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Interpreting spatial biodiversity patterns requires understanding how spatial structure affects speciation. Many studies have shown that spatial structure resulting from environmental heterogeneity facilitates speciation. However, the consequences of spatial self-structuring have received little attention so far. Using an individual-based model we study phenotypic evolution of asexually reproducing organisms inhabiting spatially continuous and intrinsically homogeneous environments. To investigate the effects of spatial self-structuring on waiting time until adaptive radiation, we varied dispersal rates and distances to examine differently mixed populations. The results show that dispersal rates and distances do not affect these waiting times independently, but their joint effects are well approximated through the resultant diffusion coefficient. We establish, that over a wide parameter range, spatial self-structuring delays adaptive radiation and explain this finding by two effects. First, in spatial clusters dominated by different phenotypes, fitness minima occur at different phenotypes. Averaged over clusters, this results in a flatter global fitness landscape and, therefore, in weaker disruptive selection pressures. The second effect arises from source-sink dynamics among spatial clusters, with some clusters expanding while others are shrinking. This causes a blurred fitness landscape and slows down the population's response to disruptive selection.
Implications of spatial structure have played a prominent role in speciation theory throughout its history. It is widely accepted that restricted dispersal, and hence limited gene flow, facilitates speciation. The well-known geographical classification of speciation modes, into allopatric and sympatric speciation, is directly based on the extent of spatial mixing. Allopatric speciation, which occurs when parts of a population become geographically isolated, no longer exchange genes, and thus gradually become genetically incompatible, has long been recognized as the dominant mode of speciation. In contrast, speciation in sympatry requires that reproductive isolation evolves in the absence of geographic isolation, a process that, until recently, was considered unlikely to occur in nature.
Geographic isolation, typical for allopatric speciation, can occur in different forms (Mayr 1942, 1963; White 1978). In vicariant speciation, a new natural geographic barrier arises and splits a population into two parts of similar size. In peripatric speciation, a small peripheral part of a population becomes separated, often through the colonization of a new area (Mayr 1954; Templeton 1980). In parapatric speciation, geographical isolation is incomplete and some gene flow remains (Endler 1977; Gavrilets 2004). Parapatric speciation can occur through isolation by distance as differences among local populations accumulate under spatially limited dispersal (Wright 1943).
Geographic barriers are an extreme form of spatial structure, and parapatric speciation can occur under milder and more realistic forms of externally imposed spatial heterogeneity. Many theoretical studies, for example, have focused on speciation in environments with two habitats that are connected through migration (Levene 1953; Maynard Smith 1966; Rausher 1984). Reduced migration between such habitats promotes diversification through the evolution of habitat specialists (Day 2000; Kirkpatrick 2000; Beltman & Haccou 2005; Szilágyi & Meszéna 2009). If the environmental conditions among the habitats differ gradually, spatial environmental gradients arise. Theoretical investigations (Doebeli & Dieckmann 2003; Mizera & Meszéna 2003; Heinz et al. 2009) and experimental studies (Rainey & Travisano 1998) have demonstrated that spatial environmental gradients facilitate parapatric speciation and adaptive radiation. Moreover, habitat choice has been shown to amplify the diversifying impact of externally imposed spatial heterogeneity (Ravigné et al. 2004, 2009; Postma & van Noordwijk 2005; Garant et al. 2005; Gavrilets and Vose 2005).
Spatial population structure can also arise in the absence of externally imposed spatial heterogeneity. Even in spatially uniform or homogeneous environments, the location of individuals is often not random, but is affected by the location of other individuals. Under conditions of low mobility, neighbourhood interactions affecting survival, growth, and/or fecundity, as well as the simple fact that offspring often stay close to their parents (reproductive pair-correlations; Hamilton 1964; Young et al. 2001), can spatially structure populations. Known as spatial self-structuring, these effects are particularly prominent in plant populations (Condit et al. 2000; Stoll & Weiner 2000). Recent studies of microbial evolution show that such spatial self-structuring may have evolutionary consequences, by inhibiting mutant invasion and slowing down adaptation (Habets et al. 2006, 2007; Perfeito et al. 2008) , or by causing the loss of diversity (Saxer et al. 2009). While spatial self-structuring has become a focus of interest in evolutionary ecology, so far it has received little attention in speciation studies. In fact, a recent review by Lion & van Baalen (2008) highlighted as a major open challenge the task "to elucidate the impact of spatial self-structuring on evolutionary branching". Here we address this challenge.
