  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.
