Stochastic models in seed dispersals: random walks and birth-death processes

Seed dispersals deal with complex systems through which the data collected using advanced seed tracking facilities pose challenges to conventional approaches, such as empirical and deterministic models. The use of stochastic models in current seed dispersal studies is encouraged. This review describes three existing stochastic models:the birth–death process (BDP), a 2 dimensional (2D) symmetric ran-dom walks and a 2D intermittent walks. The three models possess Markovian property, which make them flexible for studying natural phenomena. Only a few of applications in ecology are found in seed dispersals. The review illustrates how the models are to be used in seed dispersals context. Using the nonlinear BDP, we formulate the individual-based models for two competing plant species while the cover time model is formulated by the symmetric and intermittent random walks. We also show that these three stochastic models can be formulated using the Gillespie algorithm. The full cover time obtained by the symmetric random walks can approximate the Gumbel distribution pattern as the other searching strategies do.We suggest that the applications of these models in seed dispersals may lead to understanding of many complex systems, such as the seed removal experiments and behaviour of foraging agents, among others

A polynomial time approximation scheme for computing the supremum of Gaussian processes

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop {\mathbb{E}}_{X\leftarrow\mathcal{N}^d}[\sup_{v\in V}|\langle v,X\rangle|]$ deterministically in time $\operatorname {poly}(d)\cdot|V|^{O_{\varepsilon}(1)}$. Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471]. This answers an open question of Lee (2010) and Ding [Ann. Probab. 42 (2014) 464-496]. The study of supremum of Gaussian processes is of considerable importance in probability with applications in functional analysis, convex geometry, and in light of the recent breakthrough work of Ding, Lee and Peres [Ann. of Math. (2) 175 (2012) 1409-1471], to random walks on finite graphs. As such our result could be of use elsewhere. In particular, combining with the work of Ding [Ann. Probab. 42 (2014) 464-496], our result yields a PTAS for computing the cover time of bounded-degree graphs. Previously, such algorithms were known only for trees. Along the way, we also give an explicit oblivious estimator for semi-norms in Gaussian space with optimal query complexity. Our algorithm and its analysis are elementary in nature, using two classical comparison inequalities, Slepian's lemma and Kanter's lemma.Comment: Published in at http://dx.doi.org/10.1214/13-AAP997 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lock-in Problem for Parallel Rotor-router Walks

The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al.(2003) proved that a single rotor-router walk on any graph with m edges and diameter $D$ stabilizes to a traversal of an Eulerian circuit on the set of all 2m directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most 2mD steps. The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the conjecture that a system of k \textgreater{} 1 parallel walks also stabilizes with a period of length at most $2m$ steps. In this work we disprove this conjecture, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of $O(m^4 D^2 + mD \log k)$ and $O(m^5 k^2)$ on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of $k = 2$ walks