The evolution of dispersal rates in a heterogeneous time-periodic environment

A reaction-diffusion model for the evolution of dispersal rates is considered in which there is both spatial heterogeneity and temporal periodicity. The model is restricted to two phenotypes because of technical difficulties, but a wide range of mathematical techniques and computational effort are needed to obtain useful answers. We find that the question of selection is a great deal richer than in the autonomous case, where the phenotype with the lowest diffusion is selected for. In the current model either the lower or higher diffuser rate may be selected, or there may be coexistence of phenotypes. The paper raises several open questions and suggests in particular that a mutation-selection multi-phenotypic model would repay study. Received: 17 April 2000 / Revised version: 2 May 2001 / Published online: 12 October 2001     

The evolution of dispersal

A non-local model for dispersal with continuous time and space is carefully justified and discussed. The necessary mathematical background is developed and we point out some interesting and challenging problems. While the basic model is not new, a spread parameter (effectively the width of the dispersal kernel) has been introduced along with a conventional rate paramter, and we compare their competitive advantages and disadvantages in a spatially heterogeneous environment. We show that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal. However, fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results. For example, in a fairly general setting given two phenotypes with different, but small spread, the smaller spread is selected while in the case of large spread the larger spread is selected. S. Martinez was partially supported by Fondecyt 1020126 and Fondecyt Lineas Complementarias 8000010. K. Mischaikow was supported in part by NSF Grant DMS 0107396. Key words or phases:Non-local dispersal – Integral kernel – Evolution of dispersal     

Molecular evolution and polymorphism in a random environment

A model of multi-allelic selection in a random environment, the SAS-CFF model, is examined for compatability with allele frequency and genetic distance data acquired by electrophoresis. The symmetric version of the model tends to predict higher than observed levels of polymorphism unless substantial positive autocorrelations in the environment are postulated. The actual allele frequency configurations observed in nature are in rough agreement with those predicted by the SAS-CFF model. An approximate analysis of the transient properties of the SAS-CFF model shows that, in broad outline, the behavior is quite similar to that of the neutral model.     

The evolution of density-dependent dispersal

Despite a large body of empirical evidence suggesting that the dispersal rates of many species depend on population density, most metapopulation models assume a density-independent rate of dispersal. Similarly, studies investigating the evolution of dispersal have concentrated almost exclusively on density-independent rates of dispersal. We develop a model that allows density-dependent dispersal strategies to evolve. Our results demonstrate that a density-dependent dispersal strategy almost always evolves and that the form of the relationship depends on reproductive rate, type of competition, size of subpopulation equilibrium densities and cost of dispersal. We suggest that future metapopulation models should account for density-dependent dispersal     

The evolution of an 'intelligent' dispersal strategy: biased, correlated random walks in patchy landscapes

Theoretical work exploring dispersal evolution focuses on the emigration rate of individuals and typically assumes that movement occurs either at random to any other patch or to one of the nearest‐neighbour patches. There is a lack of work exploring the process by which individuals move between patches, and how this process evolves. This is of concern because any organism that can exert control over dispersal direction can potentially evolve efficiencies in locating patches, and the process by which individuals find new patches will potentially have major effects on metapopulation dynamics and gene flow. Here, we take an initial step towards filling this knowledge gap. To do this we constructed a continuous space population model, in which individuals each carry heritable trait values that specify the characteristics of the biased correlated random walk they use to disperse from their natal patch. We explore how the evolution of the random walk depends upon the cost of dispersal, the density of patches in the landscape, and the emigration rate. The clearest result is that highly correlated walks always evolved (individuals tended to disperse in relatively straight lines from their natal patch), reflecting the efficiency of straight‐line movement. In our models, more costly dispersal resulted in walks with higher correlation between successive steps. However, the exact walk that evolved also depended upon the density of suitable habitat patches, with low density habitat evolving more biased walks (individuals which orient towards suitable habitat at quite large distances from that habitat). Thus, low density habitat will tend to develop individuals which disperse efficiently between adjacent habitat patches but which only rarely disperse to more distant patches; a result that has clear implications for metapopulation theory. Hence, an understanding of the movement behaviour of dispersing individuals is critical for robust long‐term predictions of population dynamics in fragmented landscapes.     

