Principal spectrum point and application to nonlocal dispersal operators

Ramsès Djidjou-Demasse (1) , Mocatar Kandé (2) , Ousmane Seydi (3)

1 MIVEGEC - Maladies infectieuses et vecteurs : écologie, génétique, évolution et contrôle
2 UCAD - Université Cheikh Anta Diop de Dakar [Sénégal]
3 Ecole polytechnique de Thiès

We propose a general framework for a sufficient condition of the existence of the principal eigenpair to the eigenvalue problem L[u]+H[u] = λu, (λ, u) ∈ R×X, where L : X → X is a positive bounded linear operator and H : D(H) ⊂ X → X a closed, possibly unbounded linear operator in the Banach space (X, ∥ • ∥). Criteria for (i) the existence of a principal eigenpair, (ii) Asynchronous Exponential Growth (AEG) and, (iii) continuity result of the spectral bound are given, without necessarily specifying forms of operators for the above problem. The criteria are then applied to generalize some results in the existing literature in context of nonlocal dispersal operators. Our results are applied to a model of a chemostat and a model of a spatial evolution of a man-environment disease.

Mots clés: Nonlocal diffusion operators; Principal eigenvalue; Resolvent positive operator; Asynchronous exponential growth

Domaines: Equations aux dérivées partielles [math.AP]

Ramsès Djidjou-Demasse, Mocatar Kandé, Ousmane Seydi. Principal spectrum point and application to nonlocal dispersal operators. 2022. ⟨hal-03691803⟩