Estimating demographic models for the range dynamics of plant species

Aims To better understand how demographic processes shape the range dynamics of woody plants (in this case, Proteaceae), we introduce a likelihood framework for fitting process‐based models of range dynamics to spatial abundance data. Location The fire‐prone Fynbos biome (Cape Floristic Region, South Africa). Methods Our process‐based models have a spatially explicit demographic submodel (describing dispersal, reproduction, mortality and local extinction) as well as an observation submodel (describing imperfect detection of individuals), and are constrained by species‐specific predictions of habitat distribution models and process‐based models for seed dispersal by wind. Free model parameters were varied to find parameter sets with the highest likelihood. After testing this approach with simulated data, we applied it to eight Proteaceae species that differ in breeding system (monoecy versus dioecy) and adult fire survival. We assess the importance of Allee effects and negative density dependence for range dynamics, by using the Akaike information criterion to select between alternative models fitted for the same species. Results The best model for all dioecious study species included Allee effects, whereas this was true for only one of four monoecious species. As expected, sprouters (in which adults survive fire) were estimated to have lower rates of reproduction and catastrophic population extinction than related non‐sprouters. Overcompensatory population dynamics seem important for three of four non‐sprouters. We also found good quantitative agreement between independent data and most estimates of reproduction, carrying capacity and extinction probability. Main conclusions This study shows that process‐based models can quantitatively describe how large‐scale abundance distributions arise from the movement and interaction of individuals. It stresses links between the life history, demography and range dynamics of Proteaceae: dioecious species seem more susceptible to Allee effects which reduce migration ability and increase local extinction risk, and sprouters seem to have high persistence of established populations, but their low reproduction limits habitat colonization and migration.     

Optimal life‐history schedule in a metapopulation with juvenile dispersal

Previous models have predicted that when mortality increases with age, older individuals should invest more of their resources in reproduction and produce less dispersive offspring, as both their future reproductive value and their prospect of competing with their own sib decline. Those models assumed stable population sizes. We here study for the first time the evolution of age‐specific reproductive effort and of age‐specific offspring dispersal rate in a metapopulation with extinction‐recolonization dynamics and juvenile dispersal. Our model explores the evolutionary consequences of disequilibrium in the age structure of individuals in local populations, generated by disturbances. Life‐history decisions are then shaped both by changes with age in individual performances, and by changes in ecological conditions, as young and old individuals do not live on average in the same environments. Lower juvenile dispersal favours the evolution of higher reproductive effort in young adults in a metapopulation with extinction‐recolonization compared with a well‐mixed population. Contrary to previous predictions for stable structured populations, we find that offspring dispersal should generally increase with maternal age. This is because young individuals, who are overrepresented in recently colonized populations, should allocate more to reproduction and less to dispersal as a strategy to exploit abundant recruitment opportunities in such populations.     

Conditional dispersal under kin competition: extension of the Hamilton-May model to brood size-dependent dispersal

I consider a site-based model with contest competition among siblings, and assume that dispersal is conditional on the number of offspring in the natal site. Evolutionarily stable populations contain threshold dispersal strategies, which retain a certain number of offspring in the natal site and disperse the rest (if the actual number of offspring is less than the threshold, then all offspring are retained). Due to the discrete nature of the strategy set (the threshold must be integer), the ESS may not be unique or may not exist. In the latter case, two neighboring threshold strategies coexist in the evolutionarily stable population. Dispersal first decreases and then increases as a function of dispersal mortality, such that all but one offspring should be dispersed both when dispersal mortality is very small or very high. Population-level dispersal fractions are often similar to the unconditional ESS, but differ strongly when fecundity is small and dispersal mortality is high.     

Optimization and Phenotype Allocation

We study the phenotype allocation problem for the stochastic evolution of a multitype population in a random environment. Our underlying model is a multitype Galton–Watson branching process in a random environment. In the multitype branching model, different types denote different phenotypes of offspring, and offspring distributions denote the allocation strategies. Two possible optimization targets are considered: the long-term growth rate of the population conditioned on nonextinction, and the extinction probability of the lineage. In a simple and biologically motivated case, we derive an explicit formula for the long-term growth rate using the random Perron–Frobenius theorem, and we give an approximation to the extinction probability by a method similar to that developed by Wilkinson. Then we obtain the optimal strategies that maximize the long-term growth rate or minimize the approximate extinction probability, respectively, in a numerical example. It turns out that different optimality criteria can lead to different strategies.     

