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Traveling waves and minimal speed in noncooperative go-or-grow systems

Determine conditions that guarantee the existence of traveling-wave solutions and characterize the minimum invasion speed in noncooperative instances of the general go-or-grow reaction–diffusion system for migrating cells u and proliferating cells v with linear diffusion in u, proliferation g(u+v)v, and bidirectional transition rates α(u,v) and β(u,v) that generate noncooperative kinetics.

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Background

The authors develop existence and speed results using cooperative-system theory for a broad class of go-or-grow models. However, biologically motivated switching laws can induce noncooperative dynamics, outside the scope of the presented theory.

They pose the unresolved problem of establishing traveling waves and their minimum speeds for noncooperative models, which are common in applications and not covered by the cooperative framework.

References

While we have reviewed a wide array of analytical and numerical results concerning this special class of mathematical models, several important mathematical questions remain unresolved. Many biological systems can be described by models of the form general-model, with specific expressions for the transition functions $\alpha(u,v)$ and $\beta(u,v)$ that may result in noncooperative dynamics. Under what conditions is the existence of traveling waves guaranteed in these noncooperative models, and what is the minimum wave speed associated with them?

— Go-or-Grow Models in Biology: a Monster on a Leash (2412.05191 - Thiessen et al., 6 Dec 2024) in Section 8 (Discussion), Open Problems