Diauxic Growth at the Mesoscopic Scale

In the present paper, we study a diauxic growth that can be generated by a class of model at the mesoscopic scale. Although the diauxic growth can be related to the macroscopic scale, similarly to the logistic scale, one may ask whether models on mesoscopic or microscopic scales may lead to such a behavior. The present paper is the first step towards the developing of the mesoscopic models that lead to a diauxic growth at the macroscopic scale. We propose various nonlinear mesoscopic models conservative or not that lead directly to some diauxic growths.

Mathematical models for cell migration: a non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

A stochastic model for cancer metastasis: branching stochastic process with settlement

We introduce a new stochastic model for metastatic growth, which takes the form of a branching stochastic process with settlement. The moving particles are interpreted as clusters of cancer cells, while stationary particles correspond to micro-tumours and metastases. The analysis of expected particle location, their locational variance, the furthest particle distribution and the extinction probability leads to a common type of differential equation, namely, a non-local integro-differential equation with distributed delay. We prove global existence and uniqueness results for this type of equation. The solutions' asymptotic behaviour for long time is characterized by an explicit index, a metastatic reproduction number $R_0$: metastases spread for $R_{0}>1$ and become extinct for $R_{0}<1$. Using metastatic data from mouse experiments, we show the suitability of our framework to model metastatic cancer.

A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things

Wireless sensor network and industrial internet of things have been a growing area of research which is exploited in various fields such as smart home, smart industries, smart transportation, and so on. There is a need of a mechanism which can easily tackle the problems of nonlinear delay integro-differential equations for large-scale applications of Internet of Things. In this paper, Haar wavelet collocation technique is developed for the solution of nonlinear delay integro-differential equations for wireless sensor network and industrial Internet of Things. The method is applied to nonlinear delay Volterra, delay Fredholm and delay Volterra–Fredholm integro-differential equations which are based on the use of Haar wavelets. Some examples are given to show the computational efficiency of the proposed technique. The approximate solutions are compared with the exact solution. The maximum absolute and mean square roots errors for distant number of collocation points are also calculated. The results show that Haar method is efficient for solving these equations for industrial Internet of Things. The results are compared with existing methods from the literature. The results exhibit that the method is simple, precise and efficient.

Birth-jump processes and application to forest fire spotting

Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.