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

TL;DR

It is argued that there is currently no accurate numerical solver for these models, and it is emphasized that special care must be taken when dealing with the “monster on a leash”.

Abstract

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".

Figures (7)

• Figure 1: Sketch of the go-or grow dichotomy. In this framework, cells can switch between a moving phenotype and a stationary and reproducing phenotype. Created in BioRender. Conte, M. (2024) https://BioRender.com/i82o612
• Figure 2: Qualitative trend of $\lambda_1(\omega)$. Typical form of the unstable eigenvalue $\lambda_1(\omega)$ computed from \ref{['lambdaomega']} as function of the mode $\omega$. Here, the parameters are set to $d_u=1$, $k_u=-2$, $k_v=-2$, $h_v=1$, and $h_u=2$.
• Figure 3: Example of 2D instabilities emerging from model \ref{['balanced-model']}. Two-dimensional simulations of the balanced model \ref{['balanced-model']} on a rectangular domain of size $[0, 50]\times[0,50]$ for the transition and growth functions \ref{['Func_Pham']}. The first row shows the evolution of the population $u(t,x)$, while the second row illustrates $v(t,x)$, with $x=(x_1,x_2)$. The three columns illustrate the population behavior at $t=0,100,500$, respectively. The regular mesh size is set to $\Delta x_1=\Delta x_2=0.2$, while the parameter values are chosen as $\nu=4$, $K=1$, $r=1$, $\gamma=1.6$, $n^*=0.75$, and $d=1$.
• Figure 4: Example of 2D instabilities emerging from model \ref{['balanced-model']}. Two-dimensional simulations at time $t=500$ of the population $v(t,x)$, with $x=(x_1,x_2)$, for the balanced model \ref{['balanced-model']} for three increasing value of the mesh size: $\Delta x=0.25, 0.34, 0.5$. Here $\Delta x=\Delta x_i$, for $i=1,2$. The model is simulated on a rectangular domain of size $[0, 50]\times[0,50]$ for the transition and growth functions \ref{['Func_Pham']}. Parameters values are set to $\nu=4$, $K=1$, $r=1$, $\gamma=1.6$, $n^*=0.75$, and $d=1$.
• Figure 5: Examples of 1D instabilities from model \ref{['balanced-model']}. One-dimensional simulations of the balanced model \ref{['balanced-model']} on the domain $[0, 180]$ for the transition and growth functions \ref{['Func_Pham']}. The value of $n^*$ is increased from left to right: $n^*=0.57$, $n^*=0.6$, and $n^*=0.75$. The other parameters are set to $\nu=4$, $K=1$, $r=1$, $\mu=1.6$, and $d=1$.

Theorems & Definitions (18)

• Definition 1
• Theorem 1
• Theorem 2: Theorems 4.60--4.61 in britton1986reaction
• Theorem 3: Theorem 2.21 in wang2024upper
• Corollary 4
• Theorem 5: Theorem 4 in HadelerLewis2002
• Definition 2
• Definition 3
• Lemma 6
• proof