On spreading speeds and traveling waves for growth and migration models in a periodic habitat

Abstract

It is shown that the methods previously used by the author [Wei82] and by R. Lui [Lui89] to obtain asymptotic spreading results and sometimes the existence of traveling waves for a discrete-time recursion with a translation invariant order preserving operator can be extended to a recursion with a periodic order preserving operator. The operator can be taken to be the time-one map of a continuous time reaction-diffusion model, or it can be a more general model of time evolution in population genetics or population ecology in a periodic habitat. Methods of estimating the speeds of spreading in various directions will also be presented.

An integrodifference model for biological invasions in a periodically fragmented environment

“…In any event, we could successfully derive a formula for the speed that exactly fit with the speed numerically obtained. Incidentally, Weinberger [43] previously obtained a mathematical formula for speed in general periodic media in the framework of recursions of the form nt+l = Q [nt]. The formula was derived under some hypotheses, among which Q is assumed to be an order-preserving operator in the sense that if u(x) < v(x) at every point, then Q[u] < Q [v].…”

“…(23) If c has a real root of (23) for a positive s, it should be a candidate of the average frontal speed. Following Weinberger [43] (see also Discussion section), we expect that the minimal value among the candidates of c(s) gives the speed stably realized as a solution of (1):…”

“…In the framework of reactiondiffusion equations, there have been a number of work that studied propagating waves in periodic media and proved the existence of the asymptotic front speed (Gartner and Freidlin [11], Freidlin [9], [10], Xin [47]). Recently Weinberger [43] presented a more general and strict mathematical formula for the asymptotic speed in periodically varying environments (see also Berestycki et al [2], Berestycki and Hamel [1]). In the ecological context, Shigesada et al [37] first presented a reactiondiffusion model for the spread of a single species in a patchy environment with periodic variations in diffusivity and growth rate and investigated how fragmentation of a favorable habitat could retard the rate of spread (see also Shigesada and Kawasaki [34], Kinezaki et al [18], [17], Lutscher et al [24]).…”

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

“…If ν 2 = 0, then (1.1) is the classical reaction diffusion equation, so called random dispersal equation, (without loss of generality, let ν 1 = 1) u t (t, x) = ∆u(t, x) + u (t, x) f (x, u(t, x)), x ∈ R N (1.2) which is broadly used to model the population dynamics of many species in unbounded environments, where u(t, x) is the population density of the species at time t and location x, ∆u characterizes the internal interaction of the organisms, and f (x, u) represents the growth rate of the population, which satisfies that f (x, u) < 0 for u ≫ 1 and ∂ u f (x, u) < 0 for u ≥ 0 (see Aronson & Weinberger, 1957;Aronson & Weinberger, 1978;Cantrell & Cosner, 2003;Fife, 1979;Fife & Peletier, 1977;Fisher, 1937;Kolmogorov, Petrowsky, & Poscunov, 1937;Murray, 1989;Shigesada & Kawasaki, 1997;Skellam, 1951;Weinberger, 1982;Weinberger, 2002;Zhao, 2003;etc. ).…”

“…We refer to (Aronson & Weinberger, 1957;Aronson & Weinberger, 1978;Berestycki, Hamel, & Nadirashvili, 2010;Kametaka, 1976;Liang & Zhao, 2007;Liang, Yi, & Zhao, 2006;Sattinger, 1976;Uchiyama, 1978;Weinberger, 1982;etc). for the study of (1.2) in the case that f (x, u) is independent of x and refer to (Berestycki, Hamel, & Nadirashvili, 2005;Berestycki, Hamel, & Roques, 2005;Freidlin & Gärtner, 1979;Hamel, 2008;Hudson & Zinner, 1995;Nadin, 2009;Nolen, Rudd, & Xin, 2005;Weinberger, 2002;etc). for the study of (1.2) in the case that f (x, u) is periodic in x; refer to (Coville & Dupaigne, 2005;Coville, Dávila, & Martínez, 2008;Li, Sun, & Wang, 2010;etc).…”