Already Hamilton (1964) had highlighted that spatial self-structuring can influence selection pressures, with kin selection being amplified by more intensive interactions among local kin. Furthermore, spatial self-structuring often promotes phenotypic assortment, and has thus been important for understanding the evolution of cooperation (e.g., Nowak & May 1992; Killingback & Doebeli 1996). Spatial self-structuring must also be expected to modify the negative frequency-dependent selection pressures that play a central role in ecologically driven sympatric speciation (Maynard Smith 1966; Rosenzweig 1978; Slatkin 1979; Udovic 1980; Felsenstein 1981; Seger 1985; Metz et al. 1995; Geritz et al. 1997, 1998; Dieckmann & Doebeli 1999). Under negative frequency-dependent selection, a phenotype's fitness increases when it is rare. However, being locally rare is more difficult when similar phenotypes are spatially clustered, as interactions then preferentially occur among similar phenotypes.
Consequently, there are good reasons to investigate how the spatial patterns that naturally result from the self-structuring of populations influence the course and pace of evolutionary diversifications. In sexual populations, understanding the dynamics of phenotypic evolution is complicated not only by the effects of recombination, and the potential complexities of genotype-to-phenotype maps, but also by the interplay between sexual and natural selection (Kirkpatrick & Ravigné 2002; Gourbiere 2004). The present study therefore deliberately focuses on asexual populations while investigating how natural selection and the self-organized ecological interactions among individuals in spatially extended populations can alter the pattern and process of phenotypic diversification.
We investigate an asexual, individual-based model of phenotypic evolution similar to the model developed by Doebeli and Dieckmann (2003). The main difference from the original model is absence of an environmental gradient, which results in a uniform environment. We study an individual-based model so as to incorporate both realistic spatial population structure and stochasticity inherent to biological interactions (Wilson 2000; DeAngelis & Mooij 2005; Grimm & Railsback 2005).
Overview. In our model, individuals compete, reproduce, and disperse locally. New phenotypes are occasionally created through mutation. Phenotypically similar and closely located individuals compete more intensely. The model is defined in continuous time and generations are overlapping. The space is continuous with periodic boundary conditions on the unit square.
Individuals. The phenotype of each individual determines the type resource or environment to which an individual is best adapted, and varies from to . The function describes how quantitative parameter determines individual preference, where and is maximal carrying capacity density. Individuals are characterized by rates at which births, deaths, and dispersals occur; and have a spatial location with .
Reproduction. Birth occurs with a constant rate , with offspring inheriting parental phenotype unless a mutation occurs with probability, where and is the current population size. In this case the offspring phenotype is drawn from a normal distribution with a mean of parental phenotype and standard deviation. Each newborn offspring undergoes an initial dispersal event from the location of the parent.
Death. An individual's death rate changes during its lifetime and depends both on logistic competition and on individual carrying capacity. The death rate of individual at location () is defined as .
Dispersal. Dispersal occurs at a constant rate with individuals changing their location by the random distances in directions, drawn from a normal distribution with 0 mean and standard deviation.
Algorithmically, the model is implemented using the minimal process method (Gillespie 1976). After each event in the model, individuals are assigned rates , , . The time that passes until the next event is based on the total rates , , , and , and is drawn from an exponential distribution with mean . The individual for the next event is chosen based on probability . The type of the event is chosen according to probabilities , , respectively. After the chosen individual reproduces, dies, or disperses in space, the rates for the whole population are updated. Values of the parameters we use for the model runs are shown in Table 1 (unless stated otherwise).
