2018/12/29 by Jérôme Coville, Coville, Jérôme, François Hamel +1 · 1 citation
Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.1812.11412
openalex publication_date 2018/12/29 · openalex created_date 2019/01/11 · openalex updated_date 2026/07/28
This article is concerned with the following spectral problem: to find a positive function Φ ∈ C 1 (Ω) and λ ∈ R such that q(x)Φ (x) + ^ Ω J(x, y)Φ(y) dy + a(x)Φ(x) + λΦ(x) = 0 for x ∈ Ω, where Ω ⊂ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair (λ p , Φ p) and derive some of its main properties. In particular, we prove that λ p (Ω) = lim R→+∞ λ p (Ω R), where Ω R = Ω ∩ (--R, R) and λ p (Ω R) corresponds to the principal eigenvalue of the truncation operator defined in Ω R. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.