We investigate the long term behavior for a class of competition-diffusion systems of Lotka-Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially segregated state satisfying some variational inequalities.
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 …
We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. The existence and uniqueness of the solution are shown. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some free boundary problems.
One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.
This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, w…
In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion perturbations. This criticality is showcased by proving that adding an extra cross-diffusion term that breaks the symmetry of the population pressure in the system leads to completely different behaviours, namely segregation or mixing, depending on th…
Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations \[ -Δu_i(x)=-μu_i (x)\displaystyle{\sum_{j\neq i}} u_j (x) \quad i=1,...,k \] in a domain $D$ with appropriate boundary conditions. Any $u_i$ represents a population density and the parameter $μ$ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as $μ\longrightarrow+\infty$ on a pla…
This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction-diffusion systems with strongly competitive interactions of Lotka-Volterra type, when the matrix of the inter-specific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists in a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, st…
The existence of a global martingale solution to a cross-diffusion system with multiplicative Wiener noise in a bounded domain with no-flux boundary conditions is shown. The model describes the dynamics of population densities of different species due to segregation cross-diffusion effects. The diffusion matrix is generally neither symmetric nor positive semidefinite. This difficulty is overcome by exploiting the Rao entropy structure. The existence proof uses a stochastic Galerkin method, uniform estimates from the Rao entropy inequality, and the Skorokhod--Jakubowski theorem. Furthermore, an…
We study a diffusive Lotka-Volterra competition system with advection under Neumann boundary conditions. Our system models a competition relationship that one species escape from the region of high population density of their competitors in order to avoid competitions. We establish the global existence of bounded classical solutions for the system in one-dimensional domain. For multi-dimensional domains, globally bounded classical solutions are obtained for a parabolic-elliptic system under proper assumptions on the system parameters. These global existence results make it possible to study bo…
We consider the reaction-diffusion competition system in the so-called {\it critical competition case}. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the non-existence of {\it ultimately monotone} traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the "faster" species excludes the "slower" one (with a known {\it spreading speed}), but also provide a sharp description of the profil…
In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.
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.
This paper investigates the global well-posedness of a class of reaction-advection-diffusion models with nonlinear diffusion and Lotka-Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic-elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka-Volterra competition systems.
In this paper, we study a diffusive Lotka-Volterra competition model under homogeneous Dirichlet boundary conditions. We shall discuss the effects of dispersal rate and spatial heterogeneity on population dynamics. More precisely, we establish the main results about the global asymptotic stability of semitrivial as well as coexistence steady states.
Recently, J.-S. Guo, C.-H. Wu (Nonlinearity, 28(2015), 1-27) and C.-H. Wu (J. Differential Equations, 259(3)(2015), 873-897) studied a two-species competition-diffusion model with two free boundaries. The existence, uniqueness and long time behavior of global solution were established. In this note we still discuss the long time behavior of global solution and provide some new results and simpler proofs.
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.
This paper is devoted to the study of propagation phenomena for a Lotka-Volterra reaction-advection-diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trival steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reacti…
Numerical Algorithms for a Variational Problem of the Spatial Segregation of Reaction-Diffusion Systems
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Institute of Mathematics, National Academy of Science of Armenia（亚美尼亚国家科学院数学研究所）；Faculty of Sciences, Persian Gulf University（波斯湾大学理学院）
This work deals with the aggregation diffusion equation \[\partial_t ρ= Δ_pρ+ λdiv((K_a*ρ)ρ),\] where $K_a(x)=\frac{x}{|x|^a}$ is an attraction kernel and $Δ_p$ is the so called $p$-Laplacian. We show that the domain $a < p(d+1)-2d$ is subcritical with respect to the competition between the aggregation and diffusion by proving that there is existence unconditionally with respect to the mass. In the critical case we show existence of solution in a small mass regime for an $L\ln L$ initial condition.
Propagation Direction of Bistable Traveling Fronts in the Lotka--Volterra Competition--Diffusion System
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）
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…
Sharp Thresholds for Linear Determinacy and Zero Wave Speed in Lotka-Volterra Competition-Diffusion Systems
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然博物馆）；Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）
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…
In this work we deal with the local in time propagation of chaos without cut-off for some two dimensional fractional Keller Segel equations. More precisely the diffusion considered here is given by the fractional Laplacian operator $-(-Δ)^{\frac{a}{2}}$ with $a \in (1,2)$ and the singularity of the interaction is of order $|x|^{1-α}$ with $α\in ]1,a]$. In the case $α\in (1,a)$ we give a complete propagation of chaos result, proving the $Γ$-l.s.c property of the fractional Fisher information, already known for the classical Fisher information, using a result of Mischler and Hauray. In the fair …
In this paper, we study the dynamics of a two-species competition model with two different free boundaries in heterogeneous time-periodic environment, where the two species adopt a combination of random movement and advection upward or downward along the resource gradient. We show that 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. Rough estimates of the asymptotic spreading speed of free boundaries and the long time behavior…
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…
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 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 …
Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays
