Propagation dynamics for monotone evolution systems without spatial translation invariance

doi:10.1016/j.jfa.2020.108722

Journal of Functional Analysis

Volume 279, Issue 10, 1 December 2020, 108722

https://doi.org/10.1016/j.jfa.2020.108722 Get rights and content

Abstract

In this paper, under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for time-delayed nonlocal equations, reaction-diffusion equations in a cylinder, and asymptotically homogeneous KPP-type equations. We also obtain the existence of steady state solutions and asymptotic spreading properties of solutions for a time-delayed reaction-diffusion equation subject to the Dirichlet boundary condition.

Introduction

This paper is devoted to the study of the propagation dynamics for nonlinear evolution equations admitting the comparison principle. Since the seminal works of Fisher [16] and KPP [22], there have been extensive investigations on traveling wave solutions and propagation phenomena for various evolution equations (see, e.g., [15], [32], [34] and references therein). A fundamental feature of propagation problem is the asymptotic spreading speed introduced by Aronson and Weinberger [1]. Under an abstract setting, Weinberger [37] established the theory of traveling waves and spreading speeds for monotone discrete-time systems with spatial translation invariance. This theory has been greatly developed in [13], [14], [17], [26], [27], [28], [29], [38], [41], [43] for more general monotone and some non-monotone semiflows so that it can be applied to a variety of discrete and continuous-time evolution systems in homogeneous or periodic media. By using the Harnack inequality up to boundary and the strict positivity of solutions, Berestycki et al. [5], [6] studied the asymptotic spreading speed for KPP equations in periodic or non-periodic spatial domains. Differently from these two approaches, the spreading speed and asymptotic propagation were obtained in [40] for the Dirichlet problem of monostable reaction-diffusion equations on the half line by employing the iterative properties of traveling wave maps. Note that the solution maps of such a Dirichlet problem have no spatial translation invariance and the Harnack inequality cannot be extended to the boundary.

With an increasing interest in impacts of climate changes (see, e.g., [2], [21], [35]), there have been quite a few works on traveling waves and asymptotic behavior for evolution equation models with a shifting environment, see [3], [4], [7], [8], [9], [10], [11], [12], [18], [20], [24], [25], [30], [36], [39], [44] and references therein. Another class of evolution equations consists of those in locally spatially inhomogeneous media (see [23]). We should point out that these evolution equations admit the comparison principle, but their solution maps no longer possess the spatial translation invariance. This motivated us to develop the theory of spreading speeds and traveling waves for the monotone semiflows without spatial translation invariance. As a starting point, we assume that the given monotone system has two limiting systems in certain translation sense, and then establish the existence of steady state solutions and asymptotic propagation properties for monotone semiflows without translation invariance.

In order to overcome the difficulty induced by the lack of translation invariance, we first introduce two limiting systems admitting the translation invariance under an abstract setting. Then for a special class of initial functions having compact supports, we obtain certain estimates of their orbits under translations for the limiting system with the upward convergence property, and further carry them to the given system without the translation invariance by comparison arguments (see Section 2). Combining these estimates with the asymptotic annihilation property of the other limiting system, we are able to characterize the propagation dynamics for the given system.

The rest of the paper is organized as follows. In Section 2, we present notations and preliminary results. In order to avoid using traveling wave mappings, we directly establish the links between the system without translation invariance and its limiting systems. In Section 3, we prove the existence of fixed points and asymptotic propagation properties for discrete-time semiflows. In Sections 4 and 5, we extend these results to continuous-time semiflows and a class of nonautonomous evolution systems without translation invariance, respectively. In Section 6, we apply the developed theory to two types of time-delayed nonlocal equations with a shifting habitat, a reaction-diffusion equation in a cylinder, the Dirichlet problem for a time-delayed equation on the half line, and a KPP-type equation in spatially inhomogeneous media. We expect that our developed theory and methods in this paper may be applied to other monotone evolution systems including cooperative and competitive models with spatio-temporal heterogeneity.

Section snippets

Preliminaries

Let $Z$ , $N$ , $R$ , $R_{+}$ , $R^{N}$ , and $R_{+}^{N}$ be the sets of all integers, nonnegative integers, reals, nonnegative reals, N-dimensional real vectors, and N-dimensional nonnegative real vectors, respectively. We equip $R^{N}$ with the norm $| | ξ | |_{R^{N}} ≜ \sqrt{\sum_{n = 1}^{N} ξ_{n}^{2}}$ . Let $X = B C (R, R^{N})$ be the normed vector space of all bounded and continuous functions from $R$ to $R^{N}$ with the norm $| | ϕ | |_{X} ≜ \sum_{n = 1}^{\infty} 2^{- n} \sup_{| x | \leq n} {| | ϕ (x) | |_{R^{N}}}$ . Let $X_{+} = {ϕ \in X : ϕ (x) \in R_{+}^{N}, \forall x \in R}$ and $X_{+}^{\circ} = {ϕ \in X : ϕ (x) \in I n t (R_{+}^{N}), \forall x \in R}$ .

