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Biomedical subjects

R Durrett

Publications and source records attributed to R Durrett.

9 recordsLinked to original sources
PubMed2000

Lessons on pattern formation from planet WATOR.

It is well known that if reacting species experience unequal diffusion rates, then dynamics that lead to a constant steady state in a "well-mixed" environment can in a spatial setting lead to interesting patterns. In this paper, we focus on complementary pattern formation mechanisms that operate even when the diffusion rates are equal. In particular, we can say that when the mean-field ODE has an attracting periodic orbit then the stochastic spatial model will have large-scale spatial structures in equilibrium. We explore this mechanism in depth through the dynamics of the simulator WATOR.

Animals↗
PubMed2000

Spatial models for hybrid zones.

We introduce a spatially explicit model of natural hybrid zones that allows us to consider how patterns of allele frequencies and linkage disequilibria change over time. We examine the influence of hybrid zone origins on patterns of variation at two loci, a locus under selection in a two-patch environment, and a linked neutral locus. We consider several possible starting conditions that represent explicit realizations of two alternative scenarios for hybrid zone origins: primary intergradation and secondary contact. Our results indicate that in some circumstances, differences in hybrid zone origins will result in substantially different patterns of variation that may persist for thousands of generations. Our conclusions are generally similar to those previously derived from partial differential equations, but there are also some important differences.

Alleles↗
PubMed2000

Distribution and abundance of microsatellites in the yeast genome can Be explained by a balance between slippage events and point mutations.

We fit a Markov chain model of microsatellite evolution introduced by Kruglyak et al. to data on all di-, tri-, and tetranucleotide repeats in the yeast genome. Our results suggest that many features of the distribution of abundance and length of microsatellites can be explained by this simple model, which incorporates a competition between slippage events and base pair substitutions, with no need to invoke selection or constraints on the lengths. Our results provide some new information on slippage rates for individual repeat motifs, which suggest that AT-rich trinucleotide repeats have higher slippage rates. As our model predicts, we found that many repeats were adjacent to shorter repeats of the same motif. However, we also found a significant tendency of microsatellites of different motifs to cluster.

Base Sequence↗
PubMed1999

Local frequency dependence and global coexistence.

In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-dependent competition influenced whether two species could coexist. Individuals of the two species were randomly placed on a grid and allowed to interact according to local frequency-dependent rules. For four different frequency-dependent scenarios, the results indicated that over a broad parameter range the two species could coexist. Comparisons between explicit spatial simulations and the mean-field approximation indicate that coexistence occurs over a broader region in the explicit spatial simulation.

Ecosystem↗
PubMed1998

Spatial aspects of interspecific competition.

Using several variants of a stochastic spatial model introduced by Silvertown et al., we investigate the effect of spatial distribution of individuals on the outcome of competition. First, we prove rigorously that if one species has a competitive advantage over each of the others, then eventually it takes over all the sites in the system. Second, we examine tradeoffs between competition and dispersal distance in a two-species system. Third, we consider a cyclic competitive relationship between three types. In this case, a nonspatial treatment leads to densities that follow neutrally stable cycles or even unstable spiral solutions, while a spatial model yields a stationary distribution with an interesting spatial structure.

Competitive Behavior↗
PubMed1997

Allelopathy in Spatially Distributed Populations

In a homogeneously mixing population of E. coli, colicin-producing and colicin-sensitive strategies both may be evolutionarily stable for certain parameter ranges, with the outcome of competition determined by initial conditions. In contrast, in a spatially-structured population, there is a unique ESS for any given set of parameters; the outcome is determined by how effective allelopathy is in relation to its costs. Furthermore, in a spatially-structured environment, a dynamic equilibrium may be sustained among a colicin-sensitive type, a high colicin-producing type, and a "cheater" that expends less on colicin production but is resistant. Copyright 1997 Academic Press Limited

Journal Article↗
PubMed1996

From individuals to epidemics.

Heterogeneous mixing fundamentally changes the dynamics of infectious diseases; finding ways to incorporate it into models represents a critical challenge. Phenomenological approaches are deficient in their lack of attention to underlying process; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail. The challenge then is to find ways to bridge these levels of description, starting from individual-based models and deriving macroscopic descriptions from them that retain essential detail, and filter out the rest. In this paper, attempts to achieve this transformation are described for a class of models where nonrandom mixing arises from the spatial localization of interactions. In general, the epidemic threshold is found to be larger owing to spatial localization than for a homogeneously mixing population. An improved estimate of the dynamics is developed by the use of moment equations, and a simple estimate of the threshold in terms of a 'dyad heuristic'. For more general models in which local infection is not described by mass action, the connection with related partial differential equations is investigated.

Communicable Diseases↗