Invading and receding travelling waves of the Fisher-KPP equation with a mass-conserving, moving boundary

Technical Summary: Invading and Receding Travelling Waves of the Fisher-KPP Equation with a Mass-Conserving, Moving Boundary

Problem Statement
The Fisher-KPP equation is the standard reaction-diffusion model for studying invasive biological phenomena, such as the spread of species or malignant growth. However, the classical formulation on an infinite domain produces smooth travelling waves with undefined front positions and cannot model population recession (contraction). Existing extensions, such as the Fisher-Stefan model, introduce moving boundaries to define sharp fronts and allow for recession. However, the Fisher-Stefan model relies on a Stefan condition where boundary velocity is proportional to the population flux. This formulation inherently assumes mass loss at the boundary, representing scenarios like cell death or conversion to unmodeled types. The authors identify a gap in modelling scenarios where a biological population remodels its environment or moves without mass consumption (e.g., tissue healing or specific mechanical interactions), necessitating a mass-conserving boundary condition.

Methodology
The authors develop a free-boundary model by coupling the Fisher-KPP equation with a moving boundary $x = L(t)$ governed by two conditions:

1. Mass Conservation: The boundary velocity is matched to the flux into the boundary, ensuring no mass is lost or gained solely due to boundary movement: $\frac{\partial u}{\partial x}(L(t), t) = -u(L(t), t)\frac{dL}{dt}$.
2. Boundary Velocity Function: The velocity of the boundary is prescribed as a general function of the local cell density, $\frac{dL}{dt} = f(u)|_{x=L(t)}$.

The study employs three primary analytical and numerical techniques:

• Phase Plane Analysis: The system is non-dimensionalised and transformed into a travelling wave coordinate $z = x - ct$. The second-order ODE is converted into a system of first-order autonomous ODEs. The authors analyze the trajectories in the $(U, W)$ phase plane, identifying equilibrium points and determining how the mass-conserving boundary condition truncates these trajectories at a specific front density $U^*$.
• Perturbation Analysis: Analytic approximations for the relationship between wave speed $c$ and front density $U^*$ are derived for two limits: very negative wave speeds ($c \to -\infty$) and small wave speeds ($|c| \ll 1$).
• Numerical Simulation: The PDE system is solved using a finite difference method on a fixed domain via a Landau-type transformation ($\rho = x/L(t)$). The authors simulate various parameter regimes for a linear boundary velocity function $f(u) = \alpha u + \beta$ to observe the evolution of initial conditions into travelling waves or other dynamic behaviors.

Key Contributions and Results

• Existence of Mass-Conserving Travelling Waves: The analysis confirms that the model supports both invading ($c > 0$) and receding ($c < 0$) travelling wave solutions with strictly positive population densities. Unlike the Fisher-Stefan model where the front density is fixed at zero, the mass-conserving condition introduces a wave-speed-dependent front density $U^*$.
• The Wave Speed Curve: A one-to-one relationship is established between the wave speed $c$ and the front density $U^*$.
  • Invading Waves: Occur for $0 < c < 2$. The front density is below the carrying capacity ($0 < U^* < 1$). There is an inverse relationship: lower front densities correspond to higher wave speeds.
  • Receding Waves: Occur for $-\infty < c < 0$. The front density exceeds the carrying capacity ($U^* > 1$) due to "bunching" effects near the front.
  • Steady State: A uniform steady state exists at $c=0$ and $U^*=1$.
• Linear Boundary Velocity Dynamics: By prescribing $\frac{dL}{dt} = \alpha u + \beta$, the authors demonstrate that the intersection of this linear function with the non-linear wave speed curve determines the system's behavior.
  • Multiple Solutions: In specific regions of the $(\alpha, \beta)$ parameter space, the system admits two distinct travelling wave solutions (e.g., two invading, two receding, or one of each).
  • Stability: Numerical simulations reveal that when multiple solutions exist, the solution with the larger wave speed (more positive $c$) is stable, while the other is unstable. Unstable solutions can lead to a reversal of direction (e.g., an initial recession evolving into an invasion).
• Non-Travelling Wave Behaviors: In regimes where no stable travelling wave exists (e.g., $\beta > 2$), the system exhibits non-travelling dynamics. These include "runaway" recession where the front density increases indefinitely, or a scenario where the front invades faster than the bulk population, causing the front to detach from the main body.
• Stability Analysis: Linear stability analysis of the uniform steady state ($u=1$) along the line $\alpha = -\beta$ reveals a transcritical bifurcation at $\alpha = -1$. For $\alpha > -1$, the steady state is stable; for $\alpha < -1$, it becomes unstable, leading to either invading waves (if $u_0 < 1$) or runaway recession (if $u_0 > 1$).

Significance and Claims
The authors claim that their framework provides a mathematically rigorous method to study population invasion and recession without assuming mass loss at the interface, a feature critical for biologically realistic scenarios involving tissue remodeling or mechanical boundary interactions.

The paper emphasizes that the phase plane analysis is independent of the specific choice of the boundary velocity function $f(u)$. This allows the framework to be readily extended to more complex, non-linear boundary functions (e.g., Hill functions) or higher-dimensional geometries. The authors assert that their results offer a means to study increasingly complex boundary dynamics where the relationship between front density and wave speed is non-trivial, contrasting with the fixed-front-density assumption of previous models. The work is presented as a preliminary step toward understanding stability in these systems, with the full stability analysis of the general system noted as beyond the current scope.