The evolution of dispersal distance in spatially-structured populations

Most evolutionary models of dispersal have concentrated on dispersal rate, with emigration being either global or restricted to nearest neighbours. Yet most organisms fall into an intermediate region where most dispersal is local but there is a wide range of dispersal distances. We use an individual-based model with 2500 patches each with identical local dynamics and show that the dispersal distance is under selection pressure. The dispersal distance that evolves is critically dependent on the ecological dynamics. When the cost of dispersal increases linearly with distance, selection is for short-distance dispersal under stable and damped local dynamics but longer distance dispersal is favoured as local dynamics become more complex. For the cases of stable, damped and periodic patch dynamics global patch synchrony occurs even with very short-distance dispersal. Increasing the scale of dispersal for chaotic local dynamics increases the scale of synchrony but global synchrony does not neccesarily occur. We discuss these results in the light of other possible causes of dispersal and argue for the importance of incorporating non-equilibrium population dynamics into evolutionary models of dispersal distance.     

The evolution of dispersal under demographic stochasticity

Temporal and spatial variations of the environment are important factors favoring the evolution of dispersal. With few exceptions, these variations have been considered to be exclusively fluctuations of habitat quality. However, since the presence of conspecifics forms part of an individual's environment, demographic stochasticity may be a component of this variability as well, in particular when local populations are small. To study this effect, we analyzed the evolution of juvenile dispersal in a metapopulation model in which habitat quality is constant in space and time but occupancy fluctuates because of demographic stochasticity. Our analysis extends previous studies in that it includes competition of resources and competition for space. Also, juvenile dispersal is not given by a fixed probability but is made conditional on the presence of free territories in a patch, whereas individuals born in full patches will always disperse. Using a combination of analytical and numerical approaches, we show that demographic stochasticity in itself may provide enough variability to favor dispersal even from patches that are not fully occupied. However, there is no simple relationship between the evolution of dispersal and various indicators of demographic stochasticity. Selected dispersal depends on all aspects of the life-history profile, including kin selection.     

The evolution of delayed dispersal in cooperative breeders.

Why do the young of cooperative breeders--species in which more than two individuals help raise offspring at a single nest--delay dispersal and live in groups? Answering this deceptively simple question involves examining the costs and benefits of three alternative strategies: (1) dispersal and attempting to breed, (2) dispersal and floating, and (3) delayed dispersal and helping. If, all other things being equal, the fitness of individuals that delay dispersal is greater than the fitness of individuals that disperse and breed on their own, intrinsic benefits are paramount to the current maintenance of delayed dispersal. Intrinsic benefits are directly due to living with others and may include enhanced foraging efficiency and reduced susceptibility to predation. However, if individuals that disperse and attempt to breed in high-quality habitat achieve the highest fitness, extrinsic constraints on the ability of offspring to obtain such high-quality breeding opportunities force offspring to either delay dispersal or float. The relevant constraint to independent reproduction has frequently been termed habitat saturation. This concept, of itself, fails to explain the evolution of delayed dispersal. Instead, we propose the delayed-dispersal threshold model as a guide for organizing and evaluating the ecological factors potentially responsible for this phenomenon. We identify five parameters critical to the probability of delayed dispersal: relative population density, the fitness differential between early dispersal/breeding and delayed dispersal, the observed or hypothetical fitness of floaters, the distribution of territory quality, and spatiotemporal environmental variability. A key conclusion from the model is that no one factor by itself causes delayed dispersal and cooperative breeding. However, a difference in the dispersal patterns between two closely related species or populations (or between individuals in the same population in different years) may be attributable to one or a small set of factors. Much remains to be done to pinpoint the relative importance of different ecological factors in promoting delayed dispersal. This is underscored by our current inability to explain satisfactorily several patterns including the relative significance of floating, geographic biases in the incidence of cooperative breeding, sexual asymmetries in delayed dispersal, the relationship between delayed dispersal leading to helping behavior and cooperative polygamy, and the rarity of the co-occurrence of helpers and floaters within the same population. Advances in this field remain to be made along several fronts.(ABSTRACT TRUNCATED AT 400 WORDS)     

The evolution of parasite manipulation of host dispersal

We investigate the evolution of manipulation of host dispersal behaviour by parasites using spatially explicit individual-based simulations. We find that when dispersal is local, parasites always gain from increasing their hosts' dispersal rate, although the evolutionary outcome is determined by the costs-to-benefits ratio. However, when dispersal can be non-local, we show that parasites investing in an intermediate dispersal distance of their hosts are favoured even when the manipulation is not costly, due to the intrinsic spatial dynamics of the host-parasite interaction. Our analysis highlights the crucial importance of ecological spatial dynamics in evolutionary processes and reveals the theoretical possibility that parasites could manipulate their hosts' dispersal.     

