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Register a new userAsymptotic spreading of KPP reactive fronts in heterogeneous shifting environments
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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 these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable $s=x/t$. In terms of the standard Fisher-KPP equation, our results leads to a new class of asymptotically homogeneous environments which share the same spreading speed with the corresponding homogeneous environments.
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We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy [ lim_{tto -infty} sup_{|x|le c|t|} u(t,x) = 0 qquad text{for each $c<2sqrt{f(0)},$,} ] and we consider here those satisfying [ lim_{tto -infty} sup_{|x|le c|t|} u(t,x) = 0 qquad text{for some $c>2sqrt{f(0)},$.} ] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.
We establish spreading properties of the Lotka-Volterra competition-diffusion system. When the initial data vanish on a right half-line, we derive the exact spreading speeds and prove the convergence to homogeneous equilibrium states between successive invasion fronts. Our method is inspired by the geometric optics approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. Our main result settles an open question raised by Shigesada et al. in 1997, and shows that one of the species spreads to the right with a nonlocally pulled front.
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. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed.
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 $c_0$.
In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(theta_tomega)u(1-u),quad xinR, eqno(1) $$ where $omegainOmega$, $(Omega, mathcal{F},mathbb{P})$ is a given probability space, $theta_t$ is an ergodic metric dynamical system on $Omega$, and $a(omega)>0$ for every $omegainOmega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),quad xinR, eqno(2) $$ where $a_0(t)$ is a positive locally Holder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1) and (2). In this second part of the series, we investigate the existence and stability of transition fronts of (1) and (2). We first study the transition fronts of (1). Under some proper assumption on $a(omega)$, we show the existence of random transition fronts of (1) with least mean speed greater than or equal to some constant $underline{c}^*$ and the nonexistence of ranndom transition fronts of (1) with least mean speed less than $underline{c}^*$. We prove the stability of random transition fronts of (1) with least mean speed greater than $underline{c}^*$. Note that it is proved in the first part that $underline{c}^*$ is the infimum of the spreading speeds of (1). We next study the existence and stability of transition fronts of (2). It is not assumed that $a(omega)$ and $a_0(t)$ are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.
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