Equal-diffusivity Fisher reduction ($v=1-u,\beta =1-\delta$)

In $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv+v_{xx}$, the line $v=1-u$ is invariant when $\beta =1-\delta$. On it $u$ satisfies Fisher's equation with growth coefficient $\delta$. The reduction also requires compatible initial and boundary data: parameter tuning alone does not reduce arbitrary two-field solutions. Positive concentrations and positive $\beta$ in this normalization require $0<\delta <1$.