An optimal control approach to nonlinear wave speed selection in reaction-diffusion equations

Abstract

Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach through an optimal control formulation, in order to obtain a lower bound on the invasion speed of travelling wave solutions in systems of nonlinear partial differential equations. We begin by analysing single-species models, where the evolution of the density is governed by a scalar equation with a density-dependent diffusion term and a nonlinear reaction term. We show that for any admissible test function, maximising with respect to the parameter of interest yields a bound on the travelling wave speed. We apply this framework to several examples, including the porous-Fisher equation, and examine when nonlinear selection mechanisms dominate over the classical linear marginal stability criterion. Extending this approach, we then consider multi-species systems of reaction-diffusion equations and, reframed as Pontryagin-type optimality systems, we derive analogous bounds on the travelling wave speed using a variational framework under weak coupling. Finally, we employ numerical simulations to confirm the accuracy of the predicted wave speeds across a range of illustrative examples.

Figures (5)

• Figure 1: Plot of the numerically estimated minimum travelling wave speed (solid lines) and the minimum travelling wave speed of Eq. \ref{['eq:wound']} predicted using the variational principle with $F(\beta)$ as given in Eq. \ref{['eq:F_wound']} (dashed lines) as a function of $n$ (left-hand side) and $m$ (right-hand side). The numerically estimated travelling wave speed is obtained by tracing the point $X(t)$ such that $u(X(t), t)=0.1$.
• Figure 2: Plot of the numerically estimated minimum travelling wave speed (solid lines) and the minimum travelling wave speed of Eq. \ref{['eq:allee']} derived using the variational principle with $F(\beta)$ found in Eq. \ref{['eq:f_allee']} (dashed lines) as a function of $\alpha$ (left-hand side) and $a$ (right-hand side). The numerically estimated travelling wave speed is obtained by tracing the point $X(t)$ such that $u(X(t), t)=0.1$.
• Figure 3: (left) Shape of travelling wave profiles for the Fisher-Stefan model given by Eqs. \ref{['eq:fisher_stefan']} and \ref{['eq:fisher_stefan_bc']} depending on the parameter $\kappa$. (right) Plot of the numerically simulated minimum travelling wave speed (squares) and analytically calculated travelling wave speed using the variational principle in Eq. \ref{['eq: bound fisher stefan simple']} (black dashed lines) as a function of $\kappa$. The numerically estimated travelling wave speed is obtained by tracing the point $X(t)$ such that $u(X(t), t)=0.1$.
• Figure 4: Plot of numerically simulated minimum travelling wave speed (solid lines) and analytically calculated travelling wave speed of Eqs. \ref{['eq:ECM_u']} and \ref{['eq: ECM_m']} using linear theory (dotted lines) and the variational principle (dashed lines) with $M_C(\beta)$ as calculated in Eq. \ref{['eq:N1']} (left-hand side) and $M_B(\beta)$ as calculated in Eq. \ref{['eq:N2']} (right-hand side) for various initial ECM densities $\nu\geq0$. The numerically estimated travelling wave speed is obtained by tracing the point $X(t)$ such that $u_1(X(t), t)=0.1$.
• Figure 5: (left) Numerical simulations of the time-dependent cell differentiation model (Eqs. \ref{['eq: landman a']} and \ref{['eq: landman b']}) exhibiting travelling wave solutions. Solutions plotted at $t = 50,\,75,\,100$ for different parameters. (right) Wave speed $c$ as a function of $\lambda$ for various values of $K$. Solid lines correspond to predictions obtained from the approximate variational principle, while square markers indicate numerical solutions of the full PDE system. Dashed vertical lines mark the threshold $\lambda \kappa / c = 1$, beyond which the weak coupling approximation is no longer valid. The black curve shows the linear theory prediction $c = 2\sqrt{1-\lambda}$.