Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction-diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction-diffusion systems and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. As an application, our results are applied to a partially cooperative system describing interactions between ungulates and grass.
The theory of traveling waves and spreading speeds is developed for time-space periodic monotone semiflows with monostable structure. By using traveling waves of the associated Poincaré maps in a strong sense, we establish the existence of time-space periodic traveling waves and spreading speeds. We then apply these abstract results to a two species competition reaction-advection-diffusion model. It turns out that the minimal wave speed exists and coincides with the single spreading speed for such a system no matter whether the spreading speed is linearly determinate. We also obtain a set of s…
This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. In a previous paper, we established that these systems admit traveling wave solutions. It is then natural to investigate their asymptotic distributions. This is the main object of this paper. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.
We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. The problem is monostable in nature, resembling the well known Fisher-KPP equation. Such a system covers various models arising from mathematical biology, with the Fisher-KPP equation as the simplest special case, where a "spreading-vanishing" dichotomy is known to govern the long time dynamical behaviour. The q…
This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the followin…
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vecto…
This paper is concerned with the asymptotic spreading of a Lotka-Volterra cooperative system. By using the theory of asymptotic spreading of nonautonomous equations, the asymptotic speeds of spreading of unknown functions formulated by a coupled system are estimated. Our results imply that the asymptotic spreading of one species can be significantly fastened by introducing a mutual species, which indicates the role of cooperation described by the coupled nonlinearities.
This paper considers the spreading speed of cooperative nonlocal dispersal system with irreducible reaction functions and non-uniform initial data. Here the non-uniformity means that all components of initial data decay exponentially but their decay rates are different. It is well-known that in a monostable reaction-diffusion or nonlocal dispersal equation, different decay rates of initial data yield different spreading speeds. In this paper, we show that due to the cooperation and irreducibility of reaction functions, all components of the solution with non-uniform initial data will possess a…
This paper deals with the existence of traveling fronts guided by the medium for a KPP reaction-diffusion equation coming from a model in population dynamics in which there is spatial spreading as well as genetic mutation of a quantitative genetic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of a threshold value on the existence of a nonzero asymptotic profile (a stationary limiting solution). When a nonzero asymptotic profile exists, we prove the existence of a traveling front. This allows us …
This paper is concerned with front-like entire solutions for monostable reactiondiffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Combining a SIS and traveling fronts with different wave speeds and directions, the existence and various qualitative properties of entire solutions are then established using comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish some comparison arguments for the three…
Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatio-temporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes and show that under the assumption of maintenance of spatially homogeneous total population the dynamics of the frequency of one of the genotypes is approximately described by the generalized Fisher-Kolmogorov-Petrovskii-Piskunov…
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 …
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.
We determine the asymptotic spreading speed of an invasive species, which invades the territory of a native competitor, governed by a diffusive competition model with a free boundary in a spherically symmetric setting. This free boundary problem was studied recently in [7] but only rough bounds of the spreading speed was obtained there. We show in this paper that the spreading speed has an asymptotic limit (as time goes to infnity), which is given by a certain traveling wave type system of one space dimension, called a semi-wave. This appears to be the first result that gives the precise asymp…
This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined.…
The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(a-bu),\quad x\in\R\cr 0=v_{xx}- λv+μu,\quad x\in\R, \end{cases} \end{equation} where $χ$, $a$, $b$, $λ$, and $μ$ are positive constants. Assume $b>χμ$. Then if in addition, $\big(1+\frac{1}{2}\frac{(\sqrt{a}-\sqrtλ)_+}{(\sqrt{a}+\sqrt{\la})}\big)χμ{ \leq} b$ holds, it is proved that $c_0^*=2\sqrt a$ is the spreading speed of the solutions of \eqref{abstract-eq1} with nonnegative co…
We study a nonlocal reaction-diffusion-mutation equation modeling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits traveling wave solutions [7]. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2$λ$ *)) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP…
In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the …
We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+μ\left(\int_ΩM(y,z)u(t,x,z)dz-u\right) + u\left(a(y)-\int_ΩK(y,z) u(t,x,z)dz\right) , \end{equation*} where $ u=u(t,x,y) $ stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\inΩ\subset \mathbb R^n$) structure, $ M(y,z) $ is a mutation kernel acting on the phenotypic space, $ a(y) $ is a fitness function and $ K(y,z) $ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases wh…
We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotony of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems. Most notably, when the profile of the coefficient diffusion is increasing we show …
We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [{\em Nonlinearity}, 33 (2020), 4013--4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed $c^*(m,r)$ of sharp wave, and prove that the initial perturbation or the boundary of th…
We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which achieves a traveling profile with speed $c$. For various nonlinear models, the existence of a (possibly measure valued) optimal control is proved, together with necessary conditions for optimality. In the last section we study a case where the wave speed cannot be modified by any control with finite cost. The present analysis is motivated by the recent results in arXiv:2201.0…
We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.
