arXiv subjects

Matthias Hammer

Publications and source records attributed to Matthias Hammer.

8 recordsLinked to original sources

arXiv2023

How the interplay of dormancy and selection affects the wave of advance of an advantageous gene

In this paper we investigate the spread of advantageous genes in two variants of the F-KPP model with dormancy. The first variant, in which dormant individuals do not move in space and instead form "localized seed banks", has recently been introduced in Blath, Hammer and Nie (2020). However, there, only a relatively crude upper bound for the critical speed of potential travelling wave solutions has been provided. The second model variant is new and describes a situation in which the dormant forms of individuals are subject to motion, while the "active" individuals remain spatially static instead. This can be motivated e.g. by spore dispersal of fungi, where the "dormant" spores are distributed by wind, water or insects, while the "active" fungi are locally fixed. For both models, we establish the existence of monotone travelling wave solutions, determine the corresponding critical wave speed in terms of the model parameters, and characterize aspects of the asymptotic shape of the waves depending on the decay properties of the initial condition. Interestingly, the slow-down effect of dormancy on the speed of propagation of beneficial alleles is often more serious in model variant II (the "spore model") than in variant I (the "seed bank model"), and this can be understood mathematically via probabilistic representations of solutions in terms of (two variants of) "on/off branching Brownian motions". Our proofs make rather heavy use of probabilistic tools in the tradition of Watanabe (1967), McKean (1975), Bramson (1978), Neveu (1987), Lalley and Sellke (1987), Champneys et al (1995) and others. However, the two-compartment nature of the model and the special forms of dormancy also pose obstacles to the classical formalism, giving rise to a variety of open research questions that we briefly discuss at the end of the paper.

math.PR↗

arXiv2021

The stochastic Fisher-KPP Equation with seed bank and on/off-branching-coalescing Brownian motion

We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial in the proofs. Further, we show that the seed bank SPDEs give rise to an interesting class of "on/off" moment duals. In particular, in the special case of the F-KPP Equation with seed bank, the moment dual is given by an "on/off-branching Brownian motion". This system differs from a classical branching Brownian motion in the sense that independently for all individuals, motion and branching may be "switched off" for an exponential amount of time after which they get "switched on" again. Here, as an application of our duality, we show that the spread of a beneficial allele, which in the classical F-KPP Equation, started from a Heaviside intial condition, evolves as a pulled traveling wave with speed $\sqrt{2}$, is slowed down significantly in the corresponding seed bank F-KPP model. In fact, by computing bounds on the position of the rightmost particle in the dual on/off-branching Brownian motion, we obtain an upper bound for the speed of propagation of the beneficial allele given by $\sqrt{\sqrt{5}-1}\approx 1.111$ under unit switching rates. This shows that seed banks will indeed slow down fitness waves and preserve genetic variability, in line with intuitive reasoning from population genetics and ecology.

math.PR↗

arXiv2019

Ergodic branching diffusions with immigration: properties of invariant occupation measure, identification of particles under high-frequency observation, and estimation of the diffusion coefficient at nonparametric rates

In branching diffusions with immigration (BDI), particles travel on independent diffusion paths in $\mathbb{R}^d$, branch at position-dependent rates and leave offspring -- randomly scattered around the parent's death position -- according to position-dependent laws. We specify a set of conditions which grants ergodicity such that the invariant occupation measure is of finite total mass and admits a continuous Lebesgue density. Under discrete-time observation, BDI configurations being recorded at discrete times $iΔ$ only, $i\in\mathbb{N}_0$, we lose information about particle identities between successive observation times. We present a reconstruction algorithm which in a high-frequency setting (asymptotics $Δ\downarrow 0$) allows to reconstruct correctly a sufficiently large proportion of particle identities, and thus allows to recover $Δ$-increments of unobserved diffusion paths on which particles are travelling. Picking some few well-chosen observations we fill regression schemes which, on cubes $A$ where the invariant occupation density is strictly positive, allow to estimate the diffusion coefficient of the one-particle motion at nonparametric rates.

math.PR↗

arXiv2019

Entrance laws for annihilating Brownian motions and the continuous-space voter model

Consider a system of particles moving independently as Brownian motions until two of them meet, when the colliding pair annihilates instantly. The construction of such a system of annihilating Brownian motions (aBMs) is straightforward as long as we start with a finite number of particles, but is more involved for infinitely many particles. In particular, if we let the set of starting points become increasingly dense in the real line it is not obvious whether the resulting systems of aBMs converge and what the possible limit points (entrance laws) are. In this paper, we show that aBMs arise as the interface model of the continuous-space voter model. This link allows us to provide a full classification of entrance laws for aBMs. We also give some examples showing how different entrance laws can be obtained via finite approximations. Further, we discuss the relation of the continuous-space voter model to the stepping stone and other related models. Finally, we obtain an expression for the $n$-point densities of aBMs starting from an arbitrary entrance law.

math.PR↗

arXiv2018

Coexistence and duality in competing species models

We discuss some stochastic spatial generalizations of the Lotka--Volterra model for competing species. The generalizations take the forms of spin systems on general discrete sets and interacting diffusions on integer lattices. Methods for proving coexistence in these generalizations and some related open questions are discussed. We use duality as the central point of view. It relates coexistence of the models to survival of their dual processes.

math.PR↗

arXiv2016

The infinite rate symbiotic branching model: from discrete to continuous space

The symbiotic branching model describes a spatial population consisting of two types that are allowed to migrate in space and branch locally only if both types are present. We continue our investigation of the large scale behaviour of the system started in Blath, Hammer and Ortgiese (2016), where we showed that the continuum system converges after diffusive rescaling. Inspired by a scaling property of the continuum model, a series of earlier works initiated by Klenke and Mytnik (2010, 2012) studied the model on a discrete space, but with infinite branching rate. In this paper, we bridge the gap between the two models by showing that by diffusively rescaling this discrete space infinite rate model, we obtain the continuum model from Blath, Hammer and Ortgiese (2016). As an application of this convergence result, we show that if we start the infinite rate system from complementary Heaviside initial conditions, the initial ordering of types is preserved in the limit and that the interface between the types consists of a single point.

math.PR↗

arXiv2016

A new look at duality for the symbiotic branching model

The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $γ$ is sent to infinity. This limit is particularly interesting since it captures the large scale behaviour of the system. As an application of the duality, we can explicitly identify the $γ= \infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.

math.PR↗

arXiv2016

The scaling limit of the interface of the continuous-space symbiotic branching model

The continuous-space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other. If started with complementary Heaviside initial conditions, the interface where both populations coexist remains compact. Together with a diffusive scaling property, this suggests the presence of an interesting scaling limit. Indeed, in the present paper, we show weak convergence of the diffusively rescaled populations as measure-valued processes in the Skorokhod, respectively the Meyer-Zheng, topology (for suitable parameter ranges). The limit can be characterized as the unique solution to a martingale problem and satisfies a "separation of types" property. This provides an important step toward an understanding of the scaling limit for the interface. As a corollary, we obtain an estimate on the moments of the width of an approximate interface.

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