High multiplicity and chaos for an indefinite problem arising from genetic models

Abstract

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation \begin{equation*} u'' + cu' + \bigr{(} \lambda a^{+}(x) - \mu a^{-}(x) \bigr{)} g(u) = 0, \end{equation*} where $\lambda,\mu>0$ are parameters, $c\in\mathbb{R}$, $a(x)$ is a locally integrable $P$-periodic sign-changing weight function, and $g\colon\mathopen{[}0,1\mathclose{]}\to\mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(u)>0$ for all $u\in\mathopen{]}0,1\mathclose{[}$, 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 behaviour 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 $\lambda$ and $\mu$ 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 countable family of globally defined solutions with a complex behaviour, coded by (possibly non-periodic) bi-infinite sequences of $3$ symbols.