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.     

Spatial Dynamics of a Lattice Lotka-Volterra Competition Model with a Shifting Habitat

In this paper, we concern with the spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat. We study the impact of the environmental deterioration rate on the population density under the strong competition condition. Our results show that if the environment deteriorates rapidly, both species will become extinct. However, when the environmental degradation rate is not so fast, the species with slow diffusion will go extinct, while those with fast diffusion will survive. The extinction of species with slow diffusion can be divided into two situations: one is the extinction caused by environmental deterioration faster than its own diffusion speed, the other is the extinction caused by slow diffusion speed under the influence of strong competition.     

Dynamical behaviors of a classical Lotka–Volterra competition–diffusion–advection system

This paper deals with a two-species competition model in a homogeneous advective environment, where two species are subjected to a net loss of individuals at the downstream end. Under the assumption that the advection and diffusion rates of two species are proportional, we give a basic classification on the global dynamics by employing the theory of monotone dynamical system. It turns out that bistability does not happen, but coexistence and competitive exclusion may occur. Furthermore, we present a complete classification on the global dynamics in terms of the growth rates of two species. However, once the above assumption does not hold, bistability may occur. In detail, there exists a tradeoff between growth rates of two species such that competition outcomes can shift between three possible scenarios, including competitive exclusion, bistability and coexistence. These results show that growth competence is important to determine dynamical behaviors.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

The diffusive Lotka–Volterra competition model in fragmented patches I: Coexistence

It is an ecological imperative that we understand how changes in landscape heterogeneity affect population dynamics and coexistence among species residing in increasingly fragmented landscapes. Decades of research have shown the dispersal process to have major implications for individual fitness, species’ distributions, interactions with other species, population dynamics, and stability. Although theoretical models have played a crucial role in predicting population level effects of dispersal, these models have largely ignored the conditional dependency of dispersal (e.g., responses to patch boundaries, matrix hostility, competitors, and predators). This work is the first in a series where we explore dynamics of the diffusive Lotka–Volterra (L–V) competition model in such a fragmented landscape. This model has been extensively studied in isolated patches, and to a lesser extent, in patches surrounded by an immediately hostile matrix. However, little attention has been focused on studying the model in a more realistic setting considering organismal behavior at the patch/matrix interface. Here, we provide a mechanistic connection between the model and its biological underpinnings and study its dynamics via exploration of nonexistence, existence, and uniqueness of the model’s steady states. We employ several tools from nonlinear analysis, including sub-supersolutions, certain eigenvalue problems, and a numerical shooting method. In the case of weak, neutral, and strong competition, our results mostly match those of the isolated patch or immediately hostile matrix cases. However, in the case where competition is weak towards one species and strong towards the other, we find existence of a maximum patch size, and thus an intermediate range of patch sizes where coexistence is possible, in a patch surrounded by an intermediate hostile matrix when the weaker competitor has a dispersal advantage. These results support what ecologists have long theorized, i.e., a key mechanism promoting coexistence among competing species is a tradeoff between dispersal and competitive ability.     

Advection-mediated competition in general environments

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large.     

The spatial behavior of a competition–diffusion–advection system with strong competition

We study a competition–diffusion–advection system for two competitive species inhabiting a spatially heterogeneous environment. We show that they spatially segregate as the interspecific competition rate tends to infinity. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system.     

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.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Hopf bifurcation in time-delayed Lotka–Volterra competition systems with advection

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s).     

Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition

This paper is concerned with the classical two-species Lotka-Volterra diffusion system with strong competition. The sharp dynamical behavior of the solution is established in two different situations: either one species is an invasive one and the other is a native one or both are invasive species. Our results seem to be the first that provide a precise spreading speed and profile for such a strong competition system. Among other things, our analysis relies on the construction of new types of supersolution and subsolution, which are optimal in certain sense.     

Coexistence in a discrete competition model with dispersal

We propose a discrete-time competition model between two populations to study the effects of dispersal upon population interactions. It is assumed that dispersal occurs after reproduction and in synchrony. We first analyse a two-patch single species population model with no interspecific competition. Based on these results, we derive sufficient conditions for population coexistence. It is proved that the system is uniformly persistent and possesses a unique coexisting equilibrium.     