Waiting time until adaptive radiation
Widely accepted biological species concept (Mayr 1942) is not applicable for asexual organisms. Hence, we recognize adaptive radiation based on the genotypic-cluster species concept (Mallet 1995), that requires presence of disruptive selection to keep the clusters separated from each other. For detection of adaptive radiation the phenotypic space is divided into 20 equal intervals and filled with individuals accordingly to their phenotypes. From the resulting distribution we identify two peaks of heights and and denote height of intervals between the peaks as . We recognize formation of two distinct clusters if and are divided by at least two intervals with and . Waiting time until adaptive radiation is defined as . To explore stochastic variation we use different random seeds for same parameter values.
As it is clear from the model description, mobility of individuals is described by two parameters: the dispersal rate and the dispersal distance . Therefore, we analyze how the relationship of and influences on . It is well-known that a dispersal process of this kind (also known as the Wiener process) can be approximated by diffusion process (Doebeli & Dieckmann 2004), with a diffusion coefficient . Here needs to be included since an individual right after birth makes an initial dispersal from the location of the parent. The underlying mathematical derivation shows that the diffusion approximation is the more accurate the higher the dispersal rate and the smaller the dispersal step are (see van Kampen 2001).
We vary parameters and in such a way that the diffusion coefficient remains constant and investigated its influence on . First we fix the parameter and vary the to change . Then we fix the parameter and vary to make from this set of simulations equal to from the previous set (with fixed value of ).
Invasion fitness and survival probability of mutants
Fitness landscapes. Fitness landscapes (Wright 1932) summarize how reproductive success of individuals changes depending on the phenotype. Therefore, the curvature of fitness landscape characterizes how strong the disruptive selection is: the higher the curvature, the stronger disruptive selection. In non-spatial models phenotype uniquely defines fitness. However, in spatially structured settings unique fitness could be lost due to different local environment. Therefore, there is a need to look at distribution of fitness values. To obtain the frequency distribution, we use a grid of phenotypic space divided into 200 by 200 intervals. During a model run we create a number of trial mutants (trial because they were not included in the population), parents of mutants were randomly chosen from the population. For each trial mutant we record phenotype , birth rate , death rate , parental phenotype , with , is number of trial mutants. Mutant fitness we define as (i.e. if the death rate is high, the fitness is low). For revealing shape of the fitness landscape, the location of the fitness minimum is very important, and since it could differ in different model runs, we define fitness minimum as mean of phenotypic distribution in the time of generating trial mutants.
In order to compare fitness landscapes, both for well-mixed and self-structured populations we generate 20 000 trial mutants in each of 1000 simulations with different random seeds. For mutant distribution we fill a matrix indexed by and by trial mutants. To reveal frequency distribution , we normaliz each value of the matrix using sum of values across columns of the matrix indexed by . We calculate mean fitness with regard to from .
Speed of divergent evolution. Population under disruptive selection may become phenotypically bimodal due to subpopulations on each side of the fitness minimum experiencing directional selection away from the fitness minimum. Therefore, it is interesting to determine average strength of these divergent selection pressures, as well as the speed of the resultant selection response. Population situated at the maximum of the resource distribution, experiences disruptive selection. To the left of the fitness minimum, directional selection causes evolution to the left, to the right of the fitness minimum it causes evolution to the right. As shown in branching theory (Athreya & Ney 1972), approximate survival probability of rare mutant , where is the reproduction rate of the mutant. In our model, , therefore, the survival probability or . To evaluate strength of resultant disruptiveness, we estimate average directional selection from the right and from the left of , based on mutant phenotype and on phenotypic difference of a mutant from its parent. The speed of directional selection to the right from the fitness minimum equals to , where or , is the mutant phenotype, where , is number of trial mutants. The speed of directional selection on the left from the fitness minimum equals and is defined in analogous way, with or . We generate 50 000 trial mutants during each of 15 time moments during a model run. We perform 50 runs, and calculate average values of and for each time moment of generation of trial mutants.
