Point Processes for Fine‐Scale Spatial Genetics and Molecular Ecology

This paper introduces point processes into fine‐scale spatial genetics and molecular ecology. Datasets given in the form of a complete map of individuals and their genotypes can be analyzed by means of the theory of marked or multivariate point processes. Beginning with reformulation of conventional spatial autocorrelation statistics in genetics by the language of point processes, this paper first shows an example of point process models that describe spatial patterns of both tree locations and their genotypes, on the assumption of limited seed dispersal and long pollen movement. The results show that isolation‐by‐distance slightly occurs from the assumption above, and more importantly, an increment of the degree of clustering of trees reduces the degree of genetic clustering. Next, the point process model is applied to field data of secondary forest regenerated after seed tree harvesting, and tests the hypothesis that the current population was formed only from a small number of seed trees. The hypothesis was not acceptable, instead, the alternative assuming advance reproduction conducted prior to the harvesting is supported. The results of this first trial of point process models suggest that point processes can provide a useful mathematical methodology in fine‐scale spatial genetics and molecular ecology.     

Limited seed dispersal shapes fine-scale spatial genetic structure in a Neotropical dioecious large-seeded palm

Seed and pollen dispersal contribute to gene flow and shape the genetic patterns of plants over fine spatial scales. We inferred fine-scale spatial genetic structure (FSGS) and estimated realized dispersal distances in Phytelephas aequatorialis, a Neotropical dioecious large-seeded palm. We aimed to explore how seed and pollen dispersal shape this genetic pattern in a focal population. For this purpose, we genotyped 138 seedlings and 99 adults with 20 newly developed microsatellite markers. We tested if rodent-mediated seed dispersal has a stronger influence than insect-mediated pollen dispersal in shaping FSGS. We also tested if pollen dispersal was influenced by the density of male palms around mother palms in order to further explore this ecological process in large-seeded plants. Rodent-mediated dispersal of these large seeds occurred mostly over short distances (mean 34.76 ± 34.06 m) while pollen dispersal distances were two times higher (mean 67.91 ± 38.29 m). The spatial extent of FSGS up to 35 m and the fact that seed dispersal did not increase the distance at which male alleles disperse suggest that spatially limited seed dispersal is the main factor shaping FSGS and contributes only marginally to gene flow within the population. Pollen dispersal distances depended on the density of male palms, decreasing when individuals show a clumped distribution and increasing when they are scattered. Our results show that limited seed dispersal mediated by rodents shapes FSGS in P. aequatorialis, while more extensive pollen dispersal accounts for a larger contribution to gene flow and may maintain high genetic diversity. Abstract in Spanish is available with online material.     

Genetic structure of Camellia japonica L. in an old-growth evergreen forest, Tsushima, Japan

The spatial genetic structure of Camellia japonica was investigated, using microsatellite markers, in a 4-ha permanent plot within an old-growth forest. Spatial distribution of individuals was also assessed to obtain an insight into spatial relationships between individuals and alleles. Morisita's index of dispersion showed that 518 C. japonica individuals in the plot were clumped, and Moran's I spatial autocorrelation coefficient revealed weak genetic structure, indicating a low level of allele clustering. Average I correlograms showed that there was stronger genetic structure over short-distance classes. The clumped distribution of individuals and the positive autocorrelation over short-distance classes may result from the limited seed dispersal and microsite heterogeneity of the stand, while the genetic structure may be weakened by overlapping seed shadow and extensive pollen flow, mediated by animal vectors, and the high outcrossing rate found in C. japonica.     