The evolution of dispersal conditioned on migration status

We consider a model for the evolution of dispersal of offspring. Dispersal is treated as a parental trait that is expressed conditional upon a parent's own "migration status," that is, whether a parent, itself, is native or nonnative to the area in which it breeds. We compare the evolution of this kind of conditional dispersal to the evolution of unconditional dispersal, in order to determine the extent to which the former changes predictions about population-wide levels of dispersal. We use numerical simulations of an inclusive-fitness model, and individual-based simulations to predict population-average dispersal rates for the case in which dispersal based on migration status occurs. When our model predictions are compared to predictions that neglect conditional dispersal, observed differences between rates are only slight, and never exceed 0.06. While the effect of dispersal conditioned upon migration status could be detected in a carefully designed experiment, we argue that less-than-ideal experimental conditions, and factors such as dispersal conditioned on sex are likely to play a larger role that the type of conditional dispersal studied here.     

The evolution of social philopatry and dispersal in female mammals

In most social mammals, some females disperse from their natal group while others remain and breed there throughout their lives but, in a few, females typically disperse after adolescence and few individuals remain and breed in their natal group. These contrasts in philopatry and dispersal have an important consequence on the kinship structure of groups which, in turn, affects forms of social relationships between females. As yet, there is still widespread disagreement over the reasons for the evolution of habitual female dispersal, partly as a result of contrasting definitions of dispersal. This paper reviews variation in the frequency with which females leave their natal group or range (social dispersal) and argues that both the avoidance of local competition for resources and breeding opportunities and the need to find unrelated partners play an important role in contrasts between and within species.     

The color of noise and the evolution of dispersal

The process of dispersal is vital for the long-term persistence of all species and hence is a ubiquitous characteristic of living organisms. A present challenge is to increase our understanding of the factors that govern the dispersal rate of individuals. Here I extend previous work by incorporating both spatial and temporal heterogeneity in terms of patch quality into a spatially explicit lattice model. The spatial heterogeneity is modeled as a two-dimensional fractal landscape, while temporal heterogeneity is included by using one-dimensional noise. It was found that the color of both the spatial and temporal variability influences the rate of dispersal selected as reddening of the temporal noise leads to a reduction in dispersal, while reddening of spatial variability results in an increase in the dispersal rate. These results demonstrate that the color of environmental noise should be considered in future studies looking at the evolution of life history characteristics.     

The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

The evolution of density-dependent dispersal in a noisy spatial population model

It is well-known that dispersal is advantageous in many different ecological situations, e.g. to survive local catastrophes where populations live in spatially and temporally heterogeneous habitats. However, the key question, what kind of dispersal strategy is optimal in a particular situation, has remained unanswered. We studied the evolution of density-dependent dispersal in a coupled map lattice model, where the population dynamics are perturbed by external environmental noise. We used a very flexible dispersal function to enable evolution to select from practically all possible types of monotonous density-dependent dispersal functions. We treated the parameters of the dispersal function as continuously changing phenotypic traits. The evolutionary stable dispersal strategies were investigated by numerical simulations. We pointed out that irrespective of the cost of dispersal and the strength of environmental noise, this strategy leads to a very weak dispersal below a threshold density, and dispersal rate increases in an accelerating manner above this threshold. Decreasing the cost of dispersal increases the skewness of the population density distribution, while increasing the environmental noise causes more pronounced bimodality in this distribution. In case of positive temporal autocorrelation of the environmental noise, there is no dispersal below the threshold, and only low dispersal below it, on the other hand with negative autocorrelation practically all individual disperses above the threshold. We found our results to be in good concordance with empirical observations.     

The biennial life strategy in a random environment

A discrete-time population model with two age classes is studied which describes the growth of biennial plants in a randomly varying environment. A fraction of the oldest age class delays its flowering each year. The solution of the model involves products of random matrices. We calculate the exact mean and variance of the long-run geometric growth rate assuming a gamma distribution for the random number of offspring per flowering plant after one year. It is shown, both by analytical calculation and numerical examples, that it is profitable for the population to delay its flowering, in the sense that the average growth rate increases and the extinction probability decreases. The optimal values of the flowering fraction depend upon the environmental and model parameters.     

The biennial life strategy in a random environment

In a previous paper (J. Math. Biol. 26, 199–215 (1988)) we calculated the mean and variance of the long-run geometric growth rate of a discrete-time population model with two age classes in a random environment. The formula which was used in that paper as the starting point for the computation of the variance represents only the contribution of the one-period variances. Here we supplement these results by a calculation of the exact variance. All qualitative conclusions reached before are unaffected.