Invasion fitness and survival probability. Understanding speed of divergent evolution could be done by looking at distributions of invasion fitness and survival probability of mutants. We consider mutants arising separately on the left and the right from the fitness minimum, and test how they benefit if make a step in phenotypic space away from their parents. Additionally, a fraction of mutants with positive fitness could be investigated in the same way, because these particular mutants contribute to the speed and direction of evolution. We calculate fitness and survival probability of 15 000 trial mutants generated just before evolutionary branching for each of 300 model runs. We fill each of two matrixes by trial mutants accordingly to their properties. To analyse the fitness distribution the first matrix was indexed by and ; for survival probability distribution the second matrix was indexed by and . We normalize each value from matrixes by the sum of values across columns indexed by both for invasion fitness (having as a result ) and for survival probability (having as a result ). Last, we calculate both mean invasion fitness from and mean survival probability from with regard to phenotypic difference of mutant from its parent.
Adaptive radiation in well-mixed and self-structured populations
We find that phenotypic evolution results in adaptive radiation both in well-mixed (figure 1a,b,c) and self-structured (figure 1d,e,f) populations. In well-mixed populations, the initial single phenotypic cluster (denoted by the same colour in the snapshot of spatial structure in figure 1a), which is under disruptive selection, splits into two distinct clusters and then undergoes divergence (figure 1b). Individuals having newly emerged phenotypes are randomly distributed in space (figure 1c). In self-structured populations, after adaptive radiation (that occurs later, as shown in figure 1e) individuals with new phenotypes aggregate into spatial clusters (figure 1f). We find that adaptive radiation could result into more than two phenotypic clusters (like in figure 1b) under conditions of strong phenotypic competition (). This result is in agreement with previous studies, when formation of multiple clusters was caused by small phenotypic interaction radius, allowing for several ecological niches to be presented in the environment (Bolnick 2006, Doebeli et al. 2007).
Confirmation of expected parameter effects
Before we investigate the delaying effects of spatial self-structuring, we first confirm some expected effects (observed in previous studies) by systematically evaluating effect of the model parameters on waiting time until adaptive radiation (figure 2). First, adaptive radiation slows down if the phenotypic interaction radius increases, since it regulates the strength of disruptive selection. Similar to what was suggested before (Dieckmann & Doebeli 1999; Doebeli & Dieckmann 2003) we obtained evolutionary branching if . Second, both big mutation steps and increased mutation probability result in earlier adaptive radiation (figure 2a,b). This is cased by general speed up of the phenotypic evolution, which was predicted by canonical equation of adaptive dynamics (Dieckmann & Law 1996). Third, adaptive radiation is delayed when the maximal carrying capacity density is reduced (figure 2c and figure 3). This delay results from stronger demographic stochasticity in small populations (see discussion for details).
Delayed adaptive radiation caused by spatial self-structuring
There are three ways to vary the degree of spatial self-structuring in a population: by changing spatial interaction radius, dispersal distance, and dispersal rate. As explained in the methods, combined effect of two latter parameters can be approximated by the corresponding diffusion coefficient. High diffusion coefficient causes well-mixing, while high interaction radius means that individuals cannot respond to small-scale spatial structure even if it is present. As the interaction radius is increased, the populations effectively become more well-mixed, and the waiting time until adaptive radiation is reduced (figure 2d). This result is in agreement with studies of Pacala (1988) and Day (2001), in which the increase of the spatial interaction group enhances disruptive selection.
Confirmation of accuracy of diffusion coefficient. Using the observation for the average waiting time for evolutionary branching we find that the diffusion approximation by coefficient , is fully appropriate for describing spatial dispersal by the relationship of dispersal distance and dispersal rate. This approximation remains accurate when population size and intensity of phenotypic competition are varied (figure 3).
