**Proof plan.** The travelling wave ansatz reduces the PDE to a second-order ODE. Rewriting as a first-order system, the fixed points and their stability determine the admissible wave speeds. The lower bound $c \geq 2$ comes from requiring non-oscillatory (biologically meaningful) fronts, and the selection $c = 2$ for compactly supported data follows from a comparison argument using the linearised equation at the leading edge. **Step 1 (Reduction to ODE).** Substituting $u(X, \tau) = U(\xi)$ with $\xi = X - c\tau$ into the nondimensionalised Fisher-Kolmogorov equation $\partial_\tau u = u(1 - u) + \partial_{XX} u$ gives \begin{align*} -cU' = U(1 - U) + U'', \end{align*} where primes denote $d/d\xi$. Introducing $V = U'$, this becomes the planar system \begin{align*} U' &= V, \\ V' &= -cV - U(1 - U). \end{align*} **Step 2 (Fixed points and linearisation).** The fixed points are $(U, V) = (0, 0)$ and $(1, 0)$, corresponding to the two homogeneous steady states. At $(1, 0)$: the Jacobian has eigenvalues $\lambda = (-c \pm \sqrt{c^2 + 4})/2$. Since $c^2 + 4 > c^2$, the eigenvalues are always real with opposite signs — $(1, 0)$ is a saddle for all $c > 0$. At $(0, 0)$: the Jacobian has eigenvalues $\lambda = (-c \pm \sqrt{c^2 - 4})/2$. The discriminant $c^2 - 4$ determines the type: - If $c > 2$: both eigenvalues are real and negative — $(0, 0)$ is a stable node. - If $c = 2$: repeated eigenvalue $\lambda = -1$ — $(0, 0)$ is a degenerate stable node. - If $0 < c < 2$: eigenvalues are complex with negative real part — $(0, 0)$ is a stable spiral. **Step 3 (Admissibility constraint).** The travelling front connects $(1, 0)$ to $(0, 0)$ in the phase plane — a heteroclinic orbit from the saddle to the stable equilibrium. Since $U$ represents a population density, the front must satisfy $0 \leq U(\xi) \leq 1$ for all $\xi$. When $c < 2$, the stable spiral at $(0, 0)$ forces the trajectory to oscillate as it approaches the origin, producing regions where $U < 0$. Such solutions are biologically inadmissible (negative population). When $c \geq 2$, the approach to $(0, 0)$ is monotone (stable node), and the heteroclinic orbit can be shown to satisfy $U > 0$ and $V < 0$ throughout. This establishes $c \geq 2$ as the admissible range. **Step 4 (Existence of heteroclinic orbit for $c \geq 2$).** For $c \geq 2$, both equilibria are hyperbolic and the saddle $(1, 0)$ has a one-dimensional unstable manifold entering the region $\{0 < U < 1, V < 0\}$. The stable node at $(0, 0)$ attracts all trajectories in this region. A standard shooting argument (or Wazewski's principle) shows that the unstable manifold of $(1, 0)$ connects to $(0, 0)$, establishing existence for every $c \geq 2$. **Step 5 (Wave speed selection for localised initial data).** Consider the linearised equation $\partial_\tau u = u + \partial_{XX} u$ valid at the leading edge where $u \ll 1$. Substituting $u \sim Ae^{-a(X - c\tau)}$ gives $c = 1/a + a$, minimised at $a = 1$ with $c_{\min} = 2$. The Fisher-Kolmogorov equation satisfies a comparison principle: if $u_1(X, 0) \leq u_2(X, 0)$, then $u_1(X, \tau) \leq u_2(X, \tau)$ for all $\tau > 0$. Initial data satisfying the compact support condition decays at least as fast as $e^{-X}$ (in fact, faster), so can be bounded above by $Ae^{-X}$ for suitable $A$. The corresponding linear solution propagates at speed $c = 2$. By the comparison principle, the nonlinear solution cannot propagate faster than $c = 2$. Since $c \geq 2$ is required for admissibility and $c \leq 2$ by comparison, the selected speed is $c = 2$.