Schedule for: 25w5453 - Emerging Connections between Reaction-Diffusion, Branching Processes, and Biology

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We consider some simple models of evolving populations, the relationships between them, and their relationships with other mathematical models, in particular branching and coalescing processes and 'voting systems' on them. Our focus will be on different ways in which populations, or individuals within a population, compete with one another. Simple models will reveal the importance of the dimension and shape of the underlying domain in which the population evolves, and the effect of noise.

In this course, we address the question of the large-time behavior of solutions of reaction-diffusion equations. We will focus in particular on two aspects: 1) the asymptotic shape of the invasion set; 2) the profile of the solution at the invasion set's interface. After an overview of the classical results for the homogeneous equation, we will focus on spatial-periodic equations, in which the invasion set is characterized by the Freidlin-Gartner formula. We will present a PDE proof of the formula that holds true for general types of reaction terms. If time permits, we will then discuss some recent results, obtained in collaboration with H. Guo and F. Hamel, concerning the convergence of the profile of solutions for the bistable equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Ermon-Song diffusion algorithm introduced in their papers 2019 and 2021 is a basic approach to producing additional samples of a "complicated" probability distribution starting with samples from another, known distribution from which sampling is easier. Without claiming any expertise in the generative or training aspects of the model, we will discuss this algorithm in the context of PDE methods of speeding up sampling, as well as the convergence of a discrete version of the Ermon-Song algorithm when the distribution from which one needs to sample has singular support. The ingredients in the proof are all completely elementary but to the best of our knowledge some of them may be new. This is a joint work with Ayya Alieva and Gautam Iyer.

A measure-valued process which models the behaviour of interacting populations of different species types in space was established in [42]. In scaling this process we accounted for the high abundance of only some of the species types, while others remained in low abundance, and obtained a limit which for the abundant types is deterministic and is described by a system of reaction-diffusion type partial differential equations, while it remains a Markov chain for the low abundant types. We now also consider fluctuations of the original measure-valued process around this limit, and prove their rescaled limit is a process that can be described as piecewise solutions to stochastic PDEs with Markov jumps in a discrete coordinate fully coupled with these solutions. We illustrate with several examples and applications where this modelling framework is useful.

Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Here, we develop a general theory for the mean first passage time (MFPT) for velocity jump processes. We focus on two scenarios that are relevant to biological modelling; the diffusive case and the anisotropic case. For the anisotropic case we also perform a parabolic scaling, leading to a well known anisotropic MFPT equation. To illustrate the results we consider a two-dimensional circular domain under radial symmetry, where the MFPT equations can be solved explicitly. Furthermore, we consider the MFPT of a random walker in an ecological habitat that is perturbed by linear features, such as wolf movement in a forest habitat that is crossed by seismic lines (joint work with M. D’Orsogna, JC. Mantooth, A. Lindsay).

For exchangeable (i.e. neutral) ancestry models with an arbitrary sequence of population and litter sizes, an ordered representation known as the lookdown (Donnelly and Kurtz, '96, '99) is a powerful tool for studying the structure of family trees. By introducing dual notions of forward and backward neutrality, we give a more intuitive derivation of the lookdown and the coupling that relates it to the exchangeable model, and show the lookdown arranges lineages in size-biased order of the number of their descendants. We then study three properties of the exchangeable models: existence of a single asymptotically dominating lineage (takeover), uniqueness of the infinite path (fixation), and whether or not the lookdown ordering can be inferred from the family trees of the exchangeable model (identifiability). Takeover, and some aspects of identifiability, are characterized by the coalescent time scale associated to the model. Fixation is more delicate; we give sufficient conditions for it to occur.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider some simple models of evolving populations, the relationships between them, and their relationships with other mathematical models, in particular branching and coalescing processes and 'voting systems' on them. Our focus will be on different ways in which populations, or individuals within a population, compete with one another. Simple models will reveal the importance of the dimension and shape of the underlying domain in which the population evolves, and the effect of noise.

In this course, we address the question of the large-time behavior of solutions of reaction-diffusion equations. We will focus in particular on two aspects: 1) the asymptotic shape of the invasion set; 2) the profile of the solution at the invasion set's interface. After an overview of the classical results for the homogeneous equation, we will focus on spatial-periodic equations, in which the invasion set is characterized by the Freidlin-Gartner formula. We will present a PDE proof of the formula that holds true for general types of reaction terms. If time permits, we will then discuss some recent results, obtained in collaboration with H. Guo and F. Hamel, concerning the convergence of the profile of solutions for the bistable equation.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

With about two-thirds of the global population using social media, online interactions often influence offline events such as protests and violence. This talk presents two mathematical frameworks to model these online-to-offline spillovers. The first framework uses an epidemic-type model on networks, exploring mean field approximations to the stochastic processes and deriving reproductive numbers for these models. We also examine how network structure impacts the accuracy of these approximations. The second framework applies a reaction-diffusion model on networks to analyze the spreading speeds of traveling wave solutions. We identify parameter regimes for approximating these speeds on k-ary trees and characterize scenarios involving pushed, pulled, and pinned waves. These models provide insights into how information spreads across networks and triggers offline behaviors.

