In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refinement in the free boundary condition. While [9], like many works in the literature, investigates the case that unfavourable environment is shifting into the favourable habitat of the concerned species, here we examine the situation that the unfavourable habitat of an invasive species is replaced by a favourable environment with a shifting speed c. We show that a spreading-vanishing dichotomy holds, and there exists a critical speed$c_0$ such that when spreading happens in the case $c < c_0$, the spreading profile is determined by a semi-wave with forced speed c, but when $c \geq c_0$, the spreading profile is determined by the usual semi-wave with speed $c_0$.
We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable $x/t$ and in which the time and space derivatives are coupled together. We will first establish uniqueness results for …
We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin \cite{DL}, where a spreading-vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed $c_0>0$. Here we consider the situation that part of such an environment becomes unfavourable,…
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 …
Propagation dynamics for nonlocal dispersal predator-prey systems in shifting habitats: A Hamilton-Jacobi approach
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Department of Mathematics, University of Miami（迈阿密大学数学系）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
A free boundary problem for spreading of an invasive species in the territory of a native competitor under shifting climate
Ho Chi Minh City University of Technology (HCMUT)（胡志明市理工大学）；Vietnam National University, Ho Chi Minh City（越南国立大学胡志明市分校）；FPT University（FPT 大学）；University of Economics and Law, Ho Chi Minh City（胡志明市经济与法律大学）
The current paper is concerned with the persistence and spreading speeds of the following Keller-Segel chemoattraction system in shifting environments, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(r(x-ct)-bu),\quad x\in\R\cr 0=v_{xx}- νv+μu,\quad x\in\R, \end{cases} \end{equation} where $χ$, $b$, $ν$, and $μ$ are positive constants, { $c\in\R$ }, $r(x)$ is Hölder continuous, bounded, $r^*=\sup_{x\in\R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and $r(x)$ satisfies either $r(-\infty)<0<r(\infty)$, or $r(\pm\infty)<0$. Assume $b>χμ$ and $b\ge \big(1+…
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…
Many species see their range shifted poleward in response to global warming and need to keep pace in order to survive. To understand the effect of climate change on species ranges and its consequences on population dynamics, we consider a space-time heterogeneous reaction-diffusion equation in dimension 1, whose unknown~$u (t,x)$ stands for a population density. More precisely, the environment consists of two patches moving with a constant climate shift speed $c \geq 0$: in the invading patch $\{ t >0 , \, x \in \R \, | \ x < ct \}$ the growth rate is negative and, in the receding patch $\{ t …
Spreading--vanishing dynamics and speed selection in a free-boundary competition model under a shifting climate
Ho Chi Minh City University of Technology (HCMUT)（胡志明市理工大学）；Vietnam National University, Ho Chi Minh City（越南胡志明市国家大学）；FPT University（FPT大学）；University of Economics and Law（经济与法律大学）
Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analyzed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates c…
In this paper, we investigate so-called forced wave solutions of a three components reaction-diffusion system from population dynamics. Our system involves three species that are respectively two competing preys and one predator; moreover, the competition between both preys is strong, i.e. in the absence of the predator, one prey is driven to extinction and the other survives. Furthermore, our problem includes a spatio-temporal heterogeneity in a moving variable that typically stands as a model for climate shift. In this context, forced waves are special stationary solutions which are expected…
Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain where the reaction term is positive is shifting/contracting at a given speed $c$. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term,…
We consider a three-species predator-prey system involving two competing predators and one prey. The species diffuse with nonlocal dispersal kernels with possibly non-compact support, and they interact in a heterogeneous environment moving with a positive forced speed such that the environment is favorable to the prey in the absence of predators far ahead of the shifting boundary and it is unfavorable far behind. Such systems arise in the modeling of population dynamics under the effect of a shifting environment, such as climate change. We show on the one hand the existence of waves connecting…
We study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and reveal some fundamental differences from its one dimensional version considered in \cite{cdjfa} recently. Technically, this high dimensional problem is much more difficult to treat since it involves two kernel functions which arise from the original kernel function $J(|x|)$ in rather implicit ways. By introducing new techniques, we are able to determine the long-time dynamics of the model, including firstly finding the threshold condition on the kernel function that governs the onset…
Range expansion and range shifts are crucial population responses to climate change. Genetic consequences are not well understood but are clearly coupled to ecological dynamics that, in turn, are driven by shifting climate conditions. We model a population with a deterministic reaction-- diffusion model coupled to a heterogeneous environment that develops in time due to climate change. We decompose the resulting travelling wave solution into neutral genetic components to analyse the spatio-temporal dynamics of its genetic structure. Our analysis shows that range expansions and range shifts und…
We study free boundary problem of Fisher-KPP equation $u_t=u_{xx}+u(1-u),\ t>0,\ ct<x<h(t)$. The number $c>0$ is a given constant, $h(t)$ is a free boundary which is determined by the Stefan-like condition. This model may be used to describe the spreading of a non-native species over a one dimensional habitat. The free boundary $x=h(t)$ represents the spreading front. In this model, we impose zero Dirichlet condition at left moving boundary $x=ct$. This means that the left boundary of the habitat is a very hostile environment and that the habitat is eroded away by the left moving boundary at c…
