Asynchronous exponential growth for structured population models in measure space

Technical Summary: Asynchronous Exponential Growth for Structured Population Models in Measure Space

Problem Formulation
This paper investigates the asymptotic behavior of physiologically structured population models formulated on the space of nonnegative Radon measures, $\mathcal{M}_+(\mathbb{R}_+)$. While classical results for such models exist in the space of Lebesgue integrable functions ($L^1$), where convergence implies the convergence of total population size, these frameworks face limitations when solutions develop concentration phenomena (e.g., Dirac masses) or when populations are represented by discrete cohorts.

The authors consider a linear structured population model governed by a transport operator (describing development with respect to a physiological variable $x$), a linear decay term (mortality), and an integral operator (birth/recruitment). The model is defined as:
$$\begin{cases} \partial_t \mu_t + \partial_x (b(x) \mu_t) + c(x) \mu_t = \int_{\mathbb{R}_+} \eta(x)(y) \, d\mu_t(y), & (t, x) \in \mathbb{R}_+ \times \mathbb{R}_+, \\ b(0) D_\lambda \mu_t(0) = 0, & t \in \mathbb{R}_+, \\ \mu_{t=0} = \mu_0, \end{cases}$$
where $\mu_t$ is a measure-valued solution, $b(x)$ is the transport velocity, $c(x)$ is the mortality rate, and $\eta(x)$ is the recruitment kernel.

The state space is equipped with the flat norm (dual bounded Lipschitz distance), $\|\cdot\|_{BL^*}$, rather than the total variation norm. This choice is motivated by the fact that transport semigroups are not strongly continuous with respect to the total variation norm, whereas they are strongly continuous in the flat metric topology.

Methodology
The authors employ semigroup theory and spectral analysis adapted to the space of measures. The key methodological steps include:

1. Well-posedness and Semigroup Construction: The model is reformulated as an abstract Cauchy problem $\frac{d\mu_t}{dt} = (B + C + N)\mu_t$ on the Banach lattice $E = \mathcal{M}(\mathbb{R}_+)$ equipped with the flat norm. The authors establish that the problem generates a strongly continuous, positive semigroup $\{T(t)\}_{t \geq 0}$.
2. Regularity Analysis: A key technical contribution involves analyzing the regularity induced by the recruitment operator. The authors prove that for any positive time $t > 0$, the solution cannot accumulate mass near the origin faster than linearly with respect to the Lebesgue measure. This ensures the boundedness of the Radon-Nikodym derivative at the origin, even though the solution may retain atomic components elsewhere.
3. Spectral Analysis via Quasi-Compactness: Since the state space is unbounded, the semigroup is not eventually compact, rendering classical spectral theorems inapplicable. Instead, the authors utilize the concept of quasi-compactness. They decompose the generator $A = B + C + N$ and analyze the adjoint semigroup generated by $B^* + C^*$. By proving that the semigroup generated by the transport and decay parts ($B+C$) is strictly contractive (using a specific equivalent norm on the dual space $BL(\mathbb{R}_+)$), they demonstrate that the essential growth bound $\omega_{ess}(T)$ is negative.
4. Irreducibility: The authors formulate a sufficient condition for the semigroup to be irreducible, requiring that the support of the recruitment kernel $\eta(y)$ for large $y$ includes the origin. This ensures that mass can propagate from any state to the boundary, a condition biologically interpreted as large individuals producing offspring of minimal size.

Key Results
The main result, Theorem 3.7, establishes the asymptotic behavior of the measure-valued solutions under Assumptions 2.4 (which include regularity of coefficients, positivity of transport, and a lower bound on mortality relative to its derivative). The theorem states that one of the following holds:

1. Exponential Stability: The semigroup is uniformly exponentially stable ($\omega_0(T) < 0$), meaning solutions decay to zero.
2. Asynchronous Exponential Growth (AEG): If the spectral bound $s(A) \geq 0$, the semigroup decomposes into a dominant part and a remainder. Specifically, there exists a dominant eigenvalue $\lambda^* \geq 0$ and a finite-rank projection $P^*$ such that:
   $$T(t) = e^{\lambda^* t} \sum_{j=0}^{k^*-1} \frac{t^j}{j!} (A - \lambda^*)^j P^* + Q(t),$$
   where the remainder $Q(t)$ decays exponentially faster than the dominant term.
3. Convergence to a One-Dimensional Attractor: If the semigroup is additionally irreducible, the dominant eigenvalue $\lambda^*$ is simple (multiplicity one), and the projection $P^*$ is rank one. Consequently, the solution exhibits asynchronous exponential growth:
   $$\lim_{t \to \infty} e^{-\lambda^* t} T(t) = P^*,$$
   implying convergence to a one-dimensional globally attracting eigenspace.

Significance and Claims
The paper claims to extend the classical theory of asynchronous exponential growth, traditionally established in $L^1$ spaces, to the space of Radon measures endowed with the flat metric. The authors note that this extension is non-trivial because:

• It avoids the failure of strong continuity for transport semigroups in the total variation norm.
• It accommodates solutions with concentration phenomena and discrete cohorts without requiring artificial regularization.
• It handles unbounded state spaces by utilizing quasi-compactness rather than eventual compactness, removing the need for an upper bound on the structuring variable (a common restriction in earlier studies).

The authors state that, to the best of their knowledge, this is the first result establishing asynchronous exponential growth for a broad class of physiologically structured population models formulated on spaces of Radon measures. The framework is presented as generic but applicable to contexts in cell biology, evolution, and ecology, and potentially extendable to coagulation–fragmentation models.