We study a branching annihilating random walk in which particles move on the discrete lattice in discrete generations. Each particle produces a poissonian number of offspring which independently move to a uniformly chosen site within a fixed distance from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. This can be thought of as a very strong form of local competition and implies that the system is not monotone. For certain ranges of the parameters of the model we show that the system dies out almost surely or, on the other hand, survives with positive probability. In a more restricted parameter range we strengthen the survival results to complete convergence with a non-trivial invariant measure. In the same regime, we show that the process in one dimension has an asymptotic speed. A central tool in the proof is comparison with oriented percolation on a coarse-grained level, using carefully tuned density profiles which expand in time and are reminiscent of discrete travelling wave solutions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider some simple models of evolving populations, the relationships between them, and their relationships with other mathematical models, in particular branching and coalescing processes and 'voting systems' on them. Our focus will be on different ways in which populations, or individuals within a population, compete with one another. Simple models will reveal the importance of the dimension and shape of the underlying domain in which the population evolves, and the effect of noise.

In this course, we address the question of the large-time behavior of solutions of reaction-diffusion equations. We will focus in particular on two aspects: 1) the asymptotic shape of the invasion set; 2) the profile of the solution at the invasion set's interface. After an overview of the classical results for the homogeneous equation, we will focus on spatial-periodic equations, in which the invasion set is characterized by the Freidlin-Gartner formula. We will present a PDE proof of the formula that holds true for general types of reaction terms. If time permits, we will then discuss some recent results, obtained in collaboration with H. Guo and F. Hamel, concerning the convergence of the profile of solutions for the bistable equation.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The model under study is a linear transport equation in the upper half plane, together with a nonlinear and nonlocal Dirichlet condition that couples the values of the unknown function at the boundary to those inside. Its primary motivation is the study of the nonlocal Kermack-McKendrick model for the spread of epidemics, and it has received a great deal of attention in the 1980's. It can be reduced to a nonlinear integral equation, from which one can infer the development of nasion fronts, whose asymptotic propagation speed can be computed. The goal of the talk is to present a fresh look at this model and to understand its sharp asymptotics, something that had not previously been done. While the relevance of such an undertaking may be questionned from the epidemiological point of view, its structure presents specificities that make it worth studying. In particular, it is reminiscent of that of the "Road-field model" introduced by Berestycki, Rossi and the author, an analogy that will be discussed. Joint work with G. Faye and M. Zhang.

H. Berestycki and Rossi introduced in 2015 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 $\lambda''=limsup_{t\rightarrow \infty}E_x[\#\{\text{particles alive at time t}\}]$. These results should be compared to those of Kyprianou and Englander in 2004, who showed that the positivity or negativity of $\lambda$ corresponds to the different notion of local survival of the corresponding branching process. This provides a probabilistic proof of the relations established by Berestycki and Rossi between these generalised principal eigenvalues, and with the maximum principle. We prove these results in a far more general setting than uniformly elliptic diffusions, necessitating a generalisation of the Berestycki-Rossi eigenvalues. In the setting of branching uniformly elliptic diffusions, we can say more. 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. It is our understanding that H. and J. Berestycki, and Graham, are currently studying the same question through PDE methods in current work in progress. We finally turn to the relationship between global and local survival of branching processes with positive and bounded stationary solutions of the FKPP equation. The existence and uniqueness of the latter has received a lot of attention in recent years, exclusively from a PDE perspective. We provide an alternative probabilistic perspective. We show that global survival of a branching process is equivalent to the existence of a positive and bounded stationary solution of the corresponding FKPP equation, in which case the probability of global survival gives the maximal such solution. Conversely if we have local survival, then the probability of local survival provides the minimal positive and bounded stationary solution of the corresponding FKPP equation. The non-uniqueness of stationary solutions of the FKPP equation is therefore equivalent to the possibility of the branching process surviving globally but not locally. We demonstrate the application of this correspondence and outline how it might be used in future research. This is joint work with Pascal Maillard.

