. Our results demonstrate that, firstly, white noises has no effect on the persistence and extinction; secondly, the general Levy jumps could make permanence vanish as well as happen.

Front-like Entire Solutions for a Lotka-Volterra Weak Competition System with Nonlocal Dispersal

This paper is concerned with the front-like entire solutions for a Lotka-Volterra weak competition system with nonlocal dispersal. Here, an entire solution means a classical solution defined for all space and time variables. This system has traveling wavefronts and enjoys the comparison principle. Based on these traveling wavefronts, we construct some super- and sub-solutions. Then, by using the comparison principle and the super- and sub-solutions method, we establish the existence of front-like entire solutions which behave as two wavefronts coming from the both sides of x-axis. Moreover, some properties of the front-like entire solutions are obtained.

Existence and Stability of Stationary States of a Reaction–Diffusion-Advection Model for Two Competing Species

It is well known that the research of two species in the Lotka–Volterra competition system could create very interesting dynamics. In our paper, we investigate the global dynamical behavior of a classic Lotka–Volterra competition system by studying the steady states and corresponding stability by mainly employing the methods of monotone dynamical systems theory, Lyapunov–Schmidt reduction and spectral theory and so on. It illustrates that the dynamical behavior substantially relies on certain variable of the maximal growth rate. Furthermore, we obtain that one of the semi-trivial steady state solutions is a global attractor in some special cases. In biology, these results show that both of the species do not coexist and the mutant forces the extinction of resident species under some condition for two similar species system.

Existence of forced waves and gap formations for the lattice Lotka–Volterra competition system in a shifting environment

We study the existence of forced traveling waves and gap formations for the lattice Lotka–Volterra competition system in a shifting habitat. By virtue of upper–lower solution method, we establish that the system admits a forced wave provided that the shifting speed of climate change falls in a certain interval. When the shifting speed is outside the range (except for the two endpoints), gap formations are then shown by theoretical proofs.

Global dynamics of a Lotka–Volterra competitive system from river ecology: general boundary conditions

In this paper, we mainly study the population dynamics of a Lotka–Volterra competition system from river ecology. One interesting feature in this model concerns the boundary conditions at upstream and downstream ends, where the species can be exposed to a net loss of individuals, as tuned by parameters bu and bd measuring the magnitude of the loss. We establish a complete classification of all possible long time behaviors for this general model, and as an application, we further present a clear picture on the global dynamics by investigating a special case where two species are competing for the same resource.

Extinction and survival in competitive Lotka-Volterra systems with constant coefficients and infinite delays

The qualitative properties of a nonautonomous competitive Lotka-Volterra system with infinite delays are studied.By using a result of matrix theory and the fluctuation lemma, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernel, which are sufficient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.

Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems

We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature.

Evolution of dispersal in advective homogeneous environments

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

A study of a three-dimensional competitive Lotka–Volterra system

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system: $$ {\left\{ {\dot x} \right._i} = {x_i}\left( {{b_i} - \sum\limits_{i = 1}^3 {{a_{ij}}{x_j}} } \right),i = 1,2,3 $$ where xi(t) is the population size of the i-th species at time t, Ẋi denote $${{dxi} \over {dt}}$$ and aij, bi are all strictly positive real numbers. This system of ordinary differential equations represent a class of Kolmogorov systems. This kind of systems is widely used in the mathematical models for the dynamics of population, like predator-prey models or different models for the spread of diseases. A qualitative analysis of this Lotka-Volterra system based on dynamical systems theory will be performed, by studying the local behavior in equilibrium points and obtaining local dynamics properties.

Permanence for a Generalized Nonautonomous Lotka-Volterra Competition System with Delays

Permanence for a Generalized Nonautonomous Lotka-Volterra Competition System with Delays

Propagation phenomena for a bistable Lotka–Volterra competition system with advection in a periodic habitat

This paper is concerned with the following Lotka-Volterra competition system with advection in a periodic habitat \begin{equation*} \begin{cases} \frac{\partial u_1}{\partial t} =d_1(x)\frac{\partial^2 u_1}{\partial x^2}-a_1(x)\frac{\partial u_1}{\partial x}+u_1\left(b_1(x)-a_{11}(x)u_1-a_{12}(x)u_2\right),\\ \frac{\partial u_2}{\partial t} =d_2(x)\frac{\partial^2 u_2}{\partial x^2}-a_2(x)\frac{\partial u_2}{\partial x}+u_2\left(b_2(x)-a_{21}(x)u_1-a_{22}(x)u_2\right), \end{cases} t>0,~x\in\Bbb R, \end{equation*} where $d_i(\cdot)$, $a_i(\cdot)$, $b_i(\cdot)$, $a_{ij}(\cdot)$ $(i,j=1,2)$ are $L$-periodic functions in $C^\nu(\Bbb{R})$ with some $\nu\in(0,1)$. Under certain assumptions, the system admits two periodic locally stable steady states $(u_1^*(x),0)$ and $(0,u_2^*(x))$. In this work, we first establish the existence of the pulsating front $U(x,x+ct)=(U_1(x,x+ct),U_2(x,x+ct))$ connecting two periodic solutions $(0,u_2^*(x))$ and $(u_1^*(x),0)$ at infinities. By using a dynamical method, we confirm further that the pulsating front is asymptotically stable for front-like initial values. As a consequence of the global asymptotically stability, we finally show that the pulsating front is unique up to translation.

Traveling waves for nonlocal Lotka-Volterra competition systems

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium $ (0, 0) $ to some unknown positive steady state for wave speed $ c&gt;c^* = \max\left\{2, 2\sqrt{dr}\right\} $ and there is no such traveling wave solutions for $ c&lt;c^* $, where $ d $ and $ r $ respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

Permanence of hybrid competitive Lotka–Volterra system with Lévy noise

This paper concerns stochastic permanence of a hybrid competitive Lotka-Volterra system with Lévy noise. Sufficient conditions of stochastic permanence are obtained by combining stochastic analytical techniques with M-matrix analysis.

Speed selection for the wavefronts of the lattice Lotka-Volterra competition system

In this paper we study speed selection for traveling wavefronts of the lattice Lotka-Volterra competition model. For the linear speed selection, by constructing new types of upper solutions to the system, we widely extend the results in the literature. We prove that, for the nonlinear speed selection, the wavefront of the first species decays with a faster rate at the far end. This enables us to construct novel lower solutions to establish the existence of pushed wavefronts, a topic that has been understudied. We raise a new conjecture related to the classical Hosono's version of the diffusive system and our numerical simulations help to confirm it, while our rigorous results only provide a partial answer.

Global dynamics of 3D competitive Lotka-Volterra equations with the identical intrinsic growth rate

This paper focuses on investigating complete global topological classification of the three dimensional competitive Lotka-Volterra system with the identical intrinsic growth rate in the compactification of the positive octant of R3 including its infinity. The system is divided into 37 topological classes among 33 stable nullcline classes by the relation on the parameters of competitive system.