Two-species Competitive System Research Articles

Optimal boundary control for a competitive population system with size structure and time delay in a polluted environment

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

Global boundedness and large time behavior of a two-species competition system with indirect signal consumption

Global boundedness and large time behavior of a two-species competition system with indirect signal consumption

Propagation dynamics of two-species competition models in a periodic discrete habitat

This paper is devoted to the study of the propagation dynamics of a Lotka-Volterra two-species competition system in a periodic discrete habitat. Under appropriate assumptions, we show that a semi-trivial equilibrium is globally stable for the spatially periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for the spatially periodic rightward traveling waves. We further obtain sufficient conditions for the linear determinacy of the rightward spreading speed. Finally, we apply these results to a specific model of two-species competition and conduct numerical simulations for the spreading speed and traveling waves.

Bistable pulsating wave of a competition model in rapidly varying media and its homogenization limit

This paper is concerned with the existence of bistable pulsating wave for a two-species competition system in rapidly periodically varying media and its homogenization limit as the spatial period tends to zero. Assuming that the homogenized system admits a bistable traveling wave, we first show the existence of pulsating wave for all small periods and determine the homogenization limit of the wave speed. Furthermore, in the case where the homogeneous wave is not stationary, we investigate the effects of spatial heterogeneity on the wave speeds. Our estimates indicate that the deviation of the wave speeds in comparison to the homogeneous one is at most linearly along with small period in the general case, and is of second order where the heterogeneity appears only in the diffusion coefficients.

A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment

<p style='text-indent:20px;'>In this paper, we investigate a reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function <inline-formula><tex-math id="M1">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when <inline-formula><tex-math id="M2">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when <inline-formula><tex-math id="M3">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when <inline-formula><tex-math id="M4">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when <inline-formula><tex-math id="M5">\begin{document}$ m(x) $\end{document}</tex-math></inline-formula> satisfies positive everywhere condition and two species spreading occurs.</p>

On a two-species competitive predator-prey system with density-dependent diffusion.

This paper deals with a two-species competitive predator-prey system with density-dependent diffusion, i.e., \begin{eqnarray*}\label{1a} \left\{ \begin{split}{}&u_t=\Delta (d_{1}(w)u)+\gamma_{1}uF_{1}(w)-uh_{1}(u)-\beta_{1}uv,&(x,t)\in \Omega\times (0,\infty),\\&v_t=\Delta (d_{2}(w)v)+\gamma_{2}vF_{2}(w)-vh_{2}(v)-\beta_{2}uv,&(x,t)\in \Omega\times (0,\infty),\\&w_t=D\Delta w-uF_{1}(w)-vF_{2}(w)+f(w),&(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, with the nonnegative initial data $\left( {u_{0}, v_{0}, w_{0}} \right) \in (W^{1,p}(\Omega))^{3}$ with $p>2$, where the parameters $D,\gamma_{1},\gamma_{2},\beta_{1},\beta_{2}>0$, $d_{1}(w)$ and $d_{2}(w)$ are density-dependent diffusion functions, $F_{1}(w)$ and $F_{2}(w)$ are commonly called the functional response functions accounting for the intake rate of predators as the functions of prey density, $h_{1}(u)$ and $h_{2}(v)$ represent the mortality rates of predators, and $f(w)$ stands for the growth function of the prey. First, we rigorously prove the global boundedness of classical solutions for the above general model provided that the parameters satisfy some suitable conditions by means of $L^{p}$-estimate techniques. Moreover, in some particular cases, we establish the asymptotic stabilization and precise convergence rates of globally bounded solutions under different conditions on the parameters by constructing some appropriate Lyapunov functionals. Our results not only extend the previous ones, but also involve some new conclusions.

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>

When and Where We Can Expect to See Early Warning Signals in Multispecies Systems Approaching Tipping Points: Insights from Theory.

AbstractEarly warning signals (EWSs) have the potential to predict tipping points where catastrophic changes occur in ecological systems. However, EWSs are plagued by false negatives, leading to undetected catastrophes. One reason may be because EWSs do not occur equally for all species in a system, so whether and how strongly EWSs are detected depends on which species is being observed. Here, we illustrate how the strength of EWSs is determined by each species' relationship to properties of the noise, the system's response to that noise, and the occurrence of critical slowing down (the dynamical phenomenon that gives rise to EWSs). Using these relationships, we present general rules for maximizing EWS detection in ecological communities. We find that for two-species competitive and mutualistic systems, one should generally monitor the species experiencing smaller intraspecific effects to maximize EWS performance, while in consumer-resource systems, one should monitor the species imposing the smaller interspecific effects. These guidelines appear to hold for at least some larger communities as well. We close by extending the theoretical basis for our rules to systems with any number of species and more complex forms of noise. Our findings provide important guidance on how to monitor systems for EWSs to maximize detection of tipping points.