The importance of dispersal in determining seed versus safe site limitation of plant populations

The traditional dichotomy of seed versus safe site limitation of plant populations is an oversimplification. While most plant models implicitly assume that the number of safe sites colonized will increase directly with increased seed production by each plant, the number of sites colonized may also strongly depend on patterns of seed dispersal relative to the parent plant, since the majority of a plant’s seeds are deposited very close to it and so not all safe sites are equally accessible. I created a series of spatially explicit individual based plant population models exploring how seed versus safe site limitation is jointly affected by the number of seeds produced per plant and mean dispersal distances. While increased dispersal distance led to reduced seed limitation (more saturation of available safe sites) when a parent plant’s site was temporarily unsuitable following its death, increased dispersal distances could increase seed limitation, especially at low per-plant fecundities, if safe sites did not turn over through time. Models comparing localized to global seed dispersal indicated substantially different degrees of seed limitation for constant per-plant fecundities. Thus seed addition experiments need to be designed to add seeds in realistic spatial patterns to yield meaningful results.     

An explicit approach to evolutionarily stable dispersal strategies: no cost of dispersal

The evolution of dispersal is examined by looking at evolutionarily stable strategies (ESS) for dispersal parameters in discrete time multisite models without any cost of dispersal. ESS are investigated analytically, based on explicit results on sensitivity analysis of matrix models. The basic model considers an arbitrary number of sites and a single age class. An ESS for dispersal parameters is obtained when the spatial reproductive values, calculated at the density-dependent population equilibrium, are equal across sites. From this basic formulation, one derives equivalently that all local populations should be at equilibrium in the absence of migration, and that dispersal between sites should be balanced, i.e., the numbers of individuals arriving to and leaving a site are equal. These results are then generalized to a model with several age classes. Equal age-specific reproductive values do not however imply balanced dispersal in this case. Our results generalize to any number of sites and age classes those available ?M. Doebeli, Dispersal and dynamics, Theoret. Popul. 47 (1995) 82 for two sites and one age class.     

The effect of sex‐biased dispersal on opposite‐sexed spatial genetic structure and inbreeding risk

Natal sex‐biased dispersal has long been thought to reduce the risk of inbreeding by spatially separating opposite‐sexed kin. Yet, comprehensive and quantitative evaluations of this hypothesis are lacking. In this study, we quantified the effectiveness of sex‐biased dispersal as an inbreeding avoidance strategy by combining spatially explicit simulations and empirical data. We quantified the extent of kin clustering by measuring the degree of spatial autocorrelation among opposite‐sexed individuals (FM structure). This allowed us to systematically explore how the extent of sex‐biased dispersal, generational overlap, and mate searching distance, influenced both kin clustering, and the resulting inbreeding in the absence of complementary inbreeding avoidance strategies. Simulations revealed that when sex‐biased dispersal was limited, positive FM genetic structure developed quickly and increased as the mate searching distance decreased or as generational overlap increased. Interestingly, complete long‐range sex‐biased dispersal did not prevent the development of FM genetic structure when generations overlapped. We found a very strong correlation between FM genetic structure and both F_IS under random mating, and pedigree‐based measures of inbreeding. Thus, we show that the detection of FM genetic structure can be a strong indicator of inbreeding risk. Empirical data for two species with different life history strategies yielded patterns congruent with our simulations. Our study illustrates a new application of spatial genetic autocorrelation analysis that offers a framework for quantifying the risk of inbreeding that is easily extendable to other species. Furthermore, our findings provide other researchers with a context for interpreting observed patterns of opposite‐sexed spatial genetic structure.     

Neighborhood models of plant population dynamics. 2. Multi-species models of annuals

Models are developed for the dynamics of multi-species communities of annual plants that lack seed dormancy. These models explicitly include plastic plant growth, the spatial distribution of individuals, and the fact that individuals interact primarily with nearby individuals. Because the models are based on submodels of individual plants (fecundity, survivorship and dispersal, and how these are affected by inter-individual interactions), they provide explanations of community-level phenomena in terms of the biology of individuals. All model parameters and functional forms may be estimated from data obtained in simple experiments of a single years's duration. The models are used to examine the community-level consequences of some types of inter-individual interactions that have been reported in the ecological literature. In addition, the models are used to demonstrate that dispersal may markedly influence the outcome of competition among plant species, even in a physically homogeneous environment, due to an effect of dispersal on the spatial distribution of individuals.     

