Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Learning differential equation models from stochastic agent-based model simulations

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology, and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel, and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread

Spreading speeds and traveling waves in some population models.

Virtually every ecosystem has been invaded by exotic organisms with potentially drastic consequences for the native fauna or flora. Studying the forms and rates of invading species has been an important topic in spatial ecology. We investigate two two-species competition models with Allee effects in the forms of reaction-diffusion equations and integro-difference equations. We discuss the spatial transitions from a mono-culture equilibrium to a coexistence equilibrium or a different mono-culture equilibrium in these models. We provide formulas for the spreading speeds based on the linear determinacy and show the results on the existence of traveling waves. We also study a two-sex stage-structured model. We carry out initial analysis for the spreading speed and conduct numerical simulations on the traveling waves and spreading speeds in the two-sex model

A spatial age-structured model of perennial plants with a seed bank.

We formulate an integro-difference model to predict the growth and spatial spread of a perennial plant population with an age-structured seed bank. We allow the seeds in the bank to be of any age, producing an infinite system of equations. The production of new seed can be density-dependent and so the function describing this growth is allowed to be non-monotone. The functions describing the seed bank are linear. We develop properties about the non-spatial model, including the existence of a positive steady-state and conditions under which solutions converge to this steady-state. We also show that when the origin is unstable, the system has a spreading speed and that this spreading speed is characterized as the slowest speed of a class of traveling wave solutions. We conduct numerical simulations of a truncated version of this model which show that both the perennial term and the seed bank can have a stabilising effect on the population. On the other hand, traveling wave solutions may exhibit different patterns of fluctuations including periodic oscillations and chaotic tails