Analysis of the periodically fragmented environment model : I – Species persistence

Abstract

This paper is concerned with the study of the stationary solutions of the equation [Equation: see text] where the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition on the sign of the first eigenvalue of the associated linearized problem with periodicity condition. We explain the biological motivation and we also interpret the results in terms of species persistence in periodic environment. The effects of various aspects of heterogeneities, such as environmental fragmentation are also discussed.

Uniqueness and stability properties of monostable pulsating fronts

“…Under the assumptions (1.13), the existence (and uniqueness) of a positive periodic steady state p + of (1.2) is equivalent to the condition µ − < 0, that is, (1.4) (see [6]). Notice also that (1.13) implies (1.5) (see [20]), as well as (1.6).…”

Coexistence and Spread of Competitors in Heterogeneous Landscapes

“…The study of spread rates in heterogeneous environments for single-species models in continuous time began with the work by Shigesada et al (1986) and has since been extended to a variety of different situations, including two-dimensional domains (Kinezaki et al, 2003), competing species (Cruywagen et al, 1996), asymmetric dispersal , discrete-time equations (Kawasaki and Shigesada, 2007;Lutscher, 2008;Dewhirst and Lutscher, 2009), and more rigorous analytical results (Weinberger, 2002;Weinberger et al, 2008;Berestycki et al, 2005). Most of these modeling approaches represent landscape heterogeneity as two types of patches that are periodically alternating in space.…”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…Equation (1.1) with β = 0 has a lot of applications in various aspects of mathematical biology, physics, especially in population dynamics. It has been intensively studied by many authors and numerous interesting results have been obtained in [2,10,6,12,14]. In particular, biological explanation of (1.1) was meticulously discussed in [6] for p = 2 and in [14] for p > 1.…”

“…It has been intensively studied by many authors and numerous interesting results have been obtained in [2,10,6,12,14]. In particular, biological explanation of (1.1) was meticulously discussed in [6] for p = 2 and in [14] for p > 1. However, aside from [2,3,10], these papers concern only bounded or periodic domains and results involving unbounded domains (for instance R N ) are much less.…”