Establishment Probability in Fluctuating Environments: A Branching Process Model

We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods according to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctuations on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson.     

Dispersal: risk spreading versus local adaptation

I investigate how risk spreading in stochastic environments and adaptation to permanent properties of local habitats interplay in the simultaneous evolution of dispersal and habitat specialization. In a simple two-patch model, I find many types of locally evolutionarily stable attractors of dispersal and of a trait involved in habitat specialization, including a single habitat specialist and a coalition of two specialists with low dispersal, a generalist with high dispersal, and several types of dispersal polymorphisms. In general, only one attractor is a global evolutionarily stable strategy (ESS). In addition to the ESS analysis, I also present some examples of the dynamics of evolution that exhibit adaptive diversification by evolutionary branching.     

Evolution of condition-dependent dispersal: A genetic-algorithm search for the ESS reaction norm

Many insects produce two types (winged and wingless) of offspring that greatly differ in dispersal ability. The ratio of the two types often depends on the quality of the local habitat and the crowding experienced by the mother. Here we studied the condition-dependent dispersal that is evolutionarily stable. The model is also applicable to annual plants that produce two types of seeds differing in dispersal rates. The model assumptions are: the population is composed of a number of sites each occupied by a single adult. The total number of offspring produced by a mother depends on the environmental quality of the site that varies over the years and between sites. The ESS fraction of dispersing type as a function of the quality of the habitat (or ESS reaction norm) states that dispersers should not be produced if habitat qualitym is smaller than a critical valuek. Ifm is larger thank, the number of dispersers should increase withm and that of nondispersers should be kept constant. Second, we developed an alternative way of searching for the ESS: the reaction norm is represented as a three-layered neural network, and the parameters (weights and biases) are chosen by genetic algorithm (GA). This method can be extended easily to the cases of multiple environmental factors. There was an optimal (relatively wide) range of mutation rates for weights and biases, outside of which the convergence of the network to the valid ESS was likely to fail. Recombination, or crossing-over, was not effective in improving the success rate. The learned network often shows several characteristic ways of deviation from the ESS. We also examined the case in which the quality of different sites was correlated. In this case the ESS fraction of dispersers increases both with the quality of the site and with the average quality of the whole population in that year.     

THE EVOLUTIONARILY STABLE PHENOTYPE DISTRIBUTION IN A RANDOM ENVIRONMENT

In an unpredictably changing environment, phenotypic variability may evolve as a “bet-hedging” strategy. We examine here two models for evolutionarily stable phenotype distributions resulting from stabilizing selection with a randomly fluctuating optimum. Both models include overlapping generations, either survival of adults or a dormant propagule pool. In the first model (mixed-strategies model) we assume that individuals can produce offspring with a distribution of phenotypes, in which case, the evolutionarily stable population always consists of a single genotype. We show that there is a unique evolutionarily stable strategy (ESS) distribution that does not depend on the amount of generational overlap, and that the ESS distribution generically is discrete rather than continuous; that is, there are distinct classes of offspring rather than a continuous distribution of offspring phenotypes. If the probability of extreme fluctuations in the optimum is sufficiently small, then the ESS distribution is monomorphic: a single type fitted to the mean environment. At higher levels of variability, the ESS distribution is polymorphic, and we find stability conditions for dimorphic distributions. For an exponential or similarly broad-tailed distribution of the optimum phenotype, the ESS consists of an infinite number of distinct phenotypes. In the second model we assume that an individual produces offspring with a single, genetically determined phenotype (pure-strategies model). The ESS population then contains multiple genotypes when the environmental variance is sufficiently high. However the phenotype distributions are similar to those in the mixed-strategies model: discrete, with an increasing number of distinct phenotypes as the environmental variance increases.     