Stochastic epidemic models with varying infectivity and waning immunity have recently been introduced. In this talk, I will present a new model, based on the work of Forien et al. (2022), that incorporates memory of the last infection. To this end, I will introduce a parametric approach and consider a piecewise deterministic Markov process that models both the evolution of the parameter, also called the trait, and the age of infection of individuals over time. At each new infection, a new trait is randomly assigned to the infected individual according to a Markov kernel, and their age is reset to zero. In the large population limit, we derive a partial differential equation (PDE) that describes the density of traits and ages. The main goal is to study the conditions under which endemic equilibria exist for the deterministic PDE model and to establish an endemicity threshold that depends on the model parameters. This is a joint work with Arsene Brice Zotsa-Ngoufack.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the past decade there have been many results on the relationship between PDEs of porous medium type and variants of the Hele-Shaw problem. In particular, the latter arises in the limit of large diffusion exponent of the former. This is called the incompressible limit. In the first half of the talk, I will present a recent result along these lines; the novelty is that the limiting problem exhibits heterogeneity on the free boundary. This motivates the topic of the second half of the talk: stochastic homogenization of a Hele-Shaw problem with oscillation on both the interior and the free boundary. The result that I will describe holds only in one spatial dimension; however, it is new even in the setting of periodic coefficients. Based on joint works with Anthony Sulak and with Paul Yuming Zhang.

Consider a sample of individuals taken at random from a population. A quantity of interest in population genetics is the site frequency spectrum, which is the number of mutations that appear on $k$ of the $n$ sampled individuals, for $k = 1, \dots, n-1$. As long as the mutation rate is constant, this number will be roughly proportional to the total length of all branches in the genealogical tree that are on the ancestral line of $k$ sampled individuals. Suppose the population is expanding in two-dimensional space. Many such two dimensional growth models are expected to fall in the so-called KPZ universality class, where KPZ refers to a stochastic partial differential equation known as the Kardar-Parisi-Zhang equation. For such models, we adopt the perspective that the genealogical tree can be approximated by the tree formed from the infinite upward geodesics in the directed landscape, a universal scaling limit constructed in by Dauvergne, Ortmann, and Virag (2022), starting from $n$ randomly chosen points. We prove new asymptotic results for the lengths of the portions of these geodesics that are ancestral to $k$ of the $n$ sampled points. This leads to the prediction that the number of mutations inherited by $k$ of the sampled individuals should be proportional to $k^{-7/5}$ when the sample comes from the entire population, and proportional to $k^{-1/2}$ when the sample comes from the outer edge of the population. The results verify and extend nonrigorous predictions of Fusco, Gralka, Kayser, Anderson, and Hallatschek (2016) and Eghdami, Paulose, and Fusco (2022). This talk is based on joint work with Shirshendu Ganguly and Yubo Shuai.

In this talk, I will consider the KPP equation under front-like initial data of types $x^{k+1}e^{-\lambda_* x}$ and $x^{\boldsymbol{\nu}}e^{-\lambda x}$ as $x$ tends to infinity, with $0<\lambda<\lambda_*=\sqrt{f'(0)}$ and $k, \boldsymbol{\nu}\in\mathbb{R}$, and discuss the position of the level sets and the ``convergence to a traveling wave'' results.

We consider the long-time behavior of a spatial "heterogeneous" binary branching Brownian motion (BBM) in which the branching rate depends on where the branching event occurs. More precisely, for a positive function g, the instantaneous branching rate of a particle at a location x is characterized by g(x) (we refer to this as g-BBM). When g is periodic, we expect that the microscopic effects of g average out on large scales, and the process should exhibit asymptotically homogeneous behavior. Nevertheless, the heterogeneity of the branching rate introduces new technical challenges. In this talk, I will prove a shape theorem for the convex hull of the g-BBM in all dimensions, namely that there exists a deterministic set W such that almost surely as t → ∞, the convex hull of the g-BBM approximates the set tW. This talk is based on joint work in progress with Louigi Addario-Berry (McGill) and Jessica Lin (McGill).

The extrema of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields as well as to reaction-diffusion equations. Considering BBM in dimensions 2 and higher raises several novel geometrical questions. How are the angles of extremal particles distributed? What plays the role of the famous Lalley-Sellke "random shift" or "derivative martingale"? What is the shape of the cloud of particles around the extrema, e.g., around the maximal particle? In this talk, we give a full description of the limiting law of the maximum norm of multidimensional BBM as well as the point process of the extrema. We conclude with a scaling limit for the shape of the particle cloud around any extremal point to a novel random surface.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.