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.

Dynamics of Competitive Systems with Diffusion Between Source-Sink Patches.

This paper considers two-species competitive systems with one-species' diffusion between patches. Each species can persist alone in the corresponding patch (a source), while the mobile species cannot survive in the other (a sink). Using the method of monotone dynamical systems, we give a rigorous analysis on persistence of the system, prove local/global stability of the equilibria and show new types of bi-stability. These results demonstrate that diffusion could lead to results reversing those without diffusion, which extend the principle of competitive exclusion: Diffusion could lead to persistence of the mobile competitor in the sink, make it reach total abundance larger than if non-diffusing and even exclude the opponent. The total abundance is shown to be a distorted function (surface) of diffusion rates, which extends both previous theory and experimental observations. A novel strategy of diffusion is deduced in which the mobile competitor could drive the opponent into extinction, and then approach the maximal abundance. Initial population density and diffusive asymmetry play a role in the competition. Our work has potential applications in biodiversity conservation and economic competition.

Traveling wave fronts in a diffusive and competitive Lotka-Volterra system

&lt;p style='text-indent:20px;'&gt;In this paper, we consider a two-species competitive and diffusive system with nonlocal delays. We investigate the existence of traveling wave fronts of the system by employing linear chain techniques and geometric singular perturbation theory. The existence of the traveling wave fronts analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a six-dimensional system without delay.&lt;/p&gt;

Positive solution branches of two-species competition model in open advective environments

The effect of competition is an important topic in spatial ecology. This paper deals with a general two-species competition system in open advective and inhomogeneous environments. At first, the critical values on the interspecific competition coefficients are established, which determine the stability of semi-trivial steady states. Secondly, by analyzing the nonexistence of coexistence steady states and using the theory of monotone dynamical system, we find that the competitive exclusion principle holds if one of the interspecific competition coefficients is large and the other is in a certain range. Thirdly, in terms of these critical values, the structure and direction of bifurcating branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis. Finally, we show that multiple coexistence occurs under certain regimes.

Population abundance of two-patch competitive systems with asymmetric dispersal.

This paper considers two-species competitive systems with two patches, in which one of the species can move between the patches. One patch is a source where each species can persist alone, but the other is a sink where the mobile species cannot survive. Based on rigorous analysis on the model, we show global stability of equilibria and bi-stability in the first octant Int[Formula: see text]. Then total population abundance of each species is explicitly expressed as a function of dispersal rates, and the function of the mobile species displays a distorted surface, which extends previous theory. A novel prediction of this work is that appropriate dispersal could make each competitor approach total population abundance larger than if non-dispersing, while the dispersal could reverse the competitive results in the absence of dispersal and promote coexistence of competitors. It is also shown that intermediate dispersal is favorable, large or small one is not good, while extremely large or small dispersal will result in extinction of species. These results are important in ecological conservation and management.

Dynamic behaviors of a two-species competitive system with nonlinear inter-inhibition terms and infinite delay

Sufficient conditions are obtained for the global attractivity of the positive equilibrium and boundary equilibria of the following two-species competitive system with nonlinear inter-inhibition terms \[ \frac{dy_1(t)}{dt} = y_1(t)\Bigg[r_1-a_1y_1-\frac{b_1\int_{-\infty}^{t}K_2(t-s)y_2(s)ds}{1+\int_{-\infty}^{t}K_2(t-s)y_2(s)ds}\Bigg],\ \ \ \frac{dy_2(t)}{dt} = y_2(t)\Bigg[r_2-a_2y_2-\frac{b_2\int_{-\infty}^{t}K_1(t-s)y_1(s)ds}{1+\int_{-\infty}^{t}K_1(t-s)y_1(s)ds}\Bigg], \] where \(r_i, a_i, b_i, i=1,2\) are all positive constants. The results not only improve but also complement the main results of [B. G. Chen, J. Math. Computer Sci., \(\bf 16 \) (2016), 481--494].

