An indefinite nonlinear problem in population dynamics: high multiplicity of positive solutions

Abstract

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).

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

“…From the mathematical point of view, the last decade has experienced a huge interest in indefinite weight problems with a logistic-type nonlinearity from different perspectives both in the ODE setting (e.g., [1,4,5,6,10,17,18,20,22,24]) and in the PDE one (e.g., [9,13,14,15,18,21,23]). The main questions addressed were the existence, the uniqueness as well as the multiplicity of non-constant positive solutions.…”

“…However, focusing on either a fixed number of sign-changes of the weight term or on the number of inflection points of the logistic term, no complete description of global bifurcation diagrams has yet been provided. We notice that some first steps in this direction are given in [5,17,24] through numerical investigations.…”

“…In the present work, we pursue the studies undertaken in [5,6], where a general nonlinearity superlinear at u = 0 is considered in place of g. Taking advantage of the precise expression of (1.1) and using the connection times approach (see [11,12,26]), we establish the shape of the bifurcation diagrams for the solutions of (1.1) by using µ and λ as the main and the secondary bifurcation parameter, respectively. Moreover, we obtain general non-existence, existence, and multiplicity results, depending on the values of µ and λ.…”

Monotone and nonmonotone clines with partial panmixia across a geographical barrier

“…In PDE models on the evolution of gene frequencies, clines are nonconstant equilibrium solutions. The general cline theory of regular migration-selection model can be found in a series of systematic studies by Lou and Nagylaki [8,9,10]; in the review papers by Nagylaki and Lou [17], Lou et al [11], and Bürger [1]; and in recent works by Hofbauer and Su [5,6], Nakashima [22,23,24], Sovrano [26], and Feltrin and Sovrano [2,3].…”

“…In the next section, we mainly use phase-plane analysis to generalize Theorem 2.1 from (8) to a much wider class of selection functions. 3. Clines with p − < p + .…”