We can get a first idea of how spatial self-structuring delays adaptive radiation by investigating the speed of divergent evolution during the initial phase of adaptive radiation. Our results show that with transition of the spatial structure from self-structured to well-mixed, the average waiting time for evolutionary branching decreased (figure 3). Long waiting time until adaptive radiation in self-structured populations was observed for different population size (, ) and for different intensity of phenotypic competition (, ).
Speed of divergent evolution
We can get a first idea of how spatial self-structuring delays adaptive radiation by investigating the speed of divergent evolution during the initial phase of adaptive radiation. The speed of disruptive evolution gradually increases during simulation runs both in well-mixed and structured populations, as it approaches the moment of evolutionary branching (figure 4c,d). During a first period the disruptive selection is weak in general and its intensity is similar in well-mixed and structured populations. However, later on the speed of divergent evolution increases in well-mixed populations and remains high during the whole model run (figure 4c).
Shapes of global fitness landscapes
Since the curvature of global fitness landscape defines the intensity of disruptive selection, we examine mean mutant fitness and distribution of trial mutants against the mutant phenotype (see methods). Our results show that global fitness landscape is wider in self-structured populations (figure 4b) when compared with the one observed in the well-mixed populations (figure 4a). Moreover, we also observe variation of the mutant distribution in the vertical dimension of the global fitness landscape when populations are self-structured. The shape of the curves of mean mutant fitness accurately reflects shape of the trial mutant distributions.
Weakening of divergent evolution caused by flattening and blurring of global fitness landscapes
While the effect of flattened global fitness landscape is evident (it reduces the strength of disruptive selection), appreciating the effect of a blurred fitness landscape requires more explanation. To understand more generally how differences in the shape of the global fitness landscape translate into different speeds of divergent evolution, we analyse invasion fitness of trial mutants arising separately on the left and on the right side of fitness minimum (as explained in the methods). Figure 5 shows the mean invasion fitness (black curves) and distribution of trial mutants (gray scale areas) with regard to the step size that a mutant performs in the phenotypic space from its parent. We found that in self-structured populations (figure 5c) mean fitness of the mutants "going the right way" (i.e. going away from the fitness minimum) is in general lower than in the well-mixed population (figure 5a).
In addition, survival probability of mutants defines the waiting time until adaptive radiation. Figure 5 (b,d) demonstrates how mean survival probabilities (denoted by black curves) and distributions of trial mutants depend on the step size mutants perform in the phenotypic space from their parents (separately for the mutants arising on the left and on the right side from the fitness minimum). Mean survival probability of the mutants "going the right way" to avoid the fitness minimum is similar for self-structured and well-mixed populations. But the survival probability of mutants "going the wrong way" (back to fitness minimum) is higher in self-structured populations (figure 5b,d). This raise of mean survival probability for mutant returning to the fitness minimum happens because in a structured population a big number of mutants with negative fitness were excluded for calculation of the survival probability (as explained in the methods).
In the present study we investigate phenotypic evolution in a spatially distributed asexual population. Our model considers continuous homogeneous environment and allows us to investigate the phenomenon of spatial self-structuring itself. We confirm expected parameter effects (figure 2) and demonstrate the accuracy of the diffusion approximation (figure 3). Our main finding is that waiting time until adaptive radiation in spatially self-structured populations is longer than in well-mixed populations (figure 3). Our results demonstrate that self-structuring results in a slower speed of divergent evolution (figure 4a,b). We show that weakening of divergent selection results from two effects arising in spatially self-structured populations: flattening and blurring of the global fitness landscape (figure 4c,d). Flattening of the global fitness landscape results to decrease of invasion fitness of mutants (figure 5c) and blurring allows mutants located at the fitness minimum, to escape pressure of disruptive selection (figure 5d).
Why spatial self-structuring hinders evolutionary branching
To understand why the speed of divergent evolution slows down in self-structured environments, we compared conditions for mutant invasion in different spatial settings. In general, in non-spatial models the probability that a lineage established by a new mutant escapes rapid extinction is determined by the mutant's fitness. In spatially explicit models mutant fitness additionally depends on its location, because a mutant could be surrounded by a different number of resident individuals with different phenotypes.