Incorporating two coupling noises into a nonlinear competitive system with saturation effect

In this paper, a novel stochastic two-species competitive system with saturation effect is formulated, in which there exist two noise resources and their coupling mode is relatively complex and every noise source has effect on the intrinsic growth rates of both species. With the help of some suitable Lyapunov functions, sufficient conditions for stochastic permanence are established as exponential extinction, extinction, permanence in time average and asymptotic pathwise estimation of system. The effect of coupling noise on the asymptotic behaviors of the populations is shown.

Dynamical analysis of a two-species competitive system with state feedback impulsive control

In this paper, three competitive systems with different kinds of state-dependent control are presented and investigated. The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory, and the stability of the order-1 periodic solution of each system is also given. Besides, sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincaré criterion, respectively. Finally, numerical simulations are carried out to verify the theoretical results.

Extinction and stationary distribution of a competition system with distributed delays and higher order coupled noises.

A stochastic two-species competition system with saturation effect and distributed delays is formulated, in which two coupling noise sources are incorporated and every noise source has effect on two species' intrinsic growth rates in nonlinear form. By transforming the two-dimensional system with weak kernel into an equivalent four-dimensional system, sufficient conditions for extinction of two species and the existence of a stationary distribution of the positive solutions to the system are obtained. Our main results show that the two coupling noises play a significant role on the long time behavior of system.

Evolutionary Substitution and Replacement in N-Species Lotka–Volterra Systems

The successful invasion of a multi-species resident system by mutants has received a great deal of attention in theoretical ecology but less is known about what happens after the successful invasion. Here, in the framework of Lotka–Volterra (LV) systems, we consider the general question where there is one resident phenotype in each species and the evolutionary outcome after invasion remains one phenotype in each species, but these include all the mutant phenotypes. In the first case, called evolutionary substitution, a mutant appears in only one species, the resident phenotype in this species dies out, and the mutant coexists with the original phenotypes of the other species. In the second case, called evolutionary replacement, a mutant appears in each species, all resident phenotypes die out, and the evolutionary outcome is coexistence among all the mutant phenotypes. For general LV systems, we show that dominance of the resident phenotype by the mutant (i.e. the mutant is always more fit) in each species where the mutant appears leads to evolutionary substitution/replacement. However, it is shown by example that, when dominance is weakened to only assuming the average fitness of the mutants is greater than the average for the resident phenotype, the residents may not die out. We also show evolutionary substitution occurs in two-species competitive LV systems when the initial invasion of the resident system (respectively, of the new coexistence system) is successful (respectively, unsuccessful). Moreover, if sequential evolutionary substitution occurs for either order that the two mutant phenotypes appear (called historically independent replacement), then it is shown evolutionary replacement occurs using a generalization of the dominance argument.

Dynamic of a nonautonomous two-species impulsive competitive system with infinite delays

Abstract In this paper, we consider a nonautonomous two-species impulsive competitive system with infinite delays. By the impulsive comparison theorem and some mathematical analysis, we investigate the permanence, extinction and global attractivity of the system, as well as the influence of impulse perturbation on the dynamic behaviors of this system. For the logistic type impulsive equation with infinite delay, our results improve those of Xuxin Yang, Weibing Wang and Jianhua Shen [Permanence of a logistic type impulsive equation with infinite delay, Applied Mathematics Letters, 24(2011), 420-427]. For the corresponding nonautonomous two-species impulsive competitive system without delays, we discuss its permanence, extinction and global attractivity, which weaken and complement the results of Zhijun Liu and Qinglong Wang [An almost periodic competitive system subject to impulsive perturbations, Applied Mathematics and Computation, 231(2014), 377-385].

On the dynamic consistency of a two-species competitive discrete system with toxicity: Local and global analysis

Discretization of a highly nonlinear system is inevitable for good approximation of its solutions due to lack of its analytical solution. However, a discrete model constructed by standard discretization technique frequently produces spurious dynamics which are not observed in its constituent continuous model. It is therefore important to construct a discrete model that produces the exact dynamics of its continuous counterpart. Here we transform a continuous-time two species nonlinear competition model with the effect of toxic substances into a discrete-time system to show its dynamic consistencies with the continuous system by employing nonstandard finite difference scheme. Linear stability property of each fixed point of the derived model has been proven to be identical with those of the continuous system and the dynamics has been shown to be independent of the step-size. Moreover, we prove the global stability of the coexisting equilibrium point of our discrete model. Comparative simulation results demonstrate that the standard discrete system shows complex dynamics, like flip bifurcation and chaos depending on the step-size, but the proposed discrete model does not show any such spurious dynamics.