Go-or-Grow Models in Biology: a Monster on a Leash

Go-or-grow approaches represent a specific class of mathematical models used to describe populations where individuals either migrate or reproduce, but not both simultaneously. These models have a wide range of applications in biology and medicine, chiefly among those the modeling of brain cancer spread. The analysis of go-or-grow models has inspired new mathematics, and it is the purpose of this review to highlight interesting and challenging mathematical properties of reaction–diffusion models of the go-or-grow type. We provide a detailed review of biological and medical applications before focusing on key results concerning solution existence and uniqueness, pattern formation, critical domain size problems, and traveling waves. We present new general results related to the critical domain size and traveling wave problems, and we connect these findings to the existing literature. Moreover, we demonstrate the high level of instability inherent in go-or-grow models. We argue that there is currently no accurate numerical solver for these models, and emphasize that special care must be taken when dealing with the “monster on a leash”.

💡 Research Summary

The paper “Go-or-Grow Models in Biology: a Monster on a Leash” is a comprehensive review that focuses on a specific class of reaction‑diffusion models in which cells can either migrate or proliferate, but not both at the same time. The authors begin by motivating the “go‑or‑grow” hypothesis with experimental evidence from glioma biology, especially glioblastoma multiforme, where highly motile cells tend to have low proliferation rates and vice‑versa. They acknowledge that this dichotomy is not universal—some tumor cells can display both traits—but it remains a useful abstraction for many studies.

The review then surveys the extensive literature, dividing it into discrete (cell‑based, lattice‑gas cellular automata, continuous‑time random walks) and continuous (partial differential equation) approaches. In the discrete realm, early works by Fedotov & Iomin introduced CTRW frameworks with arbitrary waiting‑time distributions, while later LGCA models incorporated density‑dependent switching, oxygen‑driven phenotypic changes, and Allee‑type effects. Stochastic individual‑based models by Gerlee & Nelander derived macroscopic PDEs with nonlinear diffusion and advection terms, providing a bridge between microscopic rules and continuum descriptions.

The core of the review is the minimal go‑or‑grow system consisting of two coupled equations for a moving subpopulation (u(x,t)) and a stationary/proliferating subpopulation (v(x,t)). The moving cells diffuse linearly with coefficient (D), while the stationary cells are assumed to have negligible diffusion. General transition functions (\alpha(u,v),\beta(u,v),\gamma(u,v)) govern switching between the two phenotypes and may depend on both densities. The authors discuss several extensions: nonlinear diffusion for the moving compartment, small but non‑zero diffusion for the stationary compartment, and more complex waiting‑time distributions.

Section 4 presents a rigorous existence and uniqueness theory based on Rothe’s method. By constructing a time‑discrete approximation and employing energy estimates together with a maximum principle, the authors obtain local solutions and then extend them globally under mild growth conditions on the reaction terms. However, this framework does not preclude the emergence of high‑frequency instabilities.

Section 5 addresses precisely those instabilities. Using techniques developed by Marciniak‑Czochrą et al., the authors linearize the system around homogeneous steady states and show that, when the phenotypic switching is sufficiently fast, the Jacobian becomes cooperative. In this regime, eigenvalues associated with high spatial frequencies acquire positive real parts, leading to rapid amplification of short‑wavelength perturbations. This theoretical finding explains why numerical simulations of go‑or‑grow models often exhibit spurious oscillations even when the mesh is refined.

The critical domain size problem is tackled in Section 6. Building on classical results for the Fisher–Kolmogorov–Petrovsky–Piskunov (FKPP) equation, the authors extend the analysis to the two‑species system. They prove that a non‑trivial steady state can persist only if the spatial domain exceeds a threshold that depends non‑linearly on the diffusion coefficient, proliferation rate, and the switching rates. Notably, when switching is slow, the required domain size grows dramatically, suggesting that small, confined brain regions may be unable to sustain a tumor population.

Section 7 is devoted to traveling wave solutions. By exploiting the theory of cooperative systems, the authors derive a lower bound for the minimal wave speed and demonstrate that, in the limit of infinitely fast switching, the wave speed converges to the classical FKPP speed (c_{\text{FKPP}}=2\sqrt{Dr}). For finite switching rates, two distinct wave speeds can coexist: a faster “migration front” driven primarily by the moving cells and a slower “proliferation front” governed by the stationary cells. The authors provide explicit formulas for the minimal speed in several special cases and compare these speeds with numerical simulations, showing good agreement. They also discuss the shape of the wave front, highlighting a sharp density jump at the leading edge when switching is moderate.

Finally, the authors argue that no current numerical solver can reliably capture all the challenging features of go‑or‑grow models. The combination of high‑frequency instabilities, stiff reaction terms, and possible degeneracy in the diffusion operator requires specialized algorithms—such as spectral filtering, implicit‑explicit (IMEX) time stepping, adaptive mesh refinement, and positivity‑preserving discretizations. They term the whole situation a “monster on a leash,” emphasizing that the model’s biological richness comes with substantial mathematical and computational burdens.

In conclusion, the review synthesizes biological motivation, model taxonomy, and rigorous mathematical analysis of go‑or‑grow systems. It identifies open problems—most prominently the development of robust, accurate numerical methods and the experimental validation of model predictions—and suggests future directions, including multiscale coupling, heterogeneous tissue modeling, and incorporation of treatment effects. The paper serves as both a state‑of‑the‑art reference for researchers entering the field and a call to action for mathematicians, computational scientists, and biologists to collaborate on this challenging yet highly relevant class of models.