Determinism in a transient assemblage: the roles of dispersal and local competition

Both dispersal and local competitive ability may determine the outcome of competition among species that cannot coexist locally. I develop a spatially implicit model of two-species competition at a small spatial scale. The model predicts the relative fitness of two competitors based on local reproductive rates and regional dispersal rates in the context of the number, size, and extinction probability of habitat patches in the landscape. I test the predictions of this model experimentally using two genotypes of the bacteriophagous soil nematode Caenorhabditis elegans in patchy microcosms. One genotype has higher fecundity while the other is a better disperser. With such a fecundity-dispersal trade-off between competitors, the model predicts that relative fitness will be affected most by local population size when patches do not go extinct and by the number of patches when there is a high probability of patch extinction. The microcosm experiments support the model predictions. Both approaches suggest that competitive dominance in a patchily distributed transient assemblage will depend upon the architecture and predictability of the environment. These mechanisms, operating at a small scale with high spatial admixture, may be embedded in a larger metacommunity process.     

Evolution of dispersal in a predator-prey metacommunity

Dispersal is crucial to allowing species inhabiting patchy or spatially subdivided habitats to persist globally despite the possibility of frequent local extinctions. Theoretical studies have repeatedly demonstrated that species that exhibit a regional metapopulation structure and are subject to increasing rates of local patch extinctions should experience strong selective pressures to disperse more rapidly despite the costs such increased dispersal would entail in terms of decreased local fitness. We extend these studies to consider how extinctions arising from predator-prey interactions affect the evolution of dispersal for species inhabiting a metacommunity. Specifically, we investigate how increasing a strong extinction-prone interaction between a predator and prey within local patches affects the evolution of each species' dispersal. We found that for the predator, as expected, evolutionarily stable strategy (ESS) dispersal rates increased monotonically in response to increasing local extinctions induced by strong predator top-down effects. Unexpectedly for the prey, however, ESS dispersal rates displayed a nonmonotonic response to increasing predator-induced extinction rates-actually decreasing for a significant range of values. These counterintuitive results arise from how extinctions resulting from trophic interactions play out at different spatial scales: interactions that increase extinction rates of both species locally can, at the same time, decrease the frequency of interaction between the prey and predator at the metacommunity scale.     

Non-equilibria in small metapopulations: comparing the deterministic Levins model with its stochastic counterpart

In this paper, we examine, for small metapopulations, the stochastic analog of the classical Levins metapopulation model. We study its basic model output, the expected time to metapopulation extinction, for systems which are brought out of equilibrium by imposing sudden changes in patch number and the colonization and extinction parameters. We find that the expected metapopulation extinction time shows different behavior from the relaxation time of the original, deterministic, Levins model. This relaxation time is therefore limited in value for predicting the behavior of the stochastic model. However, predictions about the extinction time for deterministically unviable cases remain qualitatively the same. Our results further suggest that, if we want to counteract the effects of habitat loss or increased dispersal resistance, the optimal conservation strategy is not to restore the original situation, that is, to create habitat or decrease resistance against dispersal. As long as the costs for different management options are not too dissimilar, it is better to improve the quality of the remaining habitat in order to decrease the local extinction rate.     

Local extinction and the evolution of dispersal rates: causes and correlations

We present the results of individual-based simulation experiments on the evolution of dispersal rates of organisms living in metapopulations. We find conflicting results regarding the relationship between local extinction rate and evolutionarily stable (ES) dispersal rate depending on which principal mechanism causes extinction: if extinction is caused by environmental catastrophes eradicating local populations, we observe a positive correlation between extinction and ES dispersal rate; if extinction is a consequence of stochastic local dynamics and environmental fluctuations, the correlation becomes ambiguous; and in cases where extinction is caused by dispersal mortality, a negative correlation between local extinction rate and ES dispersal rate emerges. We conclude that extinction rate, which both affects and is affected by dispersal rates, is not an ideal predictor for optimal dispersal rates.     

Evolutionarily stable strategies for localized dispersal in two dimensions

Evolutionarily stable dispersal strategies are calculated for a generalized two-dimensional stepping-stone model in which dispersal is not restricted to nearest neighbour sites. It is shown analytically that to an excellent approximation the ESS is independent of the dispersal pattern and is the same as the result of an earlier paper which used the simple but unrealistic island model (the dispersal survival rate of the previous paper must be replaced by an average value). When dispersal to different distances is under separate genetic control it is also possible to make general predictions concerning the fall-off of genetic correlation with distance. Two generalizations of the model are considered: random extinction of sites (creating empty sites which are opportunities for colonization), and perennation of existing occupants. These modifications are intended to bring the model closer to two real-world circumstances: dispersal between discrete colonies of animals living on a patchy resource, and seed and pollen dispersal in evenly distributed plant species. Simulations are required to validate any extensions of the analytic model; I present a relatively fast technique for determining the ESS, which relies in part on the analytic results.     

