We investigate the classical two species ODE and PDE Lotka-Volterra competition models, where one of the competitors could potentially go extinct in finite time. We show that in this setting, classical theories and intuitions do not hold, and various counter intuitive dynamics are possible. In particular, the weaker competitor could avoid competitive exclusion, and the slower diffuser may not win. Numerical simulations are performed to verify our analytical findings.
Predicting the evolution of expanding population is critical to control biological threats such as invasive species and virus explosion. In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms and a free boundary. Such a model with a free boundary describes the spreading of new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. We first find conditions on the parameters which guarantee the global existence and boundedn…
Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among $m$ ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for $m=2$ no coexistence of the two species is usually possible, if $m \ge 3$ we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coex…
Geographic ranges of communities of species evolve in response to environmental, ecological, and evolutionary forces. Understanding the effects of these forces on species' range dynamics is a major goal of spatial ecology. Previous mathematical models have jointly captured the dynamic changes in species' population distributions and the selective evolution of fitness-related phenotypic traits in the presence of an environmental gradient. These models inevitably include some unrealistic assumptions, and biologically reasonable ranges of values for their parameters are not easy to specify. As a …
We consider the classical two-species Lotka-Volterra competition-diffusion system in the strong-weak competition case. When the corresponding minimal speed of the traveling waves is not linear determined, we establish the precise asymptotic behavior of the solution of the Cauchy problem in two different situations: (i) one species is an invasive one and the other is a native species; (ii) both two species are invasive species.
Note on a two-species competition-diffusion model with two free boundaries
最近，J. - S. Guo，C.- H. Wu（Nonlinearity，28（2015），1-27）和C.- H. Wu（J.微分方程，259（3）（2015），873-897）研究了具有两个自由边界的两种群竞争扩散模型。证明了整体解的存在性、唯一性和长时间性态。本文还讨论了整体解的长时间性态，给出了一些新的结果和较简单的证明。
This paper deals with the two-species chemotaxis-competition system. About the problem, Bai--Winkler first obtained asymptotic stability under some conditions. Recently, the conditions assumed in the previous work were improved; however, there is a gap between the conditions assumed in the previous works. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior.
We investigate the spreading properties of a three-species competition-diffusion system, which is non-cooperative. We apply the Hamilton-Jacobi approach, due to Freidlin, Evans and Souganidis, to establish upper and lower estimates of spreading speed of the slowest species, in terms of the spreading speed of two faster species, and show that the estimates are sharp in some situations. The spreading speed will first be characterized as the free boundary point of the viscosity solution for certain variational inequality cast in the space of speeds. Its exact formulas will then be derived by solv…
We study a system of PDEs modeling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviors of the system are studied both analytically in the weakly nonlinear …
This paper is concerned with a Lotka-Volterra type competition model with free boundaries in time-periodic environment. One species is assumed to adopt nonlocal dispersal and the other one adopts mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We show that this free boundary problem with more general growth functions admits a unique solution defined for all time. A spreading-vanishing dichotomy is obtained and criteria for spreading and vanishing are provided. Moreover, under the weak competition condition we provide the long-time asymptotic behavior of…
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.
In this article we investigate two free boundary problems for a Lotka-Volterra competition system in a higher space dimension with sign-changing coefficients. One may be viewed as describing how two competing species invade if they occupy an initial region, the other describes the dynamical process of a new competitor invading into the habitat of a native species. For simplicity, it is assumed that the environment is radially symmetric. The main purpose of this article is to understand the asymptotic behavior of competing species spreading via a free boundary. We derive some sufficient conditi…
In this paper, we study three two competing species Lotka-Volterra competition models on finite connected graphs, with Dirichlet, Neumann or no boundary conditions. We get that when time goes to infinity, either one specie extincts while the other becomes surviving or both competing species coexist, which depend crucially on the strengh of species' competitiveness and the size of the initial population under the Neumann boundary condition and the condition that there is no boundary condition. One of our results partially answer a question posed by Slav\'ık in [SIAM J. Appl. Dyn. Syst., 19 (202…
This paper is concerned with some spreading properties of monostable Lotka--Volterra two-species competition--diffusion systems when the initial values are null or exponentially decaying in a right half-line. Thanks to a careful construction of super-solutions and sub-solutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a …
This paper is concerned with a diffusive Lotka-Volterra type competition system with a free boundary in one space dimension. Such a system may be used to describe the invasion of a new species into the habitat of a native competitor. We show that the longtime dynamical behavior of the system is determined by a spreading-vanishing dichotomy, and provide sharp criteria for spreading and vanishing of the invasive species. Moreover, we determine the asymptotic spreading speed of the invasive species when its spreading is successful, which involves two systems of traveling wave type equations, and …
This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy examp…
We consider an interacting particle system with two species under strong competition dynamics between the two species. Then, through the hydrodynamic limit procedure for the microscopic model, we derive a one-phase Stefan type free boundary problem with non-linear diffusion, letting the competition rate divergent. Non-linearity of diffusion comes from a zero-range dynamics for one species while we impose the other species to weakly diffuse according to the Kawasaki dynamics for technical reasons, which macroscopically corresponds to the vanishing viscosity method.
