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C Cosner

Publications and source records attributed to C Cosner.

10 recordsLinked to original sources
PubMed2003

Deriving reaction-diffusion models in ecology from interacting particle systems.

We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.

Animals↗
PubMed2001

Spatial heterogeneity and critical patch size: Area effects via diffusion in closed environments.

We describe a class of mathematical models for critical patch size in which the mechanisms inducing area effects are based on source-sink population dynamics arising from dispersal throughout a closed, finite, but spatially heterogeneous environment. Our models are reaction-diffusion equations, but unlike classical KISS models for area effects they do not assume that there is dispersal across the boundary of the environment into a hostile exterior. We observe that simple rescaling has the same effects in our models as in KISS models and hence predicts the same sort of area effects, but that other sorts of rescaling may not predict area effects. The models considered here provide an alternative to the KISS models used in our previous work on species-area relationships in island biogeography.

Animals↗
PubMed2001

Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction.

Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competition mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equilibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed.

Animals↗
PubMed1999

Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.

Animals↗
PubMed1999

Effects of spatial grouping on the functional response of predators.

A unified mechanistic approach is given for the derivation of various forms of functional response in predator-prey models. The derivation is based on the principle of mass action but with the crucial refinement that the nature of the spatial distribution of predators and/or opportunities for predation are taken into account in an implicit way. If the predators are assumed to have a homogeneous spatial distribution, then the derived functional response is prey-dependent. If the predators are assumed to form a dense colony or school in a single (possibly moving) location, or if the region where predators can encounter prey is assumed to be of limited size, then the functional response depends on both predator and prey densities in a manner that reflects feeding interference between predators. Depending on the specific assumptions, the resulting functional response may be of Beddington-DeAngelis type, of Hassell-Varley type, or ratio-dependent.

Animals↗
PubMed1999

A comparison of foraging strategies in a patchy environment.

In this paper we compare foraging strategies that might be used by predators seeking prey in a patchy environment. The strategies differ in the extent to which predators aggregate in response to prey density. The approach to the comparison is suggested by the idea of evolutionarily stable strategies. A strategy is said to be evolutionarily stable if it cannot be invaded by another strategy. Thus we examine scenarios where a small number of individuals using one strategy are introduced into a situation where a large number of individuals using the other strategy are already present. However, our foraging models do not explicitly incorporate predator population dynamics, so we use net energy uptake as a surrogate for reproductive fitness. In cases where all of the patches visited by predators sustain prey populations, we find that for any pair of strategies one of them will have a higher net energy uptake than the other whether it is the resident or the introduced strain. However, which one is higher will typically depend on the total predator population, which is determined by the resident strain. If the predators leave prey densities high, the more aggregative strain will have the advantage. If the predators reduce prey densities to low levels the less aggregative strain will have the advantage. In cases where one strain of predators aggregates in response to prey density and the other does not, then there might be patches which do not contain prey but do contain (non-aggregating) predators. In those cases, there is the possibility that whichever strategy is used by the introduced strain will yield a higher energy uptake than that used by the resident strain. This suggests that if some patches are empty of prey then aggregative and non-aggregative strategies may be able to coexist.

Animals↗
PubMed1998

Competitive reversals inside ecological reserves: the role of external habitat degradation.

Habitat degradation is the slow--and often subtle--deterioration in habitat quality that accompanies human activities through increases in road density, pesticide use, hunting pressure, etc. Such degradation is of particular concern in fragmented habitats where economic or jurisdictional boundaries rather than ecological ones determine the level of exploitation adjoining habitat patches endure. To examine the consequences habitat degradation might have on species interactions, we posited a patch of pristine habitat surrounded by "matrix" habitat whose degradation level was variable. Using a coupled pair of diffusive Lotka-Volterra competition equations with Robin (mixed) boundary conditions, we modeled the dynamics of two competing species inhabiting the pristine patch and incorporated matrix degradation through a tunable "hostility" parameter representing species' mortality rates in the matrix. We found that the numerical range of competition coefficients over which one species is the competitive dominant and the other inferior may grow or shrink as matrix quality deteriorates. In some cases, degradation of the exterior habitat would bring about a complete competitive reversal inside the preserve. This result, wherein a formerly inferior species supplants a formerly dominant one--even inside the "protected" remnant patch itself--has policy implications for both nature reserve design and management of human activities outside park boundaries.

Animals↗
PubMed1996

Spatially explicit models for the population dynamics of a species colonizing an island.

We construct reaction-diffusion models for the population dynamics of a species colonizing an island from a source population on a continent. We view the source population as inducing a density or flux of immigrants onto the island and interpret colonization as succeeding if the population on the island is predicted to persist even when immigration from the continent is stopped. To capture the observation that a sufficiently large population or density must be attained for colonization to succeed, we assume Allee (i.e., bistable) dynamics rather than logistic dynamics for the colonizing population. We consider the cases of colonization in both the absence and presence of a competitor. We use reaction-diffusion theory, especially comparison methods and sub- and supersolutions, to determine how parameters such as the distance from the continent to the island and the dispersal, birth and mortality rates, carrying capacity, and minimum viable population density of the colonizing species affect the outcome of the attempted colonization. In the case of colonization in the presence of a competitor we consider a number of scenarios involving different types and strengths of competition. Our analysis permits us to draw conclusions about the characteristics of a species that make it a good colonizer.

Animals↗
PubMed1995

Models for the effects of individual size and spatial scale on competition between species in heterogeneous environments.

A spatially explicit model for competition with dispersal in a heterogeneous environment is used to study the effects of individual size and the spatial scale of the environment on the competitive interactions between species. The model is a Lotka-Volterra competition system with diffusion and with spatial variation in some coefficients. The coefficients in the model are taken to reflect a situation where the larger competitor typically disperses farther in unit time than the smaller and reproduces less rapidly, but has an advantage in contests or other forms of interference competition. The environment is assumed to be closed, i.e., it is assumed that individuals do not leave through the boundary. The environment is generally assumed to consist of a patch of favorable habitat surrounded by less favorable regions. The effects of spatial scale are studied by examining how the predictions of the model change as the size of the favorable patch is varied. The predictions turn out to be in qualitative agreement with the results of some empirical studies.

Ecosystem↗