We deal with the Neumann boundary value problemwhere the weight term has two positive humps separated by a negative one and g : [0, 1] → R is a continuous function such that g(0) = g(1) = 0, g(s) > 0 for 0 < s < 1 and lim s→0 + g(s)/s = 0. We prove the existence of three solutions when λ and µ are positive and sufficiently large.
Reaction-diffusion equations have several applications in the field of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant the search of nontrivial steady states. Our paper focuses on a one-dimensional model given by a parameter-dependent equation of the form u + λa + (t) − µa − (t) g(u) = 0, where g : [0, 1] → R is a continuous function such that g(0) = g(1) = 0, g(s) > 0 for every 0 < s < 1 and lim s→0 + g(s)/s = 0, and the weight a(t) has two positive humps separated by a negative one. In this manner, we consider bounded habitats which include two favorable food sources and an unfavorable one. We deal with various boundary conditions, including the Dirichlet and Neumann ones, and we prove the existence of eight positive solutions when λ and µ are positive and sufficiently large. Throughout the paper, numerical simulations are exploited to discuss the results and to explore some open problems. * Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2017: "Problemi differenziali con peso indefinito: tra metodi topologici e aspetti dinamici". Guglielmo Feltrin is supported by the Belgian F.R.S.-FNRS -Fonds de la Recherche Scientifique, Chargé de recherches project: "Quantitative and qualitative properties of positive solutions to indefinite problems arising from population genetic models: topological methods and numerical analysis", and partially by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14).
In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation{x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s}, wheresis a real parameter,fandgare continuous functions andgisT-periodic in the variablet. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions ong. Analogous results are also obtained for Neumann boundary conditions.
We deal with the study of the evolution of the allelic frequencies, at a single locus, for a population distributed continuously over a bounded habitat. We consider evolution which occurs under the joint action of selection and arbitrary migration, that is independent of genotype, in absence of mutation and random drift. The focus is on a conjecture, that was raised up in literature of population genetics, about the possible uniqueness of polymorphic equilibria, which are known as clines, under particular circumstances. We study the number of these equilibria, making use of topological tools, and we give a negative answer to that question by means of two examples. Indeed, we provide numerical evidence of multiplicity of positive solutions for two different Neumann problems satisfying the requests of the conjecture.
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