Fine-scale genetic structure within plots of Polygala reinii (Polygalaceae) having an ant-dispersal seed

Fine-scale genetic structure within a population was analyzed for the myrmecochorous forest perennial Polygala reinii (Polygalaceae) using allozyme loci. In the analysis, two sampling plots were established to cover the isolated patchy distribution within the study population. Size and spatial structure were also examined for the plots to assess their interaction with the genetic structuring. Allozyme analysis based on 13 putative loci encoding 10 enzyme systems showed high genetic variation and low values of fixation indices at the two plots. Spatial autocorrelation analysis based on the multilocus coancestry coefficient (f _ij ) revealed significant genetic structuring in both plots, suggesting limited gene-, especially seed-, dispersal within the population. The spatial structure within the plots, assessed by O-ring statistics, was characterized by the occurrence of spatial clustering of individuals within a few meters. In particular, the range of the spatial clustering roughly corresponded to that of the genetic structuring. While the size structure did not significantly differ between the plots, these results indicate that the fine-scale genetic structure reflects the formation of spatial clustering of related individuals within the population, partly due to the restricted ant-mediated seed dispersal in P. reinii.     

Pollen and seed flow under different predominant winds in wind-pollinated and wind-dispersed species <Emphasis Type="Italic">Engelhardia roxburghiana</Emphasis>

In most plants, the contributions of pollen and seed flow to their genetic structures are generally difficult to disentangle. For typical wind-pollinated and wind-dispersed species Engelhardia roxburghiana in a 20-ha natural forest plot in lower subtropic China, because the prevailing wind directions change during its pollen release and seed dispersal seasons, we could compare its genetic structures in different directions, which could result primarily from pollen or seed flow. Furthermore, because the plot has undergone from an open to a closed canopy stage historically, we also examined forest canopy effects on gene flow in different generations and different directions. Using 522 E. roxburghiana individuals mapped in the plot, our results revealed that greater pollen flow led to biased gene flow in the pollen dispersal-predominant direction (pollen direction), while greater seed flow generated less spatial genetic structure in the seed dispersal-predominant direction (seed direction). The results predicted from generalized additive models indicated that canopy closure enhanced resistance to gene flow from the old generation to the new generation. Analyses by landscape genetic models for the new generation revealed that gene flow associated with pollen direction was more strongly affected by canopy than with seed direction. Our study is new by proposing an alternative way to separate effects of the pollen and seed flow on spatial variation patterns in E. roxburghiana. To our knowledge, our study is also the first attempt to use landscape genetic models to represent canopy effects for different dispersal vectors in spatial scales only up to a few hundred meters.     

Inferring colonization history from analyses of spatial genetic structure within populations of Pinus strobus and Quercus rubra

Many factors interact to determine genetic structure within populations including adult density, the mating system, colonization history, natural selection, and the mechanism and spatial patterns of gene dispersal. We examined spatial genetic structure within colonizing populations of Quercus rubra seedlings and Pinus strobus juveniles and adults in an aspen-white pine forest in northern Michigan, USA. A 20-year spatially explicit demographic study of the forest enables us to interpret the results in light of recent colonization of the site for both species. We assayed 217 Q. rubra seedlings and 171 P. strobus individuals at 11 polymorphic loci using nine allozyme systems. Plant genotypes and locations were used in an analysis of spatial genetic structure. Q. rubra and P. strobus showed similar observed levels of heterozygosity, but Q. rubra seedlings have less heterozygosity than expected. Q. rubra seedlings show spatial genetic clumping of individuals on a scale to 25 m and levels of genetic relatedness expected from the clumped dispersion of half-siblings. In contrast, P. strobus has low levels of genetic relatedness at the smallest distance class and positive spatial genetic structure at scales < 10 m within the plot. The low density of adult Q. rubra outside the study plot and limited, spatially clumped rodent dispersal of acorns is likely responsible for the observed pattern of spatial genetic structure and the observed heterozygote deficit (i.e. a Wahlund effect). We attribute weaker patterns observed in P. strobus to the longer dispersal distance of seeds and the historical overlap of seed shadows from adults outside of the plot coupled with the overlap of seed shadows from younger, more recently established reproductive adults. The study demonstrates the utility of long-term demographic data in interpreting mechanisms responsible for generating contemporary patterns of genetic structure within populations.     

