Abstract We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation u ′′ + c ⁢ u ′ + ( λ ⁢ a + ⁢ ( x ) - μ ⁢ a - ⁢ ( x ) ) ⁢ g ⁢ ( u ) = 0 , u^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0, where λ , μ > 0 {\lambda,\mu>0} are parameters, c ∈ ℝ {c\in\mathbb{R}} , a ⁢ ( x ) {a(x)} is a locally integrable P-periodic sign-changing weight function, and g : [ 0 , 1 ] → ℝ {g\colon{[0,1]}\to\mathbb{R}} is a continuous function such that g ⁢ ( 0 ) = g ⁢ ( 1 ) = 0 {g(0)=g(1)=0} , g ⁢ ( u ) > 0 {g(u)>0} for all u ∈ ] 0 , 1 [ {u\in{]0,1[}} , with superlinear growth at zero. A typical example for g ⁢ ( u ) {g(u)} , that is of interest in population genetics, is the logistic-type nonlinearity g ⁢ ( u ) = u 2 ⁢ ( 1 - u ) {g(u)=u^{2}(1-u)} . Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a ⁢ ( x ) {a(x)} . More precisely, when m is the number of intervals of positivity of a ⁢ ( x ) {a(x)} in a P-periodicity interval, we prove the existence of 3 m - 1 {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.