Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function $n(x).$ Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight $n.$ We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincar\'{e} - Bikhoff fixed point theorem.