This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the linearization about the unstable state. We prove that initial data consisting of compactly supported perturbations of Heaviside step functions spreads asymptotically with the anomalous speed. The proof makes use of a comparison principle and the …
We investigate a reaction-diffusion-advection equation of the form $u_t-u_{xx}+βu_x=f(u)$ $(t>0,\,0<x<h(t))$ with mixed boundary condition at $x=0$ and a free boundary condition at $x=h(t)$. Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. When $|β|<c…
This paper is concerned with the existence of pulsating traveling fronts for the equation: $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t, x, u)$, (1) where the diffusion matrix $A$, the advection term $q$ and the reaction term $f$ are periodic in $t$ and $x$. We prove that there exist some speeds $c^*$ and $c^{**}$ such that there exists a pulsating traveling front of speed $c$ for all $c\ge c^{**}$ and that there exists no such front of speed $c<c^*$. We also give some spreading properties for front-like initial data. In the case of a KPP-type reaction term, w…
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…
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…
This paper is concerned with the spatially periodic Fisher-KPP equation $u_t=(d(x)u_x)_x+(r(x)-u)u$, $x\in \mathbb{R}$, where $d(x)$ and $r(x)$ are periodic functions with period $L>0$. We assume that $r(x)$ has positive mean and $d(x)>0$. It is known that there exists a positive number $c^*_d(r)$, called the minimal wave speed, such that a periodic traveling wave solution with average speed $c$ exists if and only if $c \geq c^*_d(r)$. In the one-dimensional case, the minimal speed $c^*_d(r)$ coincides with the ``spreading speed'', that is, the asymptotic speed of the propagating front of a so…
Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part II: Pulsating traveling waves
Laboratoire de Mathématiques Appliquées du Havre, Université Le Havre Normandie, EA 3821, FR CNRS 3335（勒阿弗尔应用数学实验室，勒阿弗尔诺曼底大学）；Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University（明治大学高等数学科学研究所）
We study invasion fronts and spreading speeds in two component reaction-diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the sprea…
We consider reaction-diffusion equations $\partial_tu=Δu+f(u)$ in the whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of solutions ranging in the interval $[0,1]$, with general unbounded initial support. Under the hypothesis of the existence of a traveling front connecting $0$ and $1$ with a positive speed, we discuss the existence of spreading speeds and spreading sets, which describe the large-time global shape of the level sets of the solutions. The spreading speed in any direction is expressed as a Freidlin-Gärtner type formula. This formula holds under general …
This paper is concerned with spatial spreading dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species will become extinct in the habitat once the speed of the shifting habitat edge $c>c^*(\infty)$, however if $c<c^*(\infty)$, the species will persist and spread along the shifting habitat at an asymptotic spreading speed $c^*(\infty)$, where $c^*(\infty)$ is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed…
This paper studies an epidemic model with nonlocal dispersals. We focus on the influences of initial data and nonlocal dispersals on its spatial propagation. Here the initial data stand for the spatial concentrations of infectious agent and infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies. Two types of initial data decaying to zero exponentially or faster are considered. For the first type, we show that the spreading speeds are two constants whose signs change with the number of elements in some set. Moreover, we find an inter…
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…
