Growth-fragmentation model for a population presenting heterogeneity in growth rate: Malthus parameter and long-time behavior

TL;DR

This paper analyzes the long-time behavior of a size-structured growth-fragmentation model with heterogeneity in growth rate, incorporating a variability kernel $\kappa$ and the equal-mitosis constraint. It develops a GRE-based framework to prove existence and uniqueness of eigenelements $(\lambda,N,\phi)$ for the heterogeneous problem and to establish convergence of renormalized solutions to a steady profile $N$ in a weighted $L^1$ space, under irreducibility assumptions. The authors derive monotonicity and bounding results for the Malthus parameter: without variability, $\lambda$ is monotone in the growth rate, while with variability $\lambda$ is bounded between the corresponding homogeneous-case eigenvalues $\lambda_1$ and $\lambda_2$. Numerically, they illustrate that sufficient mixing re-establishes asynchronicity and convergence to a steady shape, whereas insufficient mixing can lead to sustained oscillations in non-mixing scenarios. Overall, the work connects spectral theory, entropy methods, and numerical analysis to characterize when heterogeneity enhances or moderates population growth and desynchronization.

Abstract

The goal of the present paper is to explore the long-time behavior of the growth-fragmentation equation formulated in the case of equal mitosis and variability in growth rate, under fairly general assumptions on the coefficients. The first results concern the monotonicity of the Malthus parameter with respect to the coefficients. Existence of a solution to the associated eigenproblem is then stated in the case of a finite set of growth rates thanks to Kreĭn-Rutman theorem and a series of estimates on moments. Afterwards, adapting the classical general relative entropy (GRE) method enables us to ensure uniqueness of the eigenelements and derive the long-time asymptotics of the Cauchy problem. We prove convergence towards the steady state including in the case of individual exponential growth known to exhibit oscillations at large times in absence of variability. A few numerical simulations are eventually performed in the case of linear growth rate to illustrate our monotonicity results and the fact that variability, providing enough mixing in the heterogeneous population, is sufficient to re-establish asynchronicity.

Figures (5)

• Figure 1: Scheme of the growth mechanism of a cell with growth rate $\tau = \tau(x)$.
• Figure 2: Time evolution of the size distribution per feature, either the whole distribution at a few times (bottom) or at many times but only at size $x=1$ (top). Obtained with coefficients $\tau (v,x) = vx$, $\gamma(v,x) = x^2 \tau(v,x)$ and the two variability kernels $\kappa^{red}$ (\ref{['fig:approx_n_nonmixing']}) and $\kappa^{irr}$ (\ref{['fig:approx_n_mixing']}).
• Figure 3: Time evolution of the estimates of the instantaneous population growth rate obtained with coefficients $\tau (v,x) = vx$, $\gamma(v,x) = x^2 \tau(v,x)$ and the two variability kernels $\kappa^{red}$ (left) and $\kappa^{irr}$ (middle). For any $t>0$, $\lambda_n(t)$ is obtained by linear regression of the log of the total number, $s \mapsto \ln ( \int \! \! \! \int_{\mathcal{S}} n(s,v,x) \, \dd v \dd x )$ (right), on the time interval $[ \frac{t}{2}, t ]$.
• Figure 4: Time evolution of the size distribution per feature at size $x=1$, for the variability kernels $\kappa^{_{F t S}}(p_0 \pm \varepsilon)$, with $p_0$ defined by \ref{['eq:p_0']} and $\varepsilon = 0.05$ to show that $p_0$ appears as a critical value for $p$ in the case Fastest to Slowest. Obtained with the coefficients $\tau (v,x) = vx$, $\gamma(v,x) = x^2 \tau(v,x)$.
• Figure 5: Time evolution of the size distribution per feature, either the whole distribution at a few times (bottom) or at many times but only at size $x=1$ (top). Obtained with coefficients $\tau (v,x) = vx$, $\gamma(v,x) = x^2 \tau(v,x)$ and the variability kernels $\kappa^{_{S t F}}(0.5)$, $\kappa^{_{F t S}}(0.2)$ and $\kappa^{_{F t S}}(0.8)$. The corresponding Malthus parameters were estimated to be (up to $10^{-3}$ precision) $v_2=2$, $v_1=1$, and $\lambda_{2, 0.8} \approx 1.356$, respectively, in accordance with the conjecture (see \ref{['table1:conjecture']}).

Theorems & Definitions (28)

• Lemma 1
• proof
• Theorem 1: Monotonicity of the Malthus parameter
• Theorem 2: Monotonicity of the Malthus parameter with variability
• proof
• Theorem 3: Existence of eigenelements
• Proposition 1: Positivity
• proof
• Definition 1: GRE
• Proposition 2: GRE principle