CS04: Branching processes as models for structured populations

date: 7/15/2025, time: 14:00-15:30, room: ICS 13

Organizer: Julie Tourniaire & Zsófia Talyigás (Université de Franche-Comté and University of Vienna)

Chair: Julie Tourniaire & Zsófia Talyigás (Université de Franche-Comté and University of Vienna)

On the survival of branching processes and generalised principal eigenvalues

Oliver Tough (Durham)

H. Berestycki and Rossi introduced in \([1]\) two notions of generalised principal eigenvalue, \(\lambda'(L)\) and \(\lambda''(L)\), for non-divergence form uniformly elliptic operators, extending the more classical generalised principal eigenvalue \(\lambda(L)\). They studied the relationship between these different notions of generalised principal eigenvalue, and their relationship with the maximum principle. Here we relate the global survival or global extinction of branching processes to the positivity or negativity (respectively) of the corresponding Berestycki-Rossi eigenvalue \(\lambda'\), and moreover show that \[\lambda''=\limsup_{t\rightarrow\infty}E[\#\{\text{particles alive at time t}\}].\]

Berestycki and Rossi established in this setting that \(\lambda'\leq \lambda''\), conjectured that one always has equality, and proved it for self-adjoint \(L\) in either one dimension or which is radially symmetric. Using our probabilistic interpretation, we prove the Berestycki-Rossi conjecture for general self-adjoint \(L\), but provide a counterexample for non self-adjoint \(L\). This provides a sharp characterisation of the validity of the maximum principle for self-adjoint \(L\) in unbounded domains.

Bibliography

\([1]\) Jane Smith. "Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains." Communications on Pure and Applied Mathematics, vol. 68, 2015, pp. 1014-1065.

Branching Brownian motion with an inhomogeneous branching rate

Jason Schweinsberg (University of California San Diego)

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We study the empirical distribution of the positions of the particles after a sufficiently long time. We show that the bulk of the distribution is well approximated by the Gaussian distribution, but the tails of the distribution follow a profile which is asymptotically related to the Airy function. This is based on joint work with Matt Roberts and Jiaqi Liu.

Bibliography

\([1]\) M. I. Roberts and J. Schweinsberg (2021). A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate Electron. J. Probab. 26, 1-76.

\([2]\) J. Liu and J. Schweinsberg (2023). Particle configurations for branching Brownian motion with an inhomogeneous branching rate. ALEA Lat. Am. J. Probab. Math. Stat. 20, 731-803.

A branching random walk with noisy selection

Zsófia Talyigás (University of Vienna)

There have been a lot of recent progress on branching particle systems with selection, in particular on the \(N\)-particle branching random walk (\(N\)-BRW). In the \(N\)-BRW, \(N\) particles have locations on the real line at all times. At each time step, every particle generates a number of children, and each child has a random displacement from its parent’s location. Then among the children only the \(N\) rightmost are selected to survive and reproduce in the next generation. In this talk we will investigate a noisy version of the \(N\)-BRW. In this model the \(N\) surviving particles are selected at random from the children in such a way, that particles more to the right on the real line are more likely to be selected. I will present some recent results on the asymptotic behaviour of this particle system as \(N\) goes to infinity; including the distribution of the \(N\) particles on the real line and the genealogical properties of the system. Our results show that as we change the selection parameter, there is a phase transition in these asymptotic properties. This is joint work with Colin Desmarais, Bastien Mallein, Francesco Paparella and Emmanuel Schertzer.