We revealed two effects that are responsible for slowing down adaptive radiations in self-structured populations. The presence of these two effects was confirmed by individual-based simulations.
The first effect (figure 6a) is the flattening of the global fitness landscape. This flattening is caused by total contribution of local fitness landscapes, because local fitness minima of spatial clusters with different phenotypic composition could significantly differ from location of the global fitness minimum (figure 4d). In self-structured population, mutants are surrounded by individuals with very similar phenotypes, and this condition makes mutants very likely to be located around the local fitness minimum, and have negative invasion fitness. For a mutant to have positive invasion fitness inside a spatial cluster, it has to perform a large step from its parent in phenotypic space in the direction avoiding global fitness minimum (figure 5c). Therefore, self-structuring results in decrease of invasion fitness of mutants this delays adaptive radiation.
Second effect (figure 6b) is blurring of the global fitness landscape in vertical direction, along the fitness axes. This is an outcome of source-sink dynamics among spatial clusters that could shrink or expand. If a cluster is shrinking, local fitness landscape sinks lower than , although, mutant lineages that extinct as a part of shrinking clusters, do not contribute to overall population evolutionary change. But if a cluster grows, local fitness landscape elevates higher than , with most of mutants having positive invasion fitness. At the global scale, some of these mutants return to the fitness minimum, but anyway survive because they come from a growing cluster (figure 5d). This allows mutants to escape disruptive selection, and as a result, adaptive radiation slows down. Consequently, there is asymmetry between implications of growing and shrinking clusters that affects the evolutionary dynamics. Since mutants in shrinking clusters dye out anyway, they do not compensate the decelerated response to disruptive selection, arising in growing clusters, so the decelerated effect prevails.
Comparison with related work
When sexual reproduction is incorporated into spatially explicit models of adaptive radiation, many factors (as mate choice, gene flow, inheritance modes and others) could alter evolutionary dynamics of asexual diversification. Therefore, it complicates comparison of spatially explicit models with asexual and sexual reproduction. Thus, in this section we provide comparison of our analysis with results from studies assuming asexual reproduction.
Influence of population size on evolutionary branching was investigated previously in a non-spatial model (Claessen et al. 2007). It was shown that demographic stochasticity, facilitated by small population size, delays evolutionary branching by two mechanisms: by random drift of the mean trait in the population, which causes the population to spend long periods away from disruptive selection, and by extinction of incipient branches. Our result on longer waiting time until adaptive radiation in small populations extend these previous findings to self-structured populations (figure 3), and we observe signs of both of these mechanisms acting in our model runs. While in our study (as well as in the study of Claessen et al.), demographic stochasticity is directly caused by small population size, strong environmental fluctuations could as well decrease the population size and delay sympatric speciation (Johansson & Ripa 2006).
While our model here examines effects of spatial self-structuring in spatially homogeneous environment, some of our findings can be compared to results from studies focused on environmental heterogeneity, or externally imposed spatial structure. Spatial population structure could take many forms, including environmental gradient. Spatial structuring along environmental gradient is induced both by low mobility of individuals and by local adaptation to environmental conditions along the gradient. Such structuring was shown to result in strong gradient-induced frequency-dependent selection, which facilitates adaptive radiation (Doebeli & Dieckmann 2003; Heinz et al. 2009). In our study we consider homogeneous environment, and observe opposite effect, with spatial self-stucturing hindering the evolutionary branching. Thus, whether evolutionary branching would be facilitated of hindered, depends on which effect prevails in the population.