Generation of Spatial Patterns in Boreal Forest Landscapes

Boreal forests are composed of a few plant species with contrasting traits with respect to ecosystem functioning and spatial patterning. Early successional deciduous species, such as birch and aspen, disperse seeds widely, do not tolerate low light and nitrogen availabilities, have rapidly decaying litter, and are highly preferred by herbivores. These later succeed to conifers, such as spruce and fir, which disperse seeds locally, tolerate low light levels and low nitrogen availability, have litter that decays slowly, and are unpalatable to most mammalian herbivores. Although there are also early successional conifers, such as jack pine and Scots pine, the aspen-birch-spruce-fir successional sequence is the most common over much of North America, and (without fir) in Fennoscandia and Siberia. The course of succession in these forests is controlled partly by seed dispersal and selective foraging by mammalian herbivores. Both of these processes are spatially dynamic, but little is known about how their spatial dynamics may affect ecosystem processes, such as nitrogen cycling or productivity. We present spatially explicit models that demonstrate the following: (a) Spatially explicit seed dispersal results in more clumped distribution of tree species and persistence of greater paper birch biomass than uniform seed rain across the landscape. Such results are consistent with current spatially explicit population models of dispersal and coexistence. (b) With localized seed dispersal, the concentrations of available soil nitrogen are distributed in larger patches with sharp transitions from low to high nitrogen availability near patch edges. In contrast, with a uniform seed rain, the distribution of soil nitrogen availability was more uniform and “hotspots” were more localized. Thus, the spatial pattern of an ecosystem process (nitrogen cycling) is determined by seed dispersal and competition for light among competing populations. (c) A dispersing herbivore, such as moose, that selectively forages on early successional deciduous species with high quality litter, such as aspen or birch, and discriminates against late successional conifers, such as spruce or fir, imposes higher-order repeated patterns of plant species and biomass distribution on the landscape. Thus, seed dispersal and herbivore foraging correlate properties in adjacent patches but in different ways, and different spatial patterns emerge. Other processes, such as insect outbreaks, fire, and water flow, also may correlate properties between adjacent patches and result in additional patterns. Received 8 February 1999; accepted 28 May 1999.     

Spatial genetic structure of the southeastern North American endemic, Ceratiola ericoides (Empetraceae)

Understanding the spatial distribution of genetic diversity (i.e., spatial genetic structure [SGS]) within plant populations can elucidate mechanisms of seed dispersal and patterns of recruitment that may play an important role in shaping the demography and spatial distribution of individuals in subsequent generations. Here we investigate the SGS of allozyme diversity in 2 populations of the southeastern North American endemic shrub, Ceratiola ericoides. The data suggest that the 2 populations have similar patterns of SGS at distances of 0-45 m that likely reflect the isolation by distance (IBD) model of seed dispersal. However, at distances >or=50 m, the pattern of SGS differs substantially between the 2 populations. Whereas one population continues to reflect the classical IBD pattern, the second population shows a marked increase in autocorrelation coefficient (r) values at 50-75 m. Furthermore, r values at these distances are as much as 33% higher than at 0-5 m where the highest r value would be predicted by IBD. A likely explanation is the differing frequencies of 2 fruit morphologies in these populations and the greater role that birds play in seed dispersal in the second population.     

