We establish via variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for minimizers.

The existence of a positive solution for the generalized predator-prey model for two species $$\begin{array}{c}\Delta u+u(a+g(u,v\left)\right)=0\text{in}\Omega ,\\ \Delta v+v(d+h(u,v\left)\right)=0\text{in}\Omega ,\\ u=v=0\text{on}\partial \Omega ,\end{array}$$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.