Growth-fragmentation Equation Research Articles

Floquet Exponent of Solution to Homogeneous Growth-Fragmentation Equation

In this work, we establish the existence and uniqueness of solution of Floquet eigenvalue and its adjoint to homogeneous growth-fragmentation equation with positive and periodic coefficients. We study the Floquet exponent, which measures the growth rate of a population. Finally, we establish the long term behavior of solution to the homogeneous growth-fragmentation equation by entropy method [1,2,3].

Growth-fragmentation equations with McKendrick–von Foerster boundary condition

The paper concerns the well-posedness and long-term asymptotics of growth–fragmentation equation with unbounded fragmentation rates and McKendrick–von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth–fragmentation problems with McKendrick–von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.

Recovering the fragmentation rate in the growth-fragmentation equation

We consider the inverse problem of determining the fragmentation rate from noisy measurements in the growth-fragmentation equation. We use Fourier transform theory on locally compact groups to treat this problem for general fragmentation probabilities. We develop a regularization method based on spectral filtering, which allows us to deal with the inverse problem in weighted spaces. As a result, we obtain a regularization method with error of order , where ɛ is the noise level and m > 0 is the a priori regularity order of the fragmentation rate.

Spectral Gap for the Growth-Fragmentation Equation via Harris's Theorem

We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the non-conservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

On the non existence of non negative solutions to a critical Growth-Fragmentation Equation

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Watson (2016) is not necessarily satisfied. It is proved that when that happens, and as it was first suggested by these authors, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exist. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely $$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$ with $0 < a_{{+}} < a_{{-}}$. We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

A Growth-Fragmentation Approach for Modeling Microtubule Dynamic Instability.

Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division. Due to their role in cell division, they are often used as targets for chemotherapy drugs used in cancer treatment. Experimental studies of MT dynamics have played an important role in the development and administration of many novel cancer drugs; however, a complete description of MT dynamics is lacking. Here, we propose a new mathematical model for MT dynamics, that can be used to study the effects of chemotherapy drugs on MT dynamics. Our model consists of a growth-fragmentation equation describing the dynamics of a length distribution of MTs, coupled with two ODEs that describe the dynamics of free GTP- and GDP-tubulin concentrations (the individual dimers that comprise of MTs). Here, we prove the well-posedness of our system and perform a numerical exploration of the influence of certain model parameters on the systems dynamics. In particular, we focus on a qualitative description for how a certain class of destabilizing drugs, the vinca alkaloids, alter MT dynamics. Through variation of certain model parameters which we know are altered by these drugs, we make comparisons between simulation results and what is observed in in vitro studies.

Long-Time Asymptotics for Polymerization Models

This study is devoted to the long-term behavior of nucleation, growth and fragmentation equations, modeling the spontaneous formation and kinetics of large polymers in a spatially homogeneous and closed environment. Such models are, for instance, commonly used in the biophysical community in order to model in vitro experiments of fibrillation. We investigate the interplay between four processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We first revisit the well-known Lifshitz-Slyozov model, which takes into account only polymerization and depolymerization, and we show that, when nucleation is included, the system goes to a trivial equilibrium: all polymers fragmentize, going back to very small polymers. Taking into account only polymerization and fragmentation, modeled by the classical growth-fragmentation equation, also leads the system to the same trivial equilibrium, whether or not nucleation is considered. However, also taking into account a depolymer-ization reaction term may surprisingly stabilize the system, since a steady size-distribution of polymers may then emerge, as soon as polymeriza-tion dominates depolymerization for large sizes whereas depolymerization dominates polymerization for smaller ones-a case which fits the classical assumptions for the Lifshitz-Slyozov equations, but complemented with fragmentation so that " Ostwald ripening " does not happen.

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.

Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation

The aim of this study is to generalise recent results of the two last authors on en-tropy methods for measure solutions of the renewal equation to other classes of structured population problems. Specifically, we develop a generalised relative entropy inequality for the growth-fragmentation equation and prove asymptotic convergence to a steady-state solution, even when the initial datum is only a non-negative measure.

Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation

We give here an explicit formula for the following critical case of the growth-fragmentation equation ∂/∂tu(t,x)+∂/∂x(gxu(t,x))+bu(t,x)=bα2u(t,αx), u(0,x)=u0(x) for some constants g &gt; 0, b &gt; 0 and α &gt;1 - the case α = 2 being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in [8], and use them to clarify how periodicity may appear asymptotically.

Limit theorems for some branching measure-valued processes

AbstractWe consider a particle system in continuous time, a discrete population, with spatial motion, and nonlocal branching. The offspring's positions and their number may depend on the mother's position. Our setting captures, for instance, the processes indexed by a Galton–Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine the asymptotic behaviour of the particle system. We also obtain a large population approximation as a weak solution of a growth-fragmentation equation. Several examples illustrate our results. The main one describes the behaviour of a mitosis model; the population is size structured. In this example, the sizes of the cells grow linearly and if a cell dies then it divides into two descendants.

A short remark on a growth–fragmentation equation

An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in [5] about the existence of global solutions in the so-called self-similar case is not satisfied. The solution is local and blows up in finite time.

Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate

We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space is the largest one in which we can expect convergence to the steady size distribution. Although this convergence is known to occur under fairly general conditions on the coefficients of the equation, we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded. First we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the Dyson–Phillips series. Then we extend it to general kernels by using the notion of quasi-compactness and the fact that it is a topological invariant.

Probabilistic aspects of critical growth-fragmentation equations

AbstractThe self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or ∞, we exhibit unexpected spontaneous generation of mass in the solutions.

Global stability for the prion equation with general incidence.

We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [11]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted L1 spaces and the analysis of a nonlinear system of three ordinary differential equations.

Estimating the division rate of the growth-fragmentation equation with a self-similar kernel

We consider the growth-fragmentation equation and we address the problem of estimating the division rate from the stable size distribution of the population, which is easily measured, but non-smooth. We propose a method based on the Mellin transform for growth-fragmentation equations with self-similar kernels. We build a sequence of functions which converges to the density of the population in division, simultaneously in several weighted spaces, as the measurement error goes to 0. This improves the previous results for self-similar kernels and allows us to understand the partial results for general fragmentation kernels. Numerical simulations confirm the theoretical results. Moreover, our numerical method is tested on real biological data, arising from a bacteria growth and fission experiment.

Estimating the division rate for the growth-fragmentation equation

Growth-fragmentation equations arise in many different contexts, ranging from cell division, protein polymerization, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest to develop theoretical and numerical methods to recover reaction rates and parameters of the equation from indirect observation of the solution. Following the work done in Perthame and Zubelli (Inverse Probl 23:1037-1052, 2007) and Doumic et al. (2009) for the specific case of the cell division equation, we address here the general question of recovering the fragmentation rate of the equation from the observation of the time-asymptotic solution, when the fragmentation kernel and the growth rates are fully general. We give both theoretical results and numerical methods, and discuss the remaining issues.