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     

Stability in a metapopulation model with density-dependent dispersal

A spatially explicit metapopulation model with positive density-dependent migration is analysed. We obtained conditions under which a previously stable system can be driven to instability caused by a density-dependent migration mechanism. The stability boundary depends on the rate of increase of the number of migrants on each site at local equilibrium, on the intrinsic rate of increase at local level, on the number of patches, and on topological aspects regarding the connectivity between patches. A concrete example is presented illustrating the dynamics on the dispersal-induced unstable regime.     

Environmentally-controlled,density-dependent secondary dispersal in a local estuarine crab population

The mechanisms driving the pelagic secondary dispersal of aquatic organisms following initial settlement to benthic habitats are poorly characterized. We examined the physical environmental (wind, diel cycle, tidal phase) and biological (ontogenetic, density-dependent) factors that contribute to the secondary dispersal of a benthic marine invertebrate, the blue crab (Callinectes sapidus) in Pamlico Sound, NC, USA. Field studies conducted in relatively large (0.05 km²) seagrass beds determined that secondary dispersal is primarily undertaken by the earliest juvenile blue crab instar stages (J1 crabs). These crabs emigrated pelagically from seagrass settlement habitats using nighttime flood tides during average wind conditions (speed ~5 m s^–1). Moreover, the secondary dispersal of J1 crabs was density-dependent and regulated by intra-cohort (J1) crab density in seagrass. Our results suggest that dispersal occurs rapidly following settlement, and promotes blue crab metapopulation persistence by redistributing juveniles from high-density settlement habitats to areas characterized by low postlarval supply. Collectively, these data indicate that blue crab secondary dispersal is an active process under behavioral control and can alter initial distribution patterns established during settlement. This study highlights the necessity of considering secondary dispersal in ecological studies to improve our understanding of population dynamics of benthic organisms.     

Accelerating invasion rates result from the evolution of density-dependent dispersal

Evolutionary processes play an important role in shaping the dynamics of range expansions, and selection on dispersal propensity has been demonstrated to accelerate rates of advance. Previous theory has considered only the evolution of unconditional dispersal rates, but dispersal is often more complex. For example, many species emigrate in response to crowding. Here, we use an individual-based model to investigate the evolution of density dependent dispersal into empty habitat, such as during an invasion. The landscape is represented as a lattice and dispersal between populations follows a stepping-stone pattern. Individuals carry three ‘genes’ that determine their dispersal strategy when experiencing different population densities. For a stationary range we obtain results consistent with previous theoretical studies: few individuals emigrate from patches that are below equilibrium density. However, during the range expansion of a previously stationary population, we observe evolution towards dispersal strategies where considerable emigration occurs well below equilibrium density. This is true even for moderate costs to dispersal, and always results in accelerating rates of range expansion. Importantly, the evolution we observe at an expanding front depends upon fitness integrated over several generations and cannot be predicted by a consideration of lifetime reproductive success alone. We argue that a better understanding of the role of density dependent dispersal, and its evolution, in driving population dynamics is required especially within the context of range expansions.     

Density-dependent dispersal and spatial distribution of a population

Many models have been proposed to suggest that animal dispersal enhances stability of an ecological system. However, little attention has been paid on the property of a boundary and on the size of a region. In this paper, we will consider the models proposed by Gurney & Nisbet (1975), from those points of view. We will show that, if a population is dispersing in a highly density-dependent manner, a stationary distribution which does not depend on boundary conditions is established in a finite region by interactions between the population and a heterogeneous environment. We will also show that, even when a population is confined in a habