Analysis of a spatial Lotka-Volterra model with a finite range predator-prey interaction

We perform an analysis of a recent spatial version of the classical Lotka-Volterra model, where a finite scale controls individuals' interaction. We study the behavior of the predator-prey dynamics in physical spaces higher than one, showing how spatial patterns can emerge for some values of the interaction range and of the diffusion parameter.Comment: 7 pages, 7 figure

The Richit-Richards family of distributions and its use in forestry

Johnson's SB and the logit-logistic are four-parameter distribution models that may be obtained from the standard normal and logistic distributions by a four-parameter transformation. For relatively small data sets, such as diameter at breast height measurements obtained from typical sample plots, distribution models with four or less parameters have been found to be empirically adequate. However, in situations in which the distributions are complex, for example in mixed stands or when the stand has been thinned or when working with aggregated data, then distribution models with more shape parameters may prove to be necessary. By replacing the symmetric standard logistic distribution of the logit-logistic with a one-parameter “standard Richards” distribution and transforming by a five-parameter Richards function, we obtain a new six-parameter distribution model, the “Richit-Richards”. The Richit-Richards includes the “logit-Richards”, the “Richit-logistic”, and the logit-logistic as submodels. Maximum likelihood estimation is used to fit the model, and some problems in the maximum likelihood estimation of bounding parameters are discussed. An empirical case study of the Richit-Richards and its submodels is conducted on pooled diameter at breast height data from 107 sample plots of Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.). It is found that the new models provide significantly better fits than the four-parameter logit-logistic for large data sets

Approximating Fixation Probabilities in the Generalized Moran Process

We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312--316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned 'fitness' value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness $r>0$ placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when $r\geq 1$) and of extinction (for all $r>0$).Comment: updated to the final version, which appeared in Algorithmic

Commitment versus persuasion in the three-party constrained voter model

In the framework of the three-party constrained voter model, where voters of two radical parties (A and B) interact with "centrists" (C and Cz), we study the competition between a persuasive majority and a committed minority. In this model, A's and B's are incompatible voters that can convince centrists or be swayed by them. Here, radical voters are more persuasive than centrists, whose sub-population consists of susceptible agents C and a fraction zeta of centrist zealots Cz. Whereas C's may adopt the opinions A and B with respective rates 1+delta_A and 1+delta_B (with delta_A>=delta_B>0), Cz's are committed individuals that always remain centrists. Furthermore, A and B voters can become (susceptible) centrists C with a rate 1. The resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph. At mean field level, there is a continuous transition from a coexistence phase when zeta= Delta_c. In a finite population of size N, demographic fluctuations lead to centrism consensus and the dynamics is characterized by the mean consensus time tau. Because of the competition between commitment and persuasion, here consensus is reached much slower (zeta=Delta_c) than in the absence of zealots (when tau\simN). In fact, when zeta<Delta_c and there is an initial minority of centrists, the mean consensus time asymptotically grows as tau\simN^{-1/2} e^{N gamma}, where gamma is determined. The dynamics is thus characterized by a metastable state where the most persuasive voters and centrists coexist when delta_A>delta_B, whereas all species coexist when delta_A=delta_B. When zeta>=Delta_c and the initial density of centrists is low, one finds tau\simln N (when N>>1). Our analytical findings are corroborated by stochastic simulations.Comment: 25 pages, 6 figures. Final version for the Journal of Statistical Physics (special issue on the "applications of statistical mechanics to social phenomena"

Evolutionary dynamics of Lewis signaling games: signaling systems vs. partial pooling

Transfer of information between senders and receivers, of one kind or another, is essential to all life. David Lewis introduced a game theoretic model of the simplest case, where one sender and one receiver have pure common interest. How hard or easy is it for evolution to achieve information transfer in Lewis signaling?. The answers involve surprising subtleties. We discuss some if these in terms of evolutionary dynamics in both finite and infinite populations, with and without mutation

Estimating structural and functional relationships

AbstractThis paper surveys the problem of estimating a linear relationship between variables which are observed with error. These are either fixed variables (functional relationship) or random variables (structural relationship). After considering various conditions for identifiability, estimation methods are surveyed in various cases when additional information is available, including Wald's method, the use of instrumental variates, and the case of more than two variables. The paper concludes with a list of unsolved problems