Nontrivial Stationary Distribution Research Articles

A stochastic spatial model for the sterile insect control strategy

In the system we study, 1’s and 0’s represent occupied and vacant sites in the contact process with births at rate λ and deaths at rate 1. −1’s are sterile individuals that do not reproduce but appear spontaneously on vacant sites at rate α and die at rate θα. We show that the system (which is attractive but has no dual) dies out at the critical value and has a nontrivial stationary distribution when it is supercritical. Our most interesting results concern the asymptotics when α→0. In this regime the process resembles the contact process in a random environment.

Motion by mean curvature in interacting particle systems

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., Cox et al. (Astérisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:477–496, 2009, Electron J Probab 19:1–64, 2014), Durrett and Neuhauser (Ann Probab 22:289–333, 1994). These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge et al. (Electron J Probab 22:1–40, 2017) to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al. (Theoret Pop Biol 55(1999):270–282, 1999) there were two nontrivial stationary distributions.

The explicit expression of non-trivial stationary distribution of SDEs under regime switching

This paper mainly focuses on the stationary probability distribution of stochastic differential equations (SDEs) under regime switching. We give the explicit expression of stationary distribution according to the limit probability distribution of SDEs under some conditions. As a special case, we obtain the stationary distribution of SDEs with Markovian switching as the weighted mean of the stationary distribution of its subsystems. In the end, an example and some conclusions are given to illustrate our theoretical results of this paper.

The zealot voter model

Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0’s are ordinary voters and 1’s are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontrivial stationary distribution, space is represented by a tree. The dynamics are a variant of the biased voter: if $x$ has degree $d(x)$ then at rate $d(x)p_{k}$ the individual at $x$ consults $k\ge 1$ neighbors. If at least one neighbor is 1, they adopt state 1, otherwise they become 0. In addition at rate $p_{0}$ individuals with opinion 1 change to 0. As in the contact process on trees, we are interested in determining when the zealots survive and when they will survive locally.

Phase transitions for a planar quadratic contact process

We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λf for the process to survive from a finite set and λe for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the “generic population model” of Bramson and Gray (1991) [3]. An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect to reflection in each axis. There is a speed c(θ) for the expansion of the process in each direction. If c(θ)>0 in all directions, then λ>λf, while if at least one speed is positive, then λ>λe. It is a challenging open problem to show that if some speed is negative, then the system dies out from any finite set.

Coexistence of grass, saplings and trees in the Staver–Levin forest model

In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone’s model in which 0${}={}$vacant, 1${}={}$juvenile and 2${}={}$a mature individual capable of giving birth, and the Staver–Levin forest model in which 0${}={}$grass, 1${}={}$sapling and 2${}={}$tree. Our first result shows that if $(0,0)$ is an unstable fixed point of the mean-field ODE for densities of 1’s and 2’s then when the range of interaction is large, there is positive probability of survival starting from a finite set and a stationary distribution in which all three types are present. The result we obtain in this way is asymptotically sharp for Krone’s model. However, in the Staver–Levin forest model, if $(0,0)$ is attracting then there may also be another stable fixed point for the ODE, and in some of these cases there is a nontrivial stationary distribution.

The contact process with fast voting

Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site, and across each edge between nearest neighbors births occur at rate $\lambda$ and voting events occur at rate $\theta$. We are interested in the asymptotics as $\theta \to\infty$ of the critical value $\lambda_c(\theta)$ for the existence of a nontrivial stationary distribution. In $d \ge 3$, $\lambda_c(\theta) \to 1/(2d\rho_d)$ where $\rho_d$ is the probability a $d$ dimensional simple random walk does not return to its starting point.In $d=2$, $\lambda_c(\theta)/\log(\theta) \to 1/4\pi$, while in $d=1$, $\lambda_c(\theta)/\theta^{1/2}$ has $\liminf \ge 1/\sqrt{2}$ and $\limsup < \infty$.The lower bound might be the right answer, but proving this, or even getting a reasonable upper bound, seems to be a difficult problem.