For a given compact topological space M, let $C = C (M, X)$ be

Discrete-time semiflows

In this section, we study the upward convergence, asymptotic annihilation, and the existence of fixed points for discrete-time maps.

Theorem 3.1

Assume that $Q_{+}$ satisfies (UC) and either Q is a subhomogeneous map with (SP), or Q satisfies (ASH-UC-SP). If $c_{+}^{⁎} > 0$ , then for any $ε \in (0, \min {c_{+}^{⁎}, \frac{c_{+}^{⁎} + c_{-}^{⁎}}{2}})$ and $φ \in C_{+} ∖ {0}$ , there holds $\lim_{n \to \infty} \max {| | Q^{n} [φ] (\cdot, x) - r^{⁎} | | : n \max {ε, - c_{-}^{⁎} + ε} \leq x \leq n (c_{+}^{⁎} - ε)} = 0$ .

Proof

For any $ε \in (0, \min {c_{+}^{⁎}, \frac{c_{+}^{⁎} + c_{-}^{⁎}}{2}})$ , we define $I^{ε} = [\max {ε, - c_{-}^{⁎} + ε}, c_{+}^{⁎} - ε],$ $U_{-} (ε) = \underset{n \to \infty}{\lim \inf} [\sup {α \in R_{+} : Q^{n} [φ] (\cdot, x) \geq α r^{⁎} for all x \in n I^{ε}}],$

Continuous-time semiflows

In this section, we extend our results on spreading speeds and asymptotic behavior to a continuous-time semiflow on $C_{+}$ . A map $Q : R_{+} \times C_{+} \to C_{+}$ is said to be a continuous-time semiflow on $C_{+}$ if for any vector $r \in I n t (R_{+}^{N})$ , $Q |_{R_{+} \times C_{r}} : R_{+} \times C_{r} \to C_{+}$ is continuous, $Q_{0} = I d |_{C_{+}}$ , and $Q_{t} \circ Q_{s} = Q_{t + s}$ for all $t, s \in R_{+}$ , where $Q_{t} ≜ Q (t, \cdot)$ for all $t \in R_{+}$ .

In this section, we need the following assumption for some results.

(SC)
For any $ϕ_{k} \in C_{+}$ with $\lim_{k \to \infty} ϕ_{k} = 0$ and $\sup_{k \in N} | | ϕ_{k} | |_{L^{\infty} (M \times R, R)} < \infty$ , there holds $\lim_{k \to \infty} T_{- y} \circ Q_{t} \circ T_{y} [ϕ_{k}] = 0$ in C uniformly for $(t$

Nonautonomous systems

In this section, we extend our results on spreading speeds and asymptotic behavior to a class of nonautonomous evolution systems. Assume that $P : R_{+} \times C_{+} \to C_{+}$ is a map such that for any vector $r \in I n t (R_{+}^{N})$ , $P |_{R_{+} \times C_{r}} : R_{+} \times C_{r} \to C_{+}$ is continuous. For any given $c \in R$ , we define a family of mappings $Q_{t} : = T_{- c t} \circ P [t, \cdot]$ with parameter $t \in R_{+}$ .

By Theorem 4.1 and the definition of $Q_{t}$ , we have the following result.

Theorem 5.1

Assume that there exist $t_{0} > 0$ and $c \in R$ such that $Q_{t} : = T_{- c t} \circ P [t, \cdot]$ is a continuous-time semiflow on $C_{+}$ , and $Q_{t}$

Applications

In this section, we apply the results obtained in Sections 4 and 5 to four classes of monotone evolution equations. We start with the definition of KPP property (see, e.g., [5], [6]).

Definition 6.1

Let $u^{⁎} \in (0, \infty)$ and $F : R_{+} \to R_{+}$ be a continuously differentiable function. We say that F satisfies the KPP property with respect to $u^{⁎}$ , or $(F, u^{⁎})$ has the KPP property if

(i)
$F (0) = 0$ , $F (u^{⁎}) = 0$ , and $F^{'} (0) > 0$ ,
(ii)
$F (u) (u - u^{⁎}) < 0$ for all $u \in (0, \infty) \ {u^{⁎}}$ ,
(iii)
$F (u) < F^{'} (0) u$ for all $u \in (0, \infty)$ .

For simplicity, we always assume in this section that f

Acknowledgements

T. Yi's research is supported by the National Natural Science Foundation of China (Grant No. 11971494), and X.-Q. Zhao's research is supported in part by the NSERC of Canada. This work was initiated during Dr. Yi's visit to Memorial University of Newfoundland, and he would like to thank the Department of Mathematics and Statistics there for its kind hospitality.

