This paper is concerned with time-dependent reaction-diffusion equations of the following type: ∂tu = Δu + f(x - cte, u), t > 0, x ∈ ℝN. These kind of equations have been introduced in [1] in the case N = 1 for studying the impact of a climate shift on the dynamics of a biological species. In the present paper, we first extend the results of [1] to arbitrary dimension N and to a greater generality in the assumptions on f. We establish a necessary and sufficient condition for the existence of travelling wave solutions, that is, solutions of the type u(t, x) = U(x - cte). This is expressed in terms of the sign of the generalized principal eigenvalue λ1 of an associated linear elliptic operator in ℝN. With this criterion, we then completely describe the large time dynamics for this equation. In particular, we characterize situations in which there is either extinction or persistence. Moreover, we consider the problem obtained by adding a term g(x,u) periodic in x in the direction e: ∂tu - Δu + f(x - cte, u) + g(x, u), t > 0, x ∈ ℝN. Here, g can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with λ1 replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about u ≡ 0 in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form U(t,x - cte), with U(t, x) periodic in t. These results still hold if the term g is also subject to the shift, but on a different time scale, that is, if g(x, u) is replaced by g(x -c′te, u), with c′ ∈ ℝ.

Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space / Berestycki, H.; Rossi, L.. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 21:1(2008), pp. 41-67. [10.3934/dcds.2008.21.41]

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

Rossi L.
2008

Abstract

This paper is concerned with time-dependent reaction-diffusion equations of the following type: ∂tu = Δu + f(x - cte, u), t > 0, x ∈ ℝN. These kind of equations have been introduced in [1] in the case N = 1 for studying the impact of a climate shift on the dynamics of a biological species. In the present paper, we first extend the results of [1] to arbitrary dimension N and to a greater generality in the assumptions on f. We establish a necessary and sufficient condition for the existence of travelling wave solutions, that is, solutions of the type u(t, x) = U(x - cte). This is expressed in terms of the sign of the generalized principal eigenvalue λ1 of an associated linear elliptic operator in ℝN. With this criterion, we then completely describe the large time dynamics for this equation. In particular, we characterize situations in which there is either extinction or persistence. Moreover, we consider the problem obtained by adding a term g(x,u) periodic in x in the direction e: ∂tu - Δu + f(x - cte, u) + g(x, u), t > 0, x ∈ ℝN. Here, g can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with λ1 replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about u ≡ 0 in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form U(t,x - cte), with U(t, x) periodic in t. These results still hold if the term g is also subject to the shift, but on a different time scale, that is, if g(x, u) is replaced by g(x -c′te, u), with c′ ∈ ℝ.
2008
Extinction; forced speed; climate change; principal eigenvalues; reaction-diffusion equations; time periodic parabolic equations; travelling waves
01 Pubblicazione su rivista::01a Articolo in rivista
Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space / Berestycki, H.; Rossi, L.. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 21:1(2008), pp. 41-67. [10.3934/dcds.2008.21.41]
File allegati a questo prodotto
File Dimensione Formato  
Berestychi_Reaction-diffusion-equations_2008.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 363.53 kB
Formato Adobe PDF
363.53 kB Adobe PDF   Contatta l'autore
Berestychi_preprint_Reaction-diffusion-equations_2008.pdf

accesso aperto

Tipologia: Documento in Pre-print (manoscritto inviato all'editore, precedente alla peer review)
Licenza: Creative commons
Dimensione 365.84 kB
Formato Adobe PDF
365.84 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1464931
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 93
  • ???jsp.display-item.citation.isi??? 91
social impact