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Rigorous bounds on travelling wave speeds in the minimal go-or-grow model

Prove that for travelling wave solutions of the two-phenotype go-or-grow reaction–diffusion system ∂tρ1 = Δρ1 − ρ1 Γ1(ρ) + ρ2 Γ2(ρ), ∂tρ2 = ρ2(1 − ρ) + ρ1 Γ1(ρ) − ρ2 Γ2(ρ), with total density ρ = ρ1 + ρ2 and density-dependent switching functions Γ1 and Γ2, the wave speed c satisfies c_min ≤ c ≤ 1, where c_min denotes the minimal speed predicted by the dispersion relation obtained from the linearization at low density with γ1 = Γ1(0) and γ2 = Γ2(0).

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Background

The paper studies a minimal go-or-grow model where cells switch between proliferative and migratory phenotypes with density-dependent switching rates. A formal fast-switching limit connects the heterogeneous model to a single-population reaction–diffusion equation with density-dependent diffusion and growth.

A dispersion relation c(σ) is derived from the linearization at the leading edge using γ1 = Γ1(0) and γ2 = Γ2(0), providing a predicted minimal speed c_min. For nonlinear switching functions, numerical simulations indicate that observed travelling wave speeds are bounded below by c_min and above by 1, but a rigorous proof is not provided.

References

Numerically, we observe that $c_{\text{min}\leq c\leq 1$ for the model given by Eqs.~eq: tw coordinates, and that $c\rightarrow 1$ as the nonlinear switching rates increase. Although this behaviour intuitively makes sense when considering the connection to single-population models, we do not currently have a rigorous proof. We leave these conjectures for further investigation.

eq: tw coordinates:

cdU1dz+d2U1dz2−U1Γ1(U)+U2Γ2(U)=0 ,cdU2dz+U2(1−U)+U1Γ1(U)−U2Γ2(U)=0 ,\begin{split} c\frac{\mathrm{d}U_1}{\mathrm{d}z} + \frac{\mathrm{d}^2U_1}{\mathrm{d}z^2} - U_1\Gamma_1(U)+U_2\Gamma_2(U)&=0\,, \\ c\frac{\mathrm{d}U_2}{\mathrm{d}z} + U_2(1-U) + U_1\Gamma_1(U)-U_2\Gamma_2(U)&=0\,, \end{split}

— Travelling waves in a minimal go-or-grow model of cell invasion (2404.11251 - Falcó et al., 17 Apr 2024) in Discussion and open problems (Section 4)