Branching stable processes and motion by mean curvature flow

We prove a new result relating solutions of the scaled fractional Allen--Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. to describe solutions of the fractional Allen--Cahn equation in terms of ternary branching $\alpha$-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators

Branching stable processes and motion by mean curvature flow

We prove a new result relating solutions of the scaled fractional Allen–Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. [30] to describe solutions of the fractional Allen–Cahn equation in terms of ternary branching α-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators

Looking forwards and backwards: dynamics and genealogies of locally regulated populations

We introduce a broad class of spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined via the convolution of the point measure with a nonnegative kernel. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE. The classical PDE is obtained both by first scaling time and population size to pass to the nonlocal PDE, and then scaling the kernel that determines local population density; and also (when the limit is a reaction-diffusion equation) by simultaneously scaling the kernel width, timescale and population size in our individual based model. A novelty of our model is that we explicitly model a juvenile phase: offspring are thrown off in a Gaussian distribution around the location of the parent, and reach (instant) maturity with a probability that can depend on the population density at the location at which they land. Although we only record mature individuals, a trace of this two-step description remains in our population models, resulting in novel limits governed by a nonlinear diffusion. Using a lookdown representation, we retain information about genealogies and, in the case of deterministic limiting models, use this to deduce the backwards in time motion of the ancestral lineage of a sampled individual. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model. We also investigate the behaviour of lineages for three different deterministic models of a population expanding its range as a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a porous medium equation with logistic growth

Looking forwards and backwards: dynamics and genealogies of locally regulated populations

We introduce a broad class of mechanistic spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined at a location to be the convolution of the point measure with a suitable non-negative integrable kernel centred on that location. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE. The classical PDE is obtained both by a two-step convergence argument, in which we first scale time and population size and pass to the nonlocal PDE, and then scale the kernel that determines local population density; and in the important special case in which the limit is a reaction-diffusion equation, directly by simultaneously scaling the kernel width, timescale and population size in our individual based model. A novelty of our model is that we explicitly model a juvenile phase. The number of juveniles produced by an individual depends on local population density at the location of the parent; these juvenile offspring are thrown off in a (possibly heterogeneous, anisotropic) Gaussian distribution around the location of the parent; they then reach (instant) maturity with a probability that can depend on the local population density at the location at which they land. Although we only record mature individuals, a trace of this two-step description remains in our population models, resulting in novel limits in which the spatial dynamics are governed by a nonlinear diffusion. Using a lookdown representation, we are able to retain information about genealogies relating individuals in our population and, in the case of deterministic limiting models, we use this to deduce the backwards in time motion of the ancestral lineage of an individual sampled from the population. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model. We also investigate (and contrast) the behaviour of lineages for three different deterministic models of a population expanding its range as a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a porous medium equation with logistic growth

Looking forwards and backwards : dynamics and genealogies of locally regulated populations

We introduce a broad class of mechanistic spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined at a location to be the convolution of the point measure with a suitable non-negative integrable kernel centred on that location. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE. The classical PDE is obtained both by a two-step convergence argument, in which we first scale time and population size and pass to the nonlocal PDE, and then scale the kernel that determines local population density; and in the important special case in which the limit is a reaction-diffusion equation, directly by simultaneously scaling the kernel width, timescale and population size in our individual based model

Modelos probabilísticos de recombinación en genómica

Tesis para optar al grado de Magíster en Ciencias de la Ingeniería, Mención Matemáticas AplicadasMemoria para optar al título de Ingeniero Civil MatemáticoEn este trabajo se estudia la recombinación de genes a tiempo continuo, esto es la evolución bajo la dinámica de recombinación de la distribución genética de una población. Por un lado se cuenta con la generalidad de recombinaciones arbitrarias e incluso permitiendo una cantidad arbitraria de padres. Por otro lado se trabaja bajo la hipótesis de población infinita lo que lleva como ventaja, según lo visto en [17], [11] o [12], que la distribución de genes esté determinada por una ecuación diferencial determinista en el espacio de las medidas. Al resolver esta se obtiene que es la esperanza de un proceso estocástico, conocido como el proceso de fragmentación. Uno de los primeros resultados es una demostración alternativa de este hecho. Luego se busca una fórmula para la ley del proceso. En un contexto similar el trabajo reali- zado en [11] da una fórmula recursiva, bajo ciertas hipótesis sobre las tasas de recombinación. Basándonos en las técnicas desarrolladas en ese trabajo y en [12], [25], [4] se deduce otra fór- mula que sirve para tasas, recombinaciones y una cantidad de padres arbitraria, bajo hipótesis similares. La clave es relacionar el proceso de fragmentación con una familia de grafos, los cuales denominaremos bosques de fragmentación. Estos fueron propuestos originalmente por Mareike Esser en [12] como generalización de los bosques de segmentación encontrados en [4]. Aquí, salvo modificaciones necesarias para la notación, serán la herramienta principal para obtener los resultados. Además esta fórmula permite apreciar que la hipótesis sobre las tasas es para evitar ciertas singularidades que aparecen al realizar los cálculos en el grafo. Una vez que se entiende esto, se discute como extender las soluciones relajando la condición sobre las tasas. Además de lo anterior, se investiga el comportamiento asintótico del proceso de fragmen- tación. Una gran cantidad de resultados interesantes fueron obtenidos por Servet Martínez en [19] para el proceso a tiempo discreto, incluyendo distribución cuasi-estacionaria y una descripción para el Q-proceso. Aquí se obtienen los que son la adaptación natural al tiempo continuo. Es decir, se obtiene un teorema que caracteriza el comportamiento asintótico del proceso de fragmentación y de este se deduce el comportamiento cuasi-estacionario. Por último se hace una síntesis de los resultados obtenidos y se discuten posibles exten- siones a problemas relacionados.CMM - Conicyt PIA AFB17000

On the effects of a wide opening in the domain of the (stochastic) Allen-Cahn equation and the motion of hybrid zones

We are concerned with a special form of the (stochastic) Allen-Cahn equation, which can be seen as a model of hybrid zones in population genetics. Individuals in the population can be of one of three types; aa are fitter than AA, and both are fitter than the aA heterozygotes. The hybrid zone is the region separating a subpopulation consisting entirely of aa individuals from one consisting of AA individuals. We investigate the interplay between the motion of the hybrid zone and the shape of the habitat, both with and without genetic drift (corresponding to stochastic and deterministic models respectively). In the deterministic model, we investigate the effect of a wide opening and provide some explicit sufficient conditions under which the spread of the advantageous type is halted, and complementary conditions under which it sweeps through the whole population. As a standing example, we are interested in the outcome of the advantageous population passing through an isthmus. We also identify rather precise conditions under which genetic drift breaks down the structure of the hybrid zone, complementing previous work that identified conditions on the strength of genetic drift under which the structure of the hybrid zone is preserved