Interactions between spatial and temporal scales in the evolution of dispersal rate

The evolution of dispersal rate is studied with a model of several local populations linked by dispersal. Three dispersal strategies are considered where all, half or none of the offspring disperse. The spatial scale (number of patches) and the temporal scale (probability of local extinction) of the environment are critical in determining the selective advantage of the different dispersal strategies. The results from the simulations suggest that an interaction between group selection and individual selection results in a different outcome in relation to the spatial and temporal scales of the environment. Such an interaction is able to maintain a polymorphism in dispersal strategies. The maintenance of this polymorphism is also scale-dependent. This study suggests a mechanism for the short-term evolution of dispersal, and provides a testable prediction of this hypothesis, namely that loss of dispersal abilities should be more frequent in spatially more continuous environments, or in temporally more stable environments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Does social complexity lead to sex-biased dispersal in polygynous mammals? A test on ground-dwelling sciurids

Mating systems are well known to influence the dispersing sex,but the magnitude of the sex-biased dispersal has not actuallybeen measured, whereas many theoretical predictions have beenmade. In this study, we tested a new prediction about the coevolutionbetween natal dispersal and sociality from a recent evolutionarilystable strategy (ESS) approach. From a comparative approach,we showed that, in agreement with the model, the male-biaseddispersal increases with increasing level of sociality in polygynousground-dwelling sciurids. In addition, the increase in male-biaseddispersal with increasing sociality results from an increasein male dispersal rates, whereas female dispersal rates remainconstant, contrary to what is expected from the ESS model. Althoughthe mating system through the level of polygyny could act asa confounding factor, our results strengthen previous work thatstates that inbreeding avoidance plays a major role in the evolutionof dispersal for the most social mammalian species.     

Evolutionary stable sex ratios with non‐facultative male‐eggs first sex allocation in fig wasps

Sex allocation theory has long generated insights into the nature of natural selection. Classical models have elucidated causal phenomena such as local mate competition and inbreeding on the degree of female bias exhibited by various invertebrates. Typically, these models assume mothers facultatively adjust sex allocation using predictive cues of future offspring mating conditions. Here we relax this assumption by developing a sex allocation model for haplodiploid mothers experiencing local mate competition that lay a fixed number of male eggs first. Female egg number is determined by remaining oviposition sites or remaining eggs of the mother, depending on which is exhausted first. Our model includes parameters for variation in foundress number, patch size, fecundity and offspring mortality that allow us to generate secondary sex ratio predictions based on specific parameterizations for natural populations. Simulations show that: 1) in line with classical models, factors that increase sib‐mating result in mothers laying relatively more female eggs; 2) high offspring mortality leads to relatively more males as fertilization insurance; 3) unlike classical model predictions, sub‐optimal predictions, such as more males than females are possible. In addition, our model provides the first quantitative predictions for the expected number of males and females in a patch where typically only one mother utilizes a given patch. We parameterized the model with data obtained from seven species of southern African fig wasps to predict expected means and variances for numbers of male and female offspring for typical numbers of mothers utilizing a patch. These predictions were compared to secondary sex ratio data from single foundress patches, the most commonly encountered situation for these species. Our predictions matched both the observed number and variance of male and female offspring with a high degree of accuracy suggesting that facultative adjustment is not required to produce evolutionary stable sex ratios.     

