Models for spatially distributed populations: The effect of within-patch variability

This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are nonlinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study.     

Design of a nonuniformly distributed biocatalyst

The properties of a nonuniformly distributed biocatalyst, where the active enzymes are immobilized on the exterior or the interior portions o a solid support, are compared with those of a conventional biocatalyst which is uniformly distributed in a spherical geometry. To investigate the performance of nonuniformly distributed biocatalysts their effectiveness factors are computed and compared for six different enzyme distribution configurations: one-half core, one-half shell, one-third center space, one-third middle annulus, one-third outer shell, and the uniformly distributed. According to the results of numerical analysis, the biocatalyst performance of the exterior "shell" configuration is always far more effective for the immobilized enzymes with positive order reaction kinetics such as Michaelis-Menten and competitive product inhibition. However, in the case of negative order enzymatic reaction kinetics such as substrate inhibition, the interior "core" configuration of the biocatalyst can render far greater enzyme utilization efficiency.     

Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread

Summary In this paper we use Aronson's and Weinberger's [1–4] concept of asymptotic speed to estimate the asymptotic behaviour of the solution of a nonlinear integral equation (with the nonlinearity not being monotone), which describes the development of a spatially distributed population.     

Imitative behavior. A theoretical view

This article reviews the results of experimental studies on imitative behavior reported by various investigators, and then discusses the possible brain mechanisms responsible for this behavior. It was found that human infants in their first hours of life were already capable of spontaneous imitation of simple motor acts demonstrated by an adult, without previous training or reward; these observations suggest that imitative behavior is an innate process that can be considered an unconditional reflex of imitation. It was also found that satiated animals resumed eating when they saw their companions eating. In the latter case, the imitative reflex triggered the previously acquired feeding behavior. Similar mechanisms could be responsible for the phenomenon of eating more in the presence of companions than in their absence, as well as that of preferring the food chosen by companions. When followed by a reward, the imitative act can be learned--that is, transformed into an instrumental conditional response; learning by imitation of simple motor acts was observed in animals, and that of complex motor acts was observed in children who had already achieved a certain developmental stage. In animals, learning complex motor tasks was facilitated by previous observation of a companion performing this task. In this case, the presence of the observer during the session could lead to habituation of the experimental situation and production of associations between this situation and stimuli or emotions related to the reward or punishment, and might result in more efficient learning later. The imitative behavior can be inhibited by stimuli producing responses antagonistic to the act of imitation.     

Self-organized dynamics in spatially structured populations

Self-organization and pattern formation represent the emergence of order in temporal and spatial processes. Self-organization in population ecology is gaining attention due to the recent advances concerning temporal fluctuations in the population size of dispersal-linked subunits. We shall report that spatially structured models of population renewal promote the emergence of a complex power law order in spatial population dynamics. We analyse a variety of population models showing that self-organization can be identified as a temporal match in population dynamics among local units, and how the synchrony changes in time. Our theoretical results are concordant with analyses of population data on the Canada lynx.     

Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence

In order to understand the spatio-temporal structure of epidemics beyond that permitted with classical SIR (susceptible-infective-recovered)-type models, a new mathematical model for the spread of a viral disease in a population of spatially distributed hosts is described. The positions of the hosts are randomly generated in a rectangular habitat. Encounters between any pair of individuals are according to a Poisson process with a mean rate that declines exponentially as the distance between them increases. The contact rate allows the mean rates to be set at a certain number of encounters per day on average. The relevant state variables for each individual at any time are given by the solution of a pair of coupled differential equations for the viral load and the quantity of general immune system effectors which reduce the viral load. The parameters describing within-host viral-immune system dynamics are generated randomly to reflect variability across a population. Transmission is assumed to depend on the viral loads in donors and occurs with a probability ptrans. The initial conditions are such that one randomly chosen individual carries a randomly chosen amount of the virus, whereas the rest of the population is uninfected. Simulations reveal local or whole-population responses. Whole-population disease spread may be in the form of isolated or multiple occurrences, the latter often being approximately periodic. The mechanisms of this oscillatory behaviour are analyzed in terms of several parameters and the distribution of critical points in the host dynamical systems. Increased contact rate, increased probability of transmission and decreased threshold for viral transmission, decreased immune strength and increased viral growth rate all increase the probability of multiple outbreaks and the distribution of the critical points also plays a role.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

Pollen dispersal in spatially aggregated populations

We perform a theoretical study of effective pollen dispersal within plant populations exhibiting intraspecific spatial aggregation. We simulate nonuniform distributions of individuals by means of a Poisson cluster process and use an individual-based spatially explicit model of pollen dispersal to assess the effects of different aggregation patterns on the effective pollen pool size (N(ep)) and the axial variance of pollen dispersal (sigma (p)). Results show clear interactions between clumping and both N(ep) and sigma (p), whose precise form and intensity depend on the relative spatial scale of aggregation to pollen dispersal range. If clump size is small relative to dispersal range, clumping results in lower N(ep) and sigma (p) than in randomly distributed populations. Interestingly, by contrast, aggregation may actually enlarge N(ep) and has minimum impact on sigma (p) if clump size is near or above the scale of dispersal. High intraclump to global density ratios enhance the sensitivity of both N(ep) and sigma (p) to clumping, while leptokurtic pollen dispersal generates sharper reductions of both N(ep) and sigma (p) for small clump sizes and stronger increments of N(ep) for larger clump sizes. Overall, our results indicate that isolation-by-distance models in plants should not ignore the effects of intraspecific spatial aggregation on effective dispersal.     

