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The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as the Fisher–KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher–KPP model cannot replicate, such as the extinction of invasive populations. The Fisher–Stefan model is an adaptation of the Fisher–KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher–Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher–Stefan model is that it is able to simulate population extinction, giving rise to a spreading–extinction dichotomy . In this work, we revisit travelling wave solutions of the Fisher–KPP model and show that these results provide new insight into travelling wave solutions of the Fisher–Stefan model and the spreading–extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher–Stefan model and often-neglected travelling wave solutions of the Fisher–KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher–Stefan model in the limit of slow travelling wave speeds, c ≪1.

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Continuum mathematical models for collective cell motion normally involve reaction-diffusion equations, such as the Fisher-KPP equation, with a linear diffusion term to describe cell motility and a logistic term to describe cell proliferation. While the Fisher-KPP equation and its generalisations are commonplace, a significant drawback for this family of models is that they are not able to capture the moving fronts that arise in cell invasion applications such as wound healing and tumour growth. An alternative, less common, approach is to include nonlinear degenerate diffusion in the models, such as in the Porous-Fisher equation, since solutions to the corresponding equations have compact support and therefore explicitly allow for moving fronts. We consider here a hole-closing problem for the Porous-Fisher equation whereby there is initially a simply connected region (the hole) with a nonzero population outside of the hole and a zero population inside. We outline how self-similar solutions (of the second kind) describe both circular and non-circular fronts in the hole-closing limit. Further, we present new experimental and theoretical evidence to support the use of nonlinear degenerate diffusion in models for collective cell motion. Our methodology involves setting up a two-dimensional wound healing assay that has the geometry of a hole-closing problem, with cells initially seeded outside of a hole that closes as cells migrate and proliferate. For a particular class of fibroblast cells, the aspect ratio of an initially rectangular wound increases in time, so the wound becomes longer and thinner as it closes; our theoretical analysis shows that this behaviour is consistent with nonlinear degenerate diffusion but is not able to be captured with commonly used linear diffusion. This work is important because it provides a clear test for degenerate diffusion over linear diffusion in cell lines, whereas standard one-dimensional experiments are unfortunately not capable of distinguishing between the two approaches.

Two-dimensional collective cell migration assays are used to study cancer and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding. Previous discrete models of collective cell migration assays involve a nearest-neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site; and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but some experimental evidence suggests that cells do not always proliferate logistically. Motivated by these observations, we introduce a generalised proliferation mechanism which allows non-nearest neighbour proliferation events to take place over a template of [Formula: see text] concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a crowding function, f(C), which accounts for the density of agents within a group of sites rather than dealing with the occupancy of a single randomly chosen site. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, [Formula: see text], where λ is the proliferation rate, is generalised to a universal growth function, [Formula: see text]. Comparing the solution of the continuum description with averaged simulation data indicates that the continuum model performs well for many choices of f(C) and r. For nonlinear f(C), the quality of the continuum-discrete match increases with r.

A spectrogram is a useful way of using short-time discrete Fourier transforms to visualise surface height measurements taken of ship wakes in real world conditions. For a steadily moving ship that leaves behind small-amplitude waves, the spectrogram is known to have two clear linear components, a sliding-frequency mode caused by the divergent waves and a constant-frequency mode for the transverse waves. However, recent observations of high speed ferry data have identified additional components of the spectrograms that are not yet explained. We use computer simulations of linear and nonlinear ship wave patterns and apply time-frequency analysis to generate spectrograms for an idealised ship. We clarify the role of the linear dispersion relation and ship speed on the two linear components. We use a simple weakly nonlinear theory to identify higher order effects in a spectrogram and, while the high speed ferry data is very noisy, we propose that certain additional features in the experimental data are caused by nonlinearity. Finally, we provide a possible explanation for a further discrepancy between the high speed ferry spectrograms and linear theory by accounting for ship acceleration.

Collective cell spreading takes place in spatially continuous environments, yet it is often modelled using discrete lattice-based approaches. Here, we use data from a series of cell proliferation assays, with a prostate cancer cell line, to calibrate a spatially continuous individual based model (IBM) of collective cell migration and proliferation. The IBM explicitly accounts for crowding effects by modifying the rate of movement, direction of movement, and the rate of proliferation by accounting for pair-wise interactions. Taking a Bayesian approach we estimate the free parameters in the IBM using rejection sampling on three separate, independent experimental data sets. Since the posterior distributions for each experiment are similar, we perform simulations with parameters sampled from a new posterior distribution generated by combining the three data sets. To explore the predictive power of the calibrated IBM, we forecast the evolution of a fourth experimental data set. Overall, we show how to calibrate a lattice-free IBM to experimental data, and our work highlights the importance of interactions between individuals. Despite great care taken to distribute cells as uniformly as possible experimentally, we find evidence of significant spatial clustering over short distances, suggesting that standard mean-field models could be inappropriate.

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multiscale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (PME). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the PME to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models.