High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equationu^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0,where {\lambda,\mu>0} are parameters, {c\in\mathbb{R}}, {a(x)} is a locally integrable P-periodic sign-changing weight function, and {g\colon{[0,1]}\to\mathbb{R}} is a continuous function such that {g(0)=g(1)=0}, {g(u)>0} for all {u\in{]0,1[}}, with superlinear growth at zero. A typical example for {g(u)}, that is of interest in population genetics, is the logistic-type nonlinearity {g(u)=u^{2}(1-u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of {a(x)}. More precisely, when m is the number of intervals of positivity of {a(x)} in a P-periodicity interval, we prove the existence of {3^{m}-1} non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.