Transient instability and recurrent aggregation in a two-species chemotaxis-competition system under environmental periodicity
Instituto de Matemática Interdisciplinar, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid（马德里康普顿斯大学交叉数学研究所分析数学与应用数学系）；Departamento de Matemática Aplicada, ICAI, Universidad Pontificia de Comillas（科米利亚斯宗座大学ICAI学院应用数学系）
We investigate the spreading behavior of two invasive species modeled by a Lotka-Volterra diffusive competition system with two free boundaries in a spherically symmetric setting. We show that, for the weak-strong competition case, under suitable assumptions, both species in the system can successfully spread into the available environment, but their spreading speeds are different, and their population masses tend to segregate, with the slower spreading competitor having its population concentrating on an expanding ball, say Bt, and the faster spreading competitor concentrating on a spherical …
This paper investigates a reaction-advection-diffusion system modeling interspecific competition between two species over bounded domains. The kinetic terms are assumed to satisfy the Beddington-DeAngelis functional responses. We consider the situation that first species disperse by a combination of random walk and directed movement along the population density of the second species which disperse randomly within the habitat. For multi-dimensional bounded domains, we prove the global existence and boundedness of time-dependent solutions. For one-dimensional finite domains, we study the effect …
This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1- u^{κ_1-1} - a_1 v^{λ_1-1}), &\quad x \in Ω,\ t>0,\\ % v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1- a_2 u^{λ_2-1} - v^{κ_2-1}), &\quad x \in Ω,\ t>0,\\ % 0 = d_3 Δw + αu + βv - h(u,v,w), &\quad x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=γw$ or $h=\frac{1}{|Ω|}\int_Ω(αu+ βv)\,dx$. In the case that $κ_1=λ_1=κ_2=λ_2=2$ and $h=γw$, it is known that sm…
A diffusive Lotka-Volterra competition model is considered for the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion. First it is shown that in a two-species system in which the diffusion coefficients, resource functions and competition rates are all spatially heterogeneous, the positive equilibrium solution is globally asymptotically stable when it exists. Secondly the existence and global asymptotic stability of the positive and semi-trivial equilibrium solutions are obtained for the model with arbitrary number of species unde…
This paper deals with the two-species chemotaxis-competition system $u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u(1 - u - a_1 v)$, $v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v(1 - a_2 u - v)$, $0 = d_3 Δw + αu + βv - γw$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $χ_i$ and $μ_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{χ_i}{μ_i}$ are small. However, the conditions in the above…
In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms in heterogeneous media. We first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we find further conditions on the parameters which establish the persistence of the two species. Then, under the same set of conditions for the persistence of two species, we prove the existence of coexistence states. Finally we prove the extinction phenomena in the s…
We consider a two-species competition-diffusion system involving a small parameter $\varepsilon>0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\varepsilon\approx0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen--Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\varepsilo…
The current paper is devoted to the study of two species competition systems of the form \begin{equation*} \begin{cases} u_t(t,x)= \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in\RR\cr v_t(t,x)= \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in\RR \end{cases} \end{equation*} where $\mathcal{A}u=u_{xx}$, or $(\mathcal{A}u)(t,x)=\int_{\RR}κ(y-x)u(t,y)dy-u(t,x)$ ($κ(\cdot)$ is a smooth non-negative convolution kernel supported on an interval centered at the origin), $a_i(t+T,x)=a_i(t,x)$, $b_i(t+T,x)=b_i(t,x)$, $c_i(t+T,x)=c_i(t,x)$, and $a_i$, $b_i$, and $c_i$ ($i=1,2$) are spa…
We analyze a reaction-diffusion system describing the growth of microbial species in a model of flocculation type that arises in biology. Existence of global classical positive solutions is proved under general growth assumptions, with flocculation and deflocculation rates polynomially bounded above, that guarantee uniform sup norm bounds for all time t obtained by an $L^{p}-$energy functional estimate. We also show finite time blow up can occur when the yield coefficients are large enough. Also, using arguments relying on the spectral and fixed theory, we show persistence and existence of non…
This paper is concerned with the limit, as the interspecific competition rate goes to infinity, of pulsating front solutions in space-periodic media for a bistable two-species competition--diffusion Lotka--Volterra system. We distinguish two important cases: null asymptotic speed and non-null as-ymptotic speed. In the former case, we show the existence of a segregated stationary equilibrium. In the latter case, we are able to uniquely characterize the segregated pulsating front, and thus full convergence is proved. The segregated pulsating front solves an interesting free boundary problem. We …
