On the bifurcation curve for an elliptic system of FitzHugh–Nagumo type

We study a system of partial differential equations derived from the FitzHugh–Nagumo model. In one dimension solutions are required to satisfy zero Dirichlet boundary conditions on the interval Ω=(−1,1). Estimates are given to describe bounds on the range of parameters over which solutions exist; numerical computations provide the global bifurcation diagram for families of symmetric and asymmetric solutions. In the two-dimensional case we use numerical methods for zero Dirichlet boundary conditions on the square domain Ω=(−1,1)×(−1,1). Numerical computations are given both for symmetric and asymmetric, and for stable and unstable solutions.