Effects of Dispersal for a Logistic Growth Population in Random Environments

We study a stochastic logistic model with diffusion between two patches in this paper. Using the definition of stationary distribution, we discuss the effect of dispersal in detail. If the species are able to have nontrivial stationary distributions when the patches are isolated, then they continue to do so for small diffusion rates. In addition, we use some examples and numerical experiments to reflect that diffusions are capable of both stabilizing and destabilizing a given ecosystem.

Chaos in a spatial epidemic model

We investigate an interacting particle system inspired by the gypsy moth, whose populations grow until they become sufficiently dense so that an epidemic reduces them to a low level. We consider this process on a random 3-regular graph and on the $d$-dimensional lattice and torus, with $d\geq2$. On the finite graphs with global dispersal or with a dispersal radius that grows with the number of sites, we prove convergence to a dynamical system that is chaotic for some parameter values. We conjecture that on the infinite lattice with a fixed finite dispersal distance, distant parts of the lattice oscillate out of phase so there is a unique nontrivial stationary distribution.

Individual versus cluster recoveries within a spatially structured population

Stochastic modeling of disease dynamics has had a long tradition. Among the first epidemic models including a spatial structure in the form of local interactions is the contact process. In this article we investigate two extensions of the contact process describing the course of a single disease within a spatially structured human population distributed in social clusters. That is, each site of the $d$-dimensional integer lattice is occupied by a cluster of individuals; each individual can be healthy or infected. The evolution of the disease depends on three parameters, namely the outside infection rate which models the interactions between the clusters, the within infection rate which takes into account the repeated contacts between individuals in the same cluster, and the size of each social cluster. For the first model, we assume cluster recoveries, while individual recoveries are assumed for the second one. The aim is to investigate the existence of nontrivial stationary distributions for both processes depending on the value of each of the three parameters. Our results show that the probability of an epidemic strongly depends on the recovery mechanism.

Stationary distribution of finite-size systems with absorbing states

We introduce a procedure that allows us to obtain nontrivial stationary distributions of finite-size models with absorbing states. Two models are studied: the contact process and the sandpile model with height restriction. To avoid the permanence of the system in the absorbing state we create a small perturbation that drives the system out of the absorbing state. In the former model a particle is created, in the latter an active site is created. The stationary distributions around the critical point are analyzed by the use of finite-size scaling.

Epidemics with Recovery in $D = 2$

We consider a modification of the spatial epidemic with removal that has regrowth of susceptibles. We show that if the original epidemic is supercritical, then the modified process has a nontrivial stationary distribution.

Complete convergence theorem for a competition model

In this paper we consider a hierarchical competition model. Durrett and Swindle have given sufficient conditions for the existence of a nontrivial stationary distribution. Here we show that under a slightly stronger condition, the complete convergence theorem holds and hence there is a unique nontrivial stationary distribution.

Annihilating branching processes

We consider Markov processes η t ⊂ Z d in which (i) particles die at rate δ ⩾ 0, (ii) births from x to a neighboring y occur at rate 1, and (iii) when a new particle lands on an occupied site the particles annihilate each other and a vacant site results. When δ = 0 product measure with density 1 2 is a stationary distribution; we show it is the limit whenever P(η 0≠ ø) = 1. We also show that if δ is small there is a nontrivial stationary distribution, and that for any δ there are most two extremal translation invariant stationary distributions.

THE STATIONARY DISTRIBUTIONS AND THE STEADY PROBABILITY CURRENTS OF ONE-DIMENSIONAL DIFFUSION PROCESSES UNDER NON-LOCAL BOUNDARY CONDITIONS

In the present paper we study the limit distributions of one-dimensional diffusion processes under non-local boundary conditions. We obtain the existence conditions for the non-trivial stationary distributions, get their concrete formulas in all cases according to boundary classification, and give the results of some probability explanations. One-dimensional diffusion processes, which provide a simple mathematical model for studying irreversible steady systems, show that the basic characteristic of an irreversible stationary state is the occurrence of the probability current, i.e. the circulation phenomenon.