A spatially explicit analysis of seedling recruitment in the terrestrial orchid Orchis purpurea

Seed dispersal and the subsequent recruitment of new individuals into a population are important processes affecting the population dynamics, genetic diversity and spatial genetic structure of plant populations. Spatial patterns of seedling recruitment were investigated in two populations of the terrestrial orchid Orchis purpurea using both univariate and bivariate point pattern analysis, parentage analysis and seed germination experiments. Both adults and recruits showed a clustered spatial distribution with cluster radii of c. 4-5 m. The parentage analysis resulted in offspring-dispersal distances that were slightly larger than distances obtained from the point pattern analyses. The suitability of microsites for germination differed among sites, with strong constraints in one site and almost no constraints in the other. These results provide a clear and coherent picture of recruitment patterns in a tuberous, perennial orchid. Seed dispersal is limited to a few metres from the mother plant, whereas the availability of suitable germination conditions may vary strongly from one site to the next. Because of a time lag of 3-4 yr between seed dispersal and actual recruitment, and irregular flowering and fruiting patterns of adult plants, interpretation of recruitment patterns using point patterns analyses ideally should take into account the demographic properties of orchid populations.     

Modelling and analysing evolution of dispersal in populations at expanding range boundaries

Abstract 1. Species would be expected to shift northwards in response to current climate warming, but many are failing to do so because of fragmentation of breeding habitats. Dispersal is important for colonisation and an individual‐based spatially explicit model was developed to investigate impacts of habitat availability on the evolution of dispersal in expanding populations. Model output was compared with field data from the speckled wood butterfly Pararge aegeria, which currently is expanding its range in Britain. 2. During range expansion, models simulated positive linear relationships between dispersal and distance from the seed location. This pattern was observed regardless of quantity (100% to 10% habitat availability) or distribution (random vs. gradient distribution) of habitat, although higher dispersal evolved at expanding range margins in landscapes with greater quantity of habitat and in gradient landscapes. Increased dispersal was no longer evident in any landscape once populations had reached equilibrium; dispersal values returned to those of seed populations. However, in landscapes with the least quantity of habitat, reduced dispersal (below that of seed populations) was observed at equilibrium. 3. Evolutionary changes in adult flight morphology were examined in six populations of P. aegeria along a transect from the distribution core to an expanding range margin in England (spanning a latitudinal distance of >200 km). Empirical data were in agreement with model output and showed increased dispersal ability (larger and broader thoraxes, smaller abdomens, higher wing aspect ratios) with increasing distance from the distribution core. Increased dispersal ability was evident in populations from areas colonised >30 years previously, although dispersal changes were generally evident only in females. 4. Evolutionary increases in dispersal ability in expanding populations may help species track future climate changes and counteract impacts of habitat fragmentation by promoting colonisation. However, at the highest levels of habitat loss, increased dispersal was less evident during expansion and reduced dispersal was observed at equilibrium indicating that, for many species, continued habitat fragmentation is likely to outweigh any benefits from dispersal.     

Spatial patterns of tree recruitment in a relict population of Pinus uncinata: forest expansion through stratified diffusion

Aim To infer future changes in the distribution of isolated relict tree populations at the limit of a species’ geographical range, a deep understanding of the regeneration niche and the spatial pattern of tree recruitment is needed. Location A relict Pinus uncinata population located at the south‐western limit of distribution of the species in the Iberian System of north‐eastern Spain. Methods Pinus uncinata individuals were mapped within a 50 × 40‐m plot, and their size, age and reproductive status were estimated. Data on seed dispersal were obtained from a seed‐release experiment. The regeneration niche of the species was assessed based on the associations of seedling density with substrate and understorey cover. The spatial pattern of seedlings was described using point‐pattern (Ripley's K) and surface‐pattern (correlograms, Moran's I) analyses. Statistical and inverse modelling were used to characterize seedling clustering. Results Pine seedlings appeared aggregated in 6‐m patches. Inverse modelling estimated a longer mean dispersal distance (27 m), which corresponded to the size of a large cluster along the north to north‐eastward direction paralleled by an eastward trend of increasing seedling age. The two spatial scales of recruitment were related to two dispersal processes. The small‐scale clustering of seedlings was due to local seed dispersal in open areas near the edge of Calluna vulgaris mats: the regeneration niche. The long‐range expansion might be caused by less frequent medium‐distance dispersal events due to the dominant north‐westerly winds. Main conclusions To understand future range shifts of marginal tree populations, data on seed dispersal, regeneration niche and spatial pattern of recruitment at local scales should be obtained. The monitoring of understorey communities should be a priority in order to predict correctly shifts in tree species range in response to global warming.     

