Fisher's equation ($u_t=D\Delta u+ru(1-u/K)$)

Fisher's equation is a reaction-diffusion equation combining diffusion with logistic growth. For positive $D,r,K$, one-dimensional fronts joining $K$ to zero have minimum speed $2\sqrt{Dr}$. The phase-plane construction of a Fisher travelling front and the linearized leading edge explain this threshold.

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 $0<\delta <1$.

For $u_t=u_{xx}+u(1-u^q)$, $q>0$, the displayed decreasing travelling wave has speed $c=(q+4)/\sqrt{2(q+2)}$, with $a>0$ setting its translation. Writing $Y=U^{q/2}$ gives $U^{\prime }=-kU(1-Y)$, $k=\sqrt{2/(q+2)}$, which verifies the wave equation by equating constant and $Y$ coefficients. The explicit speed increases with $q$, the leading-edge decay rate decreases, and the filled-state decay rate $q/\sqrt{2(q+2)}$ increases. Aligning fronts at half height distinguishes these changes from a translation.