arXiv subjects

Shubhamoy Nandan

Publications and source records attributed to Shubhamoy Nandan.

4 recordsLinked to original sources

arXiv2023

Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium

We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising $active$ and $dormant$ individuals are structured in colonies labelled by $\mathbb{Z}^d,~d\geq 1$. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: $\heartsuit$, $\spadesuit$. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a $fixed$ environment. We also show that when the migration kernel is $recurrent$, for a.e. realization of the environment, the system with an initially $consistent$ type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-$\heartsuit$ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-$\heartsuit$ densities in dormant and active population biased by ratio of the two population sizes at the target colony. For the proofs, we use duality and environment seen by particle introduced in Dolgopyat and Goldsheid (2019) for RWRE on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with single-particle dual to a reversible ergodic distribution which we transfer to the spatial system by using duality.

math.PR↗

arXiv2021

Spatially Inhomogeneous Populations with Seed-banks: II. Clustering Regime

We consider a spatial version of the classical Moran model with seed-banks where the constituent populations have finite sizes. Individuals live in colonies labelled by $\mathbb{Z}^d$, $d\geq 1$, playing the role of a geographic space, carry one of the two $types$: $\heartsuit$ or $\spadesuit$, and change type via $resampling$ as long as they are $active$. Each colony contains a seed-bank into which individuals can enter to become $dormant$, suspending their resampling until they exit the seed-bank and become active again. Individuals resample not only from their own colony, but also from other colonies according to a symmetric random walk transition kernel. The latter is referred to as $migration$. The sizes of the populations vary across colonies and remain constant in time. It was shown in Hollander and Nandan (2021) that the system is well-defined, admits a family of equilibria parametrized by the initial density of type $\heartsuit$, and exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). This dichotomy is determined by a clustering criterion that is given in terms of a dual of the system, which consists of a system of $interacting$ coalescing random walks. In this paper we provide an alternative clustering criterion, given in terms of an auxiliary dual that is simpler than the original dual, and identify a range of parameters for which the criterion is met, which we refer to as the $clustering$ $regime$. It turns out that if the sizes of the active populations are non-clumping (i.e., do not take arbitrarily large values in finite regions of the geographic space) and the relative strengths of the seed-banks (i.e., the ratio of the sizes of dormant and active population in each colony) are bounded uniformly over the geographic space, then clustering prevails if and only if the (symmetrised) migration kernel is recurrent.

math.PR↗

arXiv2021

Switching interacting particle systems: scaling limits, uphill diffusion and boundary layer

In this paper we consider three classes of interacting particle systems on $\mathbb Z$: independent random walks, the exclusion process, and the inclusion process. We allow particles to switch their jump rate (the rate identifies the type of particle) between $1$ (fast particles) and $ε\in[0,1]$ (slow particles). The switch between the two jump rates happens at rate $γ\in(0,\infty)$. In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by $N^{-1}$, time by $N^2$, the switching rate by $N^{-2}$, and letting $N\to\infty$. The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick's law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on $[N]=\{1,\ldots,N\}$, adding boundary reservoirs at sites $1$ and $N$ of fast and slow particles, respectively. Inside $[N]$ particles move as before, but now particles are injected and absorbed at sites $1$ and $N$ with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick's law made possible by the switching between types.

math.PR↗

arXiv2021

Spatially Inhomogeneous Populations with Seed-banks: Duality, Existence and Clustering

We consider a system of interacting Moran models with seed-banks. Individuals live in colonies and are subject to resampling and migration as long as they are $active$. Each colony has a seed-bank into which individuals can retreat to become $dormant$, suspending their resampling and migration until they become active again. The colonies are labelled by $\mathbb{Z}^d$, $d \geq 1$, playing the role of a $geographic\, space$. The sizes of the active and the dormant population are $finite$ and depend on the $location$ of the colony. Migration is driven by a random walk transition kernel. Our goal is to study the equilibrium behaviour of the system as a function of the underlying model parameters. In the present paper we show that, under mild condition on the sizes of the active population, the system is well-defined and has a dual. The dual consists of a system of $interacting$ coalescing random walks in an $inhomogeneous$ environment that switch between active and dormant. We analyse the dichotomy of $coexistence$ (= multi-type equilibria) versus $clustering$ (= mono-type equilibria), and show that clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one. The presence of the seed-bank $enhances\, genetic\, diversity$. In the dual this is reflected by the presence of time lapses during which the random walks are dormant and do not move.

math.PR↗