Genetic structure of age classes in Camellia japonica (Theaceae)

Camellia japonica L. (Theaceae), an insect- and bird-pollinated, broad-leaved evergreen tree, is widely distributed in Japan and the southern Korean peninsula. The species has a relatively even age distribution within populations, which may influence the spatial genetic structure of different age classes relative to species with typical L-shaped age distributions. To determine whether the internal spatial genetic structure found in seedlings and young individuals carries over into adults, we used allozyme loci, F-statistics, spatial autocorrelation statistics (Moran's I), and coancestry measures to examine changes in genetic structure among seven age classes in a population (60-m x 100-m area) in southern Korea. In seedlings, weak but significant positive values of Moran's I-statistics and coancestry measures were found for distances less than 14 m, which is consistent with a mechanism of limited seed dispersal combined with overlapping seed shadows. This spatial structure, however, dissipates in older age classes, and in adults genetic variation has an essentially random spatial distribution. Morisita's index of dispersion of individuals in each age class showed that seedlings and juveniles are more highly clustered than are older individuals. These results suggest that self-thinning changes the spatial relationships of individuals, and thus genotypes. A multilocus estimate of FST (0.008) shows a small but statistically significant difference in allele frequencies among age classes. In summary, intrapopulation genetic structure within and among age classes of C. japonica was significant but weak. Despite presumably limited seed dispersal, weak spatial genetic structure in juveniles suggests overlapping seed shadows followed by self-thinning during recruitment. The present study also demonstrates that studies of spatial genetic structure focusing on limited numbers of generations may not be sufficient to reveal the entire picture of genetic structure in populations with overlapping generations.     

Size-class differences in genetic structure and individual distribution of Camellia japonica L. in a Japanese old-growth evergreen forest

Size-class differences in genetic structure and individual spatial distribution were investigated for Camellia japonica within a 1-ha plot in a Japanese old-growth evergreen forest using microsatellite markers. Three size-classes were considered containing plants that were: 30-32.5 cm tall, 103.8 cm-200 cm tall and those that had a diameter at breast height > or =5 cm, designated JV1, JV2, and ADL, respectively. Each size-class contained 174 individuals. Morisita's index of dispersion indicated clumping of individuals was present within all size-classes, with JV2 displaying the highest level. The clumped distribution of JV1 individuals may be a result of limited seed dispersal, while that of JV2 may be attributed to heterogenieties of favourable microsites, such as canopy gaps. There were no significant differences in allele frequencies among size-classes. There were, however, some differences in spatial genetic structure among them. Moran's I spatial autocorrelation analysis revealed clear spatial genetic structure in class JV1 probably due to limited seed dispersal. In class JV2, genetic structure was not observed. Overlapping seed shadows, probably in canopy gaps, may lead to blurred genetic structure in JV2.     