Steady state coexistence solutions of reaction-diffusion competition models

Two species of animals are competing in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? It is natural to say that they can coexist peacefully if their rates of reproduction and self-limitation are relatively larger than those of competition rates. In other words, they can survive if they interact strongly among themselves and weakly with others. We investigate this phenomena in mathematical point of view. In this paper we concentrate on coexistence solutions of the competition model $\left\{ {\begin{array}{*{20}c} {\Delta u + u(a - g(u,v)) = 0,} \\ {\Delta v + v(d - h(u,v)) = 0in\Omega ,} \\ {u|_{\partial \Omega } = v|_{\partial \Omega } = 0.} \\ \end{array} } \right.$ . This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, implicit function theorem and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.     

Global Existence and Finite Time Blow-up for a Reaction-Diffusion System with Three Components

In this paper we consider the system of reaction-diffusion-advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. In strong competition case, we study the influence of competition rates on the long time behavior of solutions and prove that two species spatially segregate as the competition rates become large. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

Global dynamics of a competition model with non-local dispersal I: The shadow system

Equations with non-local dispersal have been widely used as models in biology. In this paper we focus on logistic models with non-local dispersal, for both single and two competing species. We show the global convergence of the unique positive steady state for the single equation and derive various properties of the positive steady state associated with the dispersal rate. We investigate the effects of dispersal rates and inter-specific competition coefficients in a shadow system for a two-species competition model and completely determine the global dynamics of the system. Our results illustrate that the effect of dispersal in spatially heterogeneous environments can be quite different from that in homogeneous environments.     

How Do the Spatial Structure and Time Delay Affect the Persistence of A Polluted Species

In this article, we consider the effects of diffusion and time delay on the species in a polluted environment. Persistence-extinction thresholds are given for population in the toxicant stressed logistic growth model with discrete diffusion or time delay. It is proved that dispersal allows a larger threshold, that is, dispersal can increase the endurance effectiveness of the population subjected to toxicant, and time delay has no effect on the threshold result.     

Traveling wave front for a two-component lattice dynamical system arising in competition models

We study traveling front solutions for a two-component system on a one-dimensional lattice. This system arises in the study of the competition between two species with diffusion (or migration), if we divide the habitat into discrete regions or niches. We consider the case when the nonlinear source terms are of Lotka–Volterra type and of monostable case. We first show that there is a positive constant (the minimal wave speed) such that a traveling front exists if and only if its speed is above this minimal wave speed. Then we show that any wave profile is strictly monotone. Moreover, under some conditions, we show that the wave profile is unique (up to translations) for a given wave speed. Finally, we characterize the minimal wave speed by the parameters in the system.     

Competition in patchy space with cross-diffusion and toxic substances

The main goal of this paper is to continue our investigations of the important system (see [S. Aly, M. Farkas, Competition in patchy environment with cross diffusion, Nonlinear Analysis: Real World Applications 5 (2004) 589–595]), by considering a Lotka–Volterra competitive system affected by toxic substances in two patches in which the per capita migration rate of each species is influenced not only by its own but also by the other one’s density, i.e. there is cross-diffusion present and it is assumed that the individuals of a particular species will initiate toxin production at a rate proportional not only to its own but also to the other one’s density. In the absence of diffusion, we study the conditions of the existence and stability properties of the equilibrium point with toxic substances. For the full general model (with both toxic substances and diffusion) we show that at a critical value of the bifurcation parameter of diffusion the system undergoes a Turing bifurcation and numerical studies show that if the bifurcation parameter of diffusion is increased through a critical value the spatially homogeneous equilibrium loses its stability and two new stable equilibria emerge, i.e., the cross-migration response is an important factor that should not be ignored when a pattern emerges.     

具Ⅱ型Holling功能性反应强耦合椭圆系统的共存问题

本文研究带齐次Dirichlet边界条件的强耦合椭圆系统,首先证明了当食饵和捕食者的扩散率足够大,或者出生率足够小时,系统不存在共存现象,并给出半平凡解存在的充分条件．然后利用Schauder不动点定理,得到强耦合的椭圆问题至少有一个正解存在的充分条件．该条件说明只要捕食者的内部竞争强,物种的交叉扩散相对弱,或者捕获率足够小,物种的交叉扩散相对弱,强耦合系统就至少有一个正解存在．