Intra-deme molecular diversity in spatially expanding populations

We report here a simulation study examining the effect of a recent spatial expansion on the pattern of molecular diversity within a deme. We first simulate a range expansion in a virtual world consisting in a two-dimensional array of demes exchanging a given proportion of migrants (m) with their neighbors. The recorded demographic and migration histories are then used under a coalescent approach to generate the genetic diversity in a sample of genes. We find that the shape of the gene genealogies and the overall pattern of diversity within demes depend not only on the age of the expansion but also on the level of gene flow between neighboring demes, as measured by the product Nm, where N is the size of a deme. For small Nm values (< approximately 20 migrants sent outwards per generation), a substantial proportion of coalescent events occur early in the genealogy, whereas with larger levels of gene flow, most coalescent events occur around the time of the onset of the spatial expansion. Gene genealogies are star shaped, and mismatch distributions are unimodal after a range expansion for large Nm values. In contrast, gene genealogies present a mixture of both very short and very long branch lengths, and mismatch distributions are multimodal for small Nm values. It follows that statistics used in tests of selective neutrality like Tajima's D statistic or Fu's F(S) statistic will show very significant negative values after a spatial expansion only in demes with high Nm values. In the context of human evolution, this difference could explain very simply the fact that analyses of samples of mitochondrial DNA sequences reveal multimodal mismatch distributions in hunter-gatherers and unimodal distributions in post-Neolithic populations. Indeed, the current simulations show that a recent increase in deme size (resulting in a larger Nm value) is sufficient to prevent recent coalescent events and thus lead to unimodal mismatch distributions, even if deme sizes (and therefore Nm values) were previously much smaller. The fact that molecular diversity within deme is so dependent on recent levels of gene flow suggests that it should be possible to estimate Nm values from samples drawn from a single deme.     

Pattern formation in systems with one spatially distributed species

We describe a three-species mechanism for spatial pattern formation in which only one species spatially moves. We show that a bifurcation to traveling or standing waves occurs. We contrast this mechanism for pattern formation with the better known cases where more than one species moves.     

Evolutionary games for spatially dispersed populations

An evolutionary game model is developed that incorporates both spatial dispersion and density effects in the evolutionary dynamic. It is shown that a stable equilibrium (e.g. an evolutionarily stable strategy) of the non-dispersed frequency dynamic becomes a stable equilibrium of the larger system if population density stabilizes at these fixed frequencies. It is also shown, by example, that other equilibria, whose frequencies change from one location to another, may appear when dispersal rates are relatively small.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A6187Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A7822     

On selection criteria in spatially distributed models of competition

Discrete models of competitors (initial population and mutants) are considered in which reproduction is set by an increasing and concave function, and migration in the space consisting of a set of areas is described by a Markov matrix. This allows using the theory of monotonic operators to study problems of selection, coexistence and stability. It is shown that the higher is the number of areas, the more severe are the requirements of selective advantage to the initial population.     

Host resistance and coevolution in spatially structured populations

Natural, agricultural and human populations are structured, with a proportion of interactions occurring locally or within social groups rather than at random. This within-population spatial and social structure is important to the evolution of parasites but little attention has been paid to how spatial structure affects the evolution of host resistance, and as a consequence the coevolutionary outcome. We examine the evolution of resistance across a range of mixing patterns using an approximate mathematical model and stochastic simulations. As reproduction becomes increasingly local, hosts are always selected to increase resistance. More localized transmission also selects for higher resistance, but only if reproduction is also predominantly local. If the hosts disperse, lower resistance evolves as transmission becomes more local. These effects can be understood as a combination of genetic (kin) and ecological structuring on individual fitness. When hosts and parasites coevolve, local interactions select for hosts with high defence and parasites with low transmissibility and virulence. Crucially, this means that more population mixing may lead to the evolution of both fast-transmitting highly virulent parasites and reduced resistance in the host.     

A spatially discrete model for aggregating populations

 In this paper we introduce a spatially discrete model for aggregating populations described by a system of ODEs. We study the long time behavior of the solutions and we show that the model contains mechanisms by which individuals in the population aggregate at particular points in space. Received: 29 June 1996 / Revised version: 5 August 1997     

Endemic disease in environments with spatially heterogeneous host populations

The main interest in epidemic models stems from their use in uncovering certain qualitative features of epidemic processes. A deterministic model of a general epidemic in a population with an arbitrary number of separate population centers is presented. The mixing within each center is assumed to be homogeneous, and the usual threshold theorem holds for each population. The mixing between centers is nonhomogeneous. This model is used to identify the necessary and sufficient conditions under which a disease will become endemic in the general population when each population center is below the threshold required for establishment of the disease and does not mix with other centers. These conditions depend critically on the concavity of the infection rate function with respect to the length of exposure time. The application of these results to host-vector models is discussed.