Asymptotic behaviour of some nonlocal equations in mathematical biology and kinetic theory

Abstract

We study the long-time behaviour of solutions to some partial differential equations arising in the modelling of several biological and physical phenomena. In this work, the type of equations we consider is mainly nonlocal, in the sense that they involve integral operators. Moreover, the equations we consider describe the time evolution of either some populations structured by several traits like age, elapsed-time and size, or the distribution of dynamical states of a single particle, depending on time, space and velocity. In the latter case, they are called kinetic equations. A common trait of such nonlocal population models and kinetic equations is that the underlying dynamics is a special type of stochastic process, called a Markov process. We are interested in showing quantitatively the asynchronous behaviour of interacting neu- ron populations which are composed of large and fully connected networks. Neurons undergo a charging period followed by a sudden discharge in the form of firing a spike. We consider two nonlinear models structured by the time elapsed since the last discharge. These models are proposed in Pakdaman et al. (2010, 2014). The nonlinearity comes from the dependence of firing rate on the total neural activity at a time. In the second model, there is an addition of a fragmentation term to include the effect of the past activity of neurons by displaying adapta- tion and fatigue. With this addition, the second model shares many common properties with another class of integro-partial differential equations called the growth-fragmentation equation. This is another type of equation we look at the convergence rate to a universal profile in a quan- titative way. The growth-fragmentation equation describes a system of growing and dividing particles which may be used as a model for many processes in ecology, neuroscience, telecom- munications and cell biology. In this work, we consider two critical types of fragmentation processes, namely mitosis and constant fragmentation. We also consider nonconservative cases where eigenelements cannot be computed explicitly. For the aforementioned nonlocal models for the structured population dynamics, we present quantitative exponential convergence speeds in the weighted total variation norm. The main results concerning the long-time behaviour of two nonlinear neuron populations and the growth-fragmentation equation are summarized in Cañizo and Yoldaş (2019) and Cañizo et al. (2019) respectively. Furthermore, we also study hypocoercivity of some space inhomogeneous linear kinetic equations including the linear relaxation Boltzmann (linear BGK) and the linear Boltzmann equations which are posed either on the torus or on the whole space with a confining potential term. We prove exponential convergence in the torus or on the whole space with a potential growing quadratically at infinity. Moreover, for the weaker confining potentials (subquadratic) we present subgeometric convergence rates quantitatively Cañizo et al. (2019). The physiologically structured population models and space inhomogeneous linear kinetic equations which we deal with in this work are well-studied from various aspects in the already-

existing literature. We give a brief review as well. What differs from the past plentiful studies on the asymptotic behaviour of these equations is the techniques we use here. We consider a probabilistic approach which is developed for studying ergodic properties of discrete-time Markov processes. The method is due to Doeblin and Harris Doblin (1940); Harris (1956) (var- ious versions can be found in Meyn and Tweedie (2009); Stroock (2014); Hairer and Mattingly (2011)); based on establishing a combination of a minorisation (irreducibility) and a geometric drift (Lyapunov) conditions for a Markovian process. This method gives a quantitative con- vergence speed in a totally constructive way and existence of a unique stationary state even without having to calculate it explicitly. Moreover, it allows us to consider wider range of initial data including measures and provides convergence results with more relaxed assumptions in many cases. Therefore, the application of Harris’s Theorem into the aforementioned partial differential equations to study the long-time behaviour of solutions is the core of this thesis.