References (45)

D. Aronson et al.
Multidimensional nonlinear diffusion arising in population dynamics
Adv. Math.
(1978)
H. Berestycki et al.
Forced waves of the Fisher-KPP equation in a shifting environment
J. Differ. Equ.
(2018)
J. Fang et al.
Traveling waves and spreading speeds for time-space periodic monotone systems
J. Funct. Anal.
(2017)
C. Gomez et al.
Separation dichotomy and wavefronts for a nonlinear convolution equation
J. Math. Anal. Appl.
(2014)
C. Hu et al.
Spatial dynamics for lattice differential equations with a shifting habitat
J. Differ. Equ.
(2015)
X. Liang et al.
Spreading speeds and traveling waves for periodic evolution systems
J. Differ. Equ.
(2006)
X. Liang et al.
Spreading speeds and travelling waves for abstract monostable evolution systems
J. Funct. Anal.
(2010)
R. Lui
Biological growth and spread modeled by systems of recursions, I. Mathematical theory
Math. Biosci.
(1989)
A. Potapov et al.
Climate and competition: the effect of moving range boundaries on habitat invasibility
Bull. Math. Biol.
(2004)
C. Wu et al.
Spatial-temporal dynamics of a Lotka-Volterra competition model with nonlocal dispersal under shifting environment
J. Differ. Equ.
(2019)

T. Yi et al.

Study on monostable and bistable reaction-diffusion equations by iteration of travelling wave maps

J. Differ. Equ.

(2017)

T. Yi et al.

Asymptotic propagations of asymptotical monostable type equations with shifting habitats

J. Differ. Equ.

(2020)

G.-B. Zhang et al.

Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat

J. Differ. Equ.

(2020)

C. Bellard et al.

Impacts of climate change on the future of biodiversity

Ecol. Lett.

(2012)

H. Berestycki et al.

Can a species keep pace with a shifting climate?

Bull. Math. Biol.

(2009)

H. Berestycki et al.

The speed of propagation for KPP type problems. I: periodic framework

J. Eur. Math. Soc.

(2005)

H. Berestycki et al.

The speed of propagation for KPP type problems. II: general domains

J. Am. Math. Soc.

(2010)

H. Berestycki et al.

Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space

Discrete Contin. Dyn. Syst.

(2008)

H. Berestycki et al.

Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains

Discrete Contin. Dyn. Syst.

(2009)

J. Bouhours et al.

Spreading and vanishing for a monostable reaction-diffusion equation with forced speed

J. Dyn. Differ. Equ.

(2019)

Y. Du et al.

Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary

J. Dyn. Differ. Equ.

(2018)

J. Fang et al.

Can pathogen spread keep pace with its host invasion?

SIAM J. Appl. Math.

(2016)

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Citation Excerpt :
It was shown that the spreading speed coincides with the spreading speed in the limiting homogeneous systems with KPP structure. In particular, [55] generalizes [33] in the context of (1.3). Asmptotically homogeneous environments
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 equations 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 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 of reaction-diffusion type, we give explicit formulas of the spreading speed when the environment has one or two shifting speeds. As a byproduct, we also introduce a novel class of “asymptotically homogeneous” environments which share the same spreading speed with the corresponding homogeneous environments.
On étudie les propriétés de propagation des équations du type Kolmogorov-Petrovsky-Piskunov (KPP) en milieu hétérogène et changeant, avec différentes vitesses, en utilisant la théorie des èquations de Hamilton-Jacobi. Notre cadre s'appliquent aux de l'équation réaction-diffusion et intégro-différentielle avec un retard, qui conduit à une classe d'équations de Hamilton-Jacobi dépendant de la variable $x / t$ . On montre l'existence et l'unicité d'une solution de viscosité avec des arguments élémentaires, et puis détermine de la vitesse de propagation en terme d'équation de Hamilton-Jacobi du premier ordre dans l'espaces de la vitesse. Concernant l'équation KPP classique, on invente une nouvelle classe d'environnements asymptotiquement homogènes qui ont la même vitesse de propagation avec d'environnements homogènes.

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Propagation dynamics for monotone evolution systems without spatial translation invariance

Abstract

Introduction

Section snippets

Preliminaries

Discrete-time semiflows

Continuous-time semiflows

Nonautonomous systems

Applications

Acknowledgements

Adv. Math.

J. Differ. Equ.

J. Funct. Anal.

J. Math. Anal. Appl.

J. Differ. Equ.

J. Differ. Equ.

J. Funct. Anal.

Math. Biosci.

Bull. Math. Biol.

J. Differ. Equ.

J. Differ. Equ.

J. Differ. Equ.

J. Differ. Equ.

Impacts of climate change on the future of biodiversity

Ecol. Lett.

Can a species keep pace with a shifting climate?

Bull. Math. Biol.

The speed of propagation for KPP type problems. I: periodic framework

J. Eur. Math. Soc.

The speed of propagation for KPP type problems. II: general domains

J. Am. Math. Soc.

Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space

Discrete Contin. Dyn. Syst.

Reaction-diffusion equations for population dynamics with forced speed. II. Cylindrical-type domains

Discrete Contin. Dyn. Syst.

Spreading and vanishing for a monostable reaction-diffusion equation with forced speed

J. Dyn. Differ. Equ.

Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary

J. Dyn. Differ. Equ.

Can pathogen spread keep pace with its host invasion?

SIAM J. Appl. Math.