Spatial population structure also could be externally imposed, with space divided into a set of patches/demes. Such model was explicitly investigated by Day (2001). In the Day's model competition for resources occurs within groups of finite number of individuals, with limited dispersal of individuals between groups resulting in genetic structuring of the population. Day found that low dispersal can inhibit disruptive selection, and explained this finding by rarely realized benefit that a mutant gains from being different when it is surrounded by very similar individuals. Accordingly, we find that average mutant invasion fitness is lower in self-structured populations (figure 5c). Our analysis confirms and extends results of Day (2001): spatial clusters with different local phenotypic minima in our study correspond to Day's patches where mutants do not gain a benefit from being different. However, our study extends Day's approach, we investigate continuous habitat where self-structuring generates clusters with characteristic dynamic boundaries and average diameter. Day investigated competition inside a patch of externally imposed size, whereas in our study we also consider interactions between clusters that are dynamic and have self-organized size. As a result, his model allowed to identify one of the two effects that influence on the waiting time of adaptive radiation in self-structured populations: decrease of mutant advantage inside spatial cluster (seen as widening of the global fitness landscape observed in our study). Considering spatial clusters to be dynamic and self-organized, allowed us to reveal a second effect that spatial self-structuring imposes on evolutionary branching, since a mosaic of growing and shrinking spatial clusters blurs the global fitness landscape.
The role of spatial interactions for competition, co-existence and evolutionary branching earlier was investigated in a lattice model (Mágori et al. 2005). Spatial interactions between individuals were reported to hamper co-existence and reduce possibility of evolutionary branching. The latter observation is in agreement with our results. However, initial conditions of our study differ from those reported by Mágori et al. While in our model we consider evolution of the phenotypic trait, and initially the system is represented by monomorphic population, Mágori et al. start their numerical experiments by filling the lattice cites with individuals from two different populations, with consequent analysis of co-existence of two competing population. The authors translate their results on reduced co-existence of two populations into diminished possibility of evolutionary branching. Here we challenge this particular interpretation by showing how spatial structure affects the onset of evolutionary branching. In other words, we have shown how spatial self-structuring hinders long-term co-existence of established branches with significantly different phenotypes, but also the very establishment of these branches during initial stages on evolutionary branching.
Limitations and extensions
Our study is limited to asexual reproduction, and introducing sexual reproduction and local mate choice into spatially self-structured populations could cause substantial effects on the waiting time until evolutionary branching. In sexually-reproducing populations, offspring stays close to its parent (causing spatial self-structuring), and this effectively results in assortative mating. Assortative mating prevents gene flow between incipient species, and thus could promote earlier evolutionary branching. In our study, dispersal rates and distances are fixed, but if one would consider them evolving (as has been done by Heinz et al. 2009 in a model with environmental gradient), the degree of spatial self-structuring could evolve as well. Another interesting extension would be to assume that individuals disperse depending on local environmental conditions (or selective pressures), since the evolution of conditional dispersal was shown to make a significant impact on parapatric speciation along environmental gradients (Payne et al. in press). Moreover, it could be interesting to study effects of spatial variation in environmental conditions, since effects of spatial self-structuring could interact with effects of gradual patchiness of environment and change possibility of adaptive radiation (Haller et al. in preparation).
Spatial interactions of individuals are important to most ecological systems. We made an attempt to understand how self-structuring interactions influence the evolutionary process, and we not only investigated in detail influence of spatial structure on the waiting time until adaptive radiation, but also discovered the mechanisms that are responsible for this effect. We revealed that spatial fluctuations in self-structuring systems delay evolutionary branching by flattening the global fitness landscape. Another influence of the spatial fluctuations occurs through blurring of the global fitness landscape, favouring the survival of mutants that escape disruptive selection. We suggest that these two evolutionary implications of spatial self-structuring are noteworthy, because they apply in a wide variety of ecological settings.
We are much indebted to the late Dr. Sergey Semovski for initiating the collaboration that led to this study. We thank for valuable discussions R. Mazzucco, G. Meszéna, JAJ. Metz, B. Nevado, J. Ripa, A. Sasaki and D. Sherbakov. Financial support by the Vienna Science and Technology Fund (WWTF) is gratefully acknowledged. U.D. gratefully acknowledges additional support by European Commission, European Science Foundation, Austrian Science Fund, and Austrian Ministry for Science and Research.