Weak Competition Case Research Articles

Coexistence and competitive exclusion in a time-periodic Lotka-Volterra competition-diffusion system

The evolutionary impact of both temporal periodicity and spatial heterogeneity on population ecology is a challenging problem. Hutson et al. (2001) [12] and Bai et al. (2022) [2] studied the effects of diffusion rates on the dynamics of a time-periodic Lotka-Volterra competition-diffusion system with identical interspecific competition abilities. In this paper, we consider the case of different interspecific competition coefficients. The obtained results can be divided into two groups. The first part focuses on the interplay between the diffusion intensities and the interspecific competition coefficients on the dynamics in the weak competition case. We provide some sufficient conditions on coexistence and competitive exclusion. The second part considers the dynamics for fixed diffusion coefficients and varying competition coefficients. We not only provide a complete classification on the dynamics when one of the interspecific competition coefficients is small, but also show the multiple coexistence periodic patterns by applying the bifurcation method.

Two species nonlocal diffusion systems with free boundaries

<p style='text-indent:20px;'>We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [<xref ref-type="bibr" rid="b7">7</xref>], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.</p>

Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$. Specially, when the competition rate exerted on one species converges to zero, then for any $c>c_0$, where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.

Stability analysis of a stage‐structure model with spatial heterogeneity

A stage‐structure competition‐diffusion model with spatial heterogeneity is investigated in this paper. Under some suitable assumptions, the existence of spatially non‐homogeneous steady‐state solutions is established by investigating eigenvalue problems with indefinite weight and employing Lyapunov‐Schmidt reduction. The stability of spatially nonhomogeneous steady‐state solutions is obtained by analyzing the set of the spectrum of the associated infinitesimal generator. In particular, the global stability of the steady‐state solutions is described in weak competition case.

Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary

In this paper, we investigate a two-species competitive chemotaxis Keller-Segel system equipped with a free boundary. The local existence and global existence of classical solutions are established, and then the asymptotic behavior of the solutions is described. Some spreading-vanishing dichotomies have been obtained both for the strong competition case: 0<a1<1≤a2, and weak competition case: 0<a1, a2<1. Finally, an obstructed spreading case has been found, which could happen due to the effect of the chemotaxis.

Global dynamics and complex patterns in Lotka-Volterra systems: The effects of both local and nonlocal intraspecific and interspecific competitions

We consider a Lotka-Volterra system with both local and nonlocal intraspecific and interspecific competitions, where nonlocal competitions depend on both spatial and temporal effects in a general form. Firstly, global stability of two constant semi-trivial equilibria and global convergence of the coexistence equilibrium are derived by using the functional and energy method, which implies that strengths of nonlocal intraspecific competitions have great effects on these global dynamics but the nonlocal interspecific competitions not and extends global results of Gourley and Ruan (2003) [11]. Secondly, global attracting region of each constant semi-trivial equilibrium is limited by its environment capacity regardless of the distinction of local and nonlocal intraspecific competitions. Thirdly, in the weak competition case, the coexistence equilibrium becomes Turing unstable when the kernels are chosen as generally distributed delay functions in temporal and the nonlocal intraspecific competitions are suitably strong. Additionally, spatially homogeneous and inhomogeneous periodic solutions are found numerically.

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

&lt;p style='text-indent:20px;'&gt;We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.&lt;/p&gt;

Entire solutions of diffusive Lotka-Volterra system

This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system on the real line. We prove the existence of some entire solutions that are asymptotic, as t→−∞, to a traveling wave consisting of a single species. Depending on system parameters, these entire solutions evolve into two ore more stacked invasion waves as t→+∞. Our results cover both the weak and strong competition case. In the weak competition case, the existence of a class of entire solutions that forms a 4-dimensional manifold is proved.

Global asymptotic stability in a parabolic–elliptic chemotaxis system with competitive kinetics and loop

ABSTRACT This paper deals with the initial-boundary value problem for the two-species chemotaxis-competition system with two signals under the homogeneous Neumann boundary condition, where , , , , , , and is a smooth bounded domain. If , , and are sufficiently small, then the system possesses a globally bounded classical solution for any suitably regular initial data . Furthermore, by constructing some appropriate functionals, it is shown that For the weak competition case, if are sufficiently large, then the solution converges to exponentially as . For the strong-weak competition case, if is sufficiently large, then the solution converges to with exponential decay when , and with algebraic decay when .

Global dynamics of the diffusive Lotka–Volterra competition model with stage structure

The global asymptotic behavior of the classical diffusive Lotka–Volterra competition model with stage structure is studied. A complete classification of the global dynamics is given for the weak competition case. It is shown that under otherwise same conditions, the species with shorter maturation time prevails. The method is also applied to the global dynamics of another competition models with time delays.

The impacts of dispersal on the competition outcome of multi-patch competition models.

In this paper, we study a two-species competition model over patchy environments. One species is assumed to disperse randomly between patches with a constant dispersal delay. We show that the dispersal does not affect the stability and instability of the homogeneous coexistence equilibrium in two configurations (fully connected configuration and ring-structured configuration) of an arbitrary number of patches. For the weak competition case, we show that the homogeneous coexistence equilibrium is the unique coexistence equilibrium and both species can coexist. However, for the strong competition case, we show that the homogeneous coexistence equilibrium is unstable, in addition, small dispersal rate can induce multiple coexistence equilibria and the dispersal (including the dispersal rate and the dispersal delay) does have impacts on determining the competition outcome and can induce multi-stability. As a result, transient coexistence of both species can be observed in all patches, and long-term coexistence of both species in some patches, though not in all patches, becomes possible.

Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case

In this paper, we will prove the uniqueness of traveling front solutions with critical and noncritical speeds, connecting the origin and the positive equilibrium, for the classical competitive Lotka-Volterra system with diffusion in the weak competition, which partially answers the open problem presented by Tang and Fife in [ 17 ]. In fact, once these traveling front solutions have the same wave speed and the same asymptotic behavior at $ξ = ±∞$, they are unique up to translation.

Dynamics of a diffusive competition model in spatially heterogeneous environment

We discuss the effects of dispersal rate and spatial heterogeneity on population dynamics via a diffusive Lotka–Volterra competition model under homogeneous Dirichlet boundary conditions. By the theory of monotone dynamical system, we show that in the weak competition case, for some ranges of dispersal rate of competing species 1, if dispersal rate of competing species 2 is sufficiently small, there exists a unique coexistence steady state of the model which is globally asymptotically stable. Moreover, for certain ranges of dispersal rate of competing species 1, we derive the limit behavior of the coexistence steady state as dispersal rate of competing species 2 tends to zero. On the other hand, we demonstrate that for some larger scopes of dispersal rate of competing species 1, if dispersal rate of competing species 2 is small enough, there is no coexistence steady state, whereas one of the semi-trivial steady states of the model is globally asymptotically stable.

Global existence and stability in a two-species chemotaxis system

<p style='text-indent:20px;'>This paper deals with the following two-species chemotaxis system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{v}_{t}}=\Delta v-v+h(w), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{z}_{t}}=\Delta z-z+h(u),&amp; x\in \Omega ,t&gt;0, &amp; \\\end{array} \right.$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M393">\begin{document}$Ω\subset\mathbb{R}^{n}$\end{document}</tex-math></inline-formula> with smooth boundary. The parameters in the system are positive and the signal production function <i>h</i> is a prescribed <i>C</i>¹-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium <inline-formula><tex-math id="M394">\begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document}</tex-math></inline-formula> may be globally attractive in the weak competition case (i.e., <inline-formula><tex-math id="M395">\begin{document}$0 &lt; a_{1}, a_{2} &lt; 1$\end{document}</tex-math></inline-formula>), while the constant stationary solution (0, <i>h</i>(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., <inline-formula><tex-math id="M396">\begin{document}$a_{1}&gt;1&gt;a_{2}&gt;0$\end{document}</tex-math></inline-formula>). In the fully strong competition case (i.e. <inline-formula><tex-math id="M397">\begin{document}$a_{1}, a_{2}&gt;1$\end{document}</tex-math></inline-formula>), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, <i>h</i>(1), 1, 0) and (1, 0, 0, <i>h</i>(1)) and the instability of the positive spatially homogeneous <inline-formula><tex-math id="M398">\begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document}</tex-math></inline-formula>. The matter which species ultimately wins out depends crucially on the starting advantage each species has.

Dynamics for a nonlocal competition system with a free boundary

ABSTRACTIn this paper, we investigate a nonlocal reaction–diffusion competition model with a free boundary and discuss the long time behavior of species. The main objective is to understand the effect of the nonlocal term in the form of an integral convolution on the dynamics of competing species. Specially, for the weak competition case, when spreading occurs, we provide some sufficient conditions to prove that two competing species stabilize at a positive constant equilibrium state. Furthermore, for the case of successful spreading, we estimate the asymptotic spreading speed of the free boundary.

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.

Entire solutions for the classical competitive Lotka–Volterra system with diffusion in the weak competition case

In this paper we are concerned with the construction of entire solutions for the classical competitive Lotka–Volterra system with diffusion in the weak competition. For this purpose we firstly analyze the asymptotic behavior of traveling front solutions for this system connecting the origin and the positive equilibrium. Then, by using two different ways to construct pairs of coupled super–sub solutions of this system, we obtain two different kinds of entire solutions. The construction of the first kind of entire solutions is based on these fronts, and their reflects as well as the solutions of the system without diffusion. One component of the solution starts from 0 at t≈−∞, and as t goes to +∞, the two components of the entire solution will eventually stay in a conformed region. Another kind of entire solutions is related to some traveling front solutions of scalar equations.

An evolutional free-boundary problem of a reaction–diffusion–advection system

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) &lt; ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

Qualitative behavior of solutions to cross-diffusion systems from population dynamics

A general class of cross-diffusion systems for two population species in a bounded domain with no-flux boundary conditions and Lotka–Volterra-type source terms is analyzed. Although the diffusion coefficients are assumed to depend linearly on the population densities, the equations are strongly coupled. Generally, the diffusion matrix is neither symmetric nor positive definite. Three main results are proved: the existence of global uniformly bounded weak solutions, their convergence to the constant steady state in the weak competition case, and the uniqueness of weak solutions. The results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified Shigesada–Kawasaki–Teramoto population models as a special case. The proofs are based on entropy methods, which rely on convexity properties of suitable Lyapunov functionals.

The minimal habitat size for spreading in a weak competition system with two free boundaries

In this paper, we focus on the dynamics for a Lotka–Volterra type weak competition system with two free boundaries, where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species, respectively. In the weak competition case, the dynamics of this model can be classified into four cases, which forms a spreading–vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Some sufficient conditions for spreading and vanishing are established. Also, when spreading occurs, some rough estimates for spreading speed and the long-time behavior of solutions are established.