Does disturbance favor dispersal? An analysis of ant migration using the colony-based lattice model

Spatially explicit models that simulate the evolution of dispersal strategies have not considered colonial organisms. Here we develop the colony-based lattice model, in which a colony, rather than an individual, occupies each lattice site. With this model we investigate why invasive tramp ant species usually lack long-distance dispersal, despite living in frequently disturbed habitats. We assume a new trade-off between the dispersal distance and the offspring colony size in the competition between two extreme strategies: the short-distance dispersal strategy (the S strategy, simulating budding or fission), which splits a colony in half with one of the two halves moving to a neighboring site, and the long-distance dispersal strategy (the L strategy, assuming colony-founding by winged queens), which allocates a minimal resource to an offspring colony that disperses to a randomly chosen site. Mortality of a colony is assumed to depend on the size; the L strategy suffers from costs due to small initial colony size (i.e., high mortality and late colony maturity). Disturbance causes additional mortality to both types of colonies and is controlled by disturbance frequency, p, and a stochastic parameter determining the spatial autocorrelation of disturbance, q. Simulations show that the S strategy is favored under frequent but spatially small-scale disturbances (high p and low q), whereas large-scale disturbances (low p and high q) favor the L strategy. When mortality is generally high or particularly high in small colonies, the S strategy tends to be advantageous. In contrast, when colony mortality is generally low, the L strategy is favored. We discuss the importance of colony size dynamics and the trade-off between colony size and the dispersal distance in the evolution of dispersal strategies in ants and other more or less sessile organisms.     

Evolution of sex ratios in social hymenoptera: Consequences of finite brood size

Evolutionarily stable sex ratios are determined for social hymenoptera under local mate competition (LMC) and when the brood size is finite. LMC is modelled by the parameterd. Of the reproductive progeny from a single foundress nest, a fractiond disperses (outbreeding), while (l-d) mate amongst themselves (sibmating). When the brood size is finite,d is taken to be the probability of an offspring dispersing, and similarly,r, the proportion of male offspring, the probability of a haploid egg being laid. Under the joint influence of these two stochastic processes, there is a nonzero probability that some females remain unmated in the nest. As a result, the optimal proportion of males (corresponding to the evolutionarily stable strategy, ESS) is higher than that obtained when the brood size is infinite. When the queen controls the sex ratio, the ESS becomes more female biased under increased inbreeding (lowerd). However, the ESS under worker control shows an unexpected pattern, including anincrease in the proportion ofmales withincreased inbreeding. This effect is traced to the complex interaction between inbreeding and local mate competition.     

Group selection among alternative evolutionarily stable strategies

Many important models of the evolution of social behavior have more than one evolutionarily stable strategy (ESS). Examples include co-ordination games, contests, mutualism, reciprocity, and sexual selection. Here we show that when there are multiple evolutionarily stable strategies, selection among groups can cause the spread of the strategy that has the lowest extinction rate or highest probability of contributing to the colonization of empty habitats, and that this may occur even when groups are usually very large, migration rates are substantial, and "extinction" entails only the disruption of the group and the dispersal of its members. The main requirements are: (1) individuals drawn from a single surviving group make up a sufficiently large fraction newly formed groups, and (2) the processes increasing the frequency of successful strategies within groups are strong compared to rate of migration among groups. The latter condition suggests that this form of group selection will be particularly important when behavioral variation is culturally acquired.     

EVOLUTION OF DISPERSAL RATES IN METAPOPULATION MODELS: BRANCHING AND CYCLIC DYNAMICS IN PHENOTYPE SPACE

We study the evolution of dispersal rates in a two patch metapopulation model. The local dynamics in each patch are given by difference equations, which, together with the rate of dispersal between the patches, determine the ecological dynamics of the metapopulation. We assume that phenotypes are given by their dispersal rate. The evolutionary dynamics in phenotype space are determined by invasion exponents, which describe whether a mutant can invade a given resident population. If the resident metapopulation is at a stable equilibrium, then selection on dispersal rates is neutral if the population sizes in the two patches are the same, while selection drives dispersal rates to zero if the local abundances are different. With non-equilibrium metapopulation dynamics, non-zero dispersal rates can be maintained by selection. In this case, and if the patches are ecologically identical, dispersal rates always evolve to values which induce synchronized metapopulation dynamics. If the patches are ecologically different, evolutionary branching into two coexisting dispersal phenotypes can be observed. Such branching can happen repeatedly, leading to polymorphisms with more than two phenotypes. If there is a cost to dispersal, evolutionary cycling in phenotype space can occur due to the dependence of selection pressures on the ecological attractor of the resident population, or because phenotypic branching alternates with the extinction of one of the branches. Our results extend those of Holt and McPeek (1996), and suggest that phenotypic branching is an important evolutionary process. This process may be relevant for sympatric speciation.