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

By Henri Berestycki, Luca Rossi. 2008

140 citations recorded by the source corpus.

This paper is concerned with time-dependent reaction-diffusion equations of the following type: $\partial_t u=$Δ$u+f(x-cte,u),t>0,x\in\R^N.$ 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 $\l$ of an associated linear elliptic operator in $\R^N$. 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 $e$: $\partial_t u=$Δ$u+f(x-cte,u)+g(x,u),t>0,x\in\R^N.$ 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 $\l$ replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about $u\equiv0$ 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 $c'\in\R$.

Subfields: Genetics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/e64f3436571f0fe43260708fbe98baee48111ea9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Genetics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

Diffusive and nondiffusive population models (2010, background): “Indeed, it is shown in [12] that traveling wave solutions of the type u(x, t) = v(x− cte) exist if the climate shift is not too large (i.”
Asymptotic spreading in heterogeneous diffusive excitable media (2008, background): “On the other hand, there may be propagation along an unbounded path but not persistence in the stationary frame (see [12, 13]).”
Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains (2010, unknown): “Statement (ii) was announced and used in our previous paper [12].”
Propagation dynamics for monotone evolution systems without spatial translation invariance (2020, background): “In particular, it was assumed that lim r→∞ sup |x|>r,y∈Ω ∂uf(x, y, 0) < 0 in [8] and that limx→±∞ f(x, y, u) = f ∞ ± (u) uniformly for y ∈ Ω in [9].”
Spreading and Vanishing for a Monostable Reaction–Diffusion Equation with Forced Speed (2018, background): “This result also highlights qualitative differences with the KPP framework of [5, 6, 7, 17, 29], where the persistence of the population depends only on the value of c.”
Reaction-diffusionequations for population dynamics with forced speed II -cylindrical-type domains (2009, methodology): “In [6], we proved that existence and uniqueness of travelling wave solutions of the type u(x, t) = U(x − cte) and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized principal eigenvalue in R of an associated linear operator.”