Determining the location of a deceased mother tree and estimating forest regeneration variables by use of microsatellites and spatial genetic models

In this paper we first mathematically formulate spatial genetic models that rely on dispersal kernels, using the genetic inhomogeneous Poisson process. On the basis of mapped and genotyped data pertaining to adult and juvenile trees we are able to estimate three fundamental variables of population dynamics: individual female reproductive success, seed dispersal, and pollen flow. The model was applied to a secondary Fagus crenata stand in northern Japan, regenerated after shelterwood harvesting. Highly polymorphic microsatellite data revealed that most of the juveniles around one adult tree were not that tree’s progeny and that some minor alleles were clustered there. These data suggested that another mother tree had formerly been present in the vicinity, produced offspring there and died. Inferring its genotype and location, we applied the genetic inhomogeneous Poisson process. Results confirmed that we would have wrongly assessed the regeneration if we had been unaware of the existence of the dead mother. The average distances for seed dispersal and pollen flow were 18 and 193 m, respectively. The contribution of outside mothers, simultaneously assessed using the dispersal variables in the models, ranged from 10 to 50% depending on their positions relative to preserved adults. Individual female reproductive success varied as much as fiftyfold among the eight preserved adults. Our comprehensive approach, utilizing currently available genetic information, mathematical models, and previous forestry records, helped elucidation of the past forest-regeneration processes.     

Isolation-by-distance and outbreeding depression are sufficient to drive parapatric speciation in the absence of environmental influences

A commonly held view in evolutionary biology is that speciation (the emergence of genetically distinct and reproductively incompatible subpopulations) is driven by external environmental constraints, such as localized barriers to dispersal or habitat-based variation in selection pressures. We have developed a spatially explicit model of a biological population to study the emergence of spatial and temporal patterns of genetic diversity in the absence of predetermined subpopulation boundaries. We propose a 2-D cellular automata model showing that an initially homogeneous population might spontaneously subdivide into reproductively incompatible species through sheer isolation-by-distance when the viability of offspring decreases as the genomes of parental gametes become increasingly different. This simple implementation of the Dobzhansky-Muller model provides the basis for assessing the process and completion of speciation, which is deemed to occur when there is complete postzygotic isolation between two subpopulations. The model shows an inherent tendency toward spatial self-organization, as has been the case with other spatially explicit models of evolution. A well-mixed version of the model exhibits a relatively stable and unimodal distribution of genetic differences as has been shown with previous models. A much more interesting pattern of temporal waves, however, emerges when the dispersal of individuals is limited to short distances. Each wave represents a subset of comparisons between members of emergent subpopulations diverging from one another, and a subset of these divergences proceeds to the point of speciation. The long-term persistence of diverging subpopulations is the essence of speciation in biological populations, so the rhythmic diversity waves that we have observed suggest an inherent disposition for a population experiencing isolation-by-distance to generate new species.     

The effect of dispersal and neighbourhood in games of cooperation

The prisoner's dilemma (PD) and the snowdrift (SD) games are paradigmatic tools to investigate the origin of cooperation. Whereas spatial structure (e.g. nonrandom spatial distribution of strategies) present in the spatially explicit models facilitates the emergence of cooperation in the PD game, recent investigations have suggested that spatial structure can be unfavourable for cooperation in the SD game. The frequency of cooperators in a spatially explicit SD game can be lower than it would be in an infinitely large well-mixed population. However, the source of this effect cannot be identified with certainty as spatially explicit games differ from well-mixed games in two aspects: (i) they introduce spatial correlations, (ii) and limited neighbourhood. Here we extend earlier investigations to identify the source of this effect, and thus accordingly we study a spatially explicit version of the PD and SD games with varying degrees of dispersal and neighbourhood size. It was found that dispersal favours selfish individuals in both games. We calculated the frequency of cooperators at strong dispersal limit, which in concordance with the numerical results shows that it is the short range of interactions (i.e. limited neighbourhood) and not spatial correlations that decreases the frequency of cooperators in spatially explicit models of populations. Our results demonstrate that spatial correlations are always beneficial to cooperators in both the PD and SD games. We explain the opposite effect of dispersal and neighbourhood structure, and discuss the relevance of distinguishing the two effects in general.