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Asymptotic unit-speed limit under increasing nonlinear switching rates

Establish that for 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 density-dependent switching functions Γ1 and Γ2, the travelling wave speed c converges to 1 in the regime where the nonlinear switching rates are increased (e.g., as parameters controlling the magnitudes of Γ1 and Γ2 grow).

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Background

The authors connect the heterogeneous go-or-grow model to a single-population model and show that, for constant switching rates, speeds are bounded by those of the FKPP equation. Numerical results for nonlinear switching suggest that as the switching becomes large, the observed speeds approach 1, consistent with intuition from the single-population limit.

They explicitly state that while this behavior is observed numerically and is intuitively reasonable, there is currently no rigorous proof, and they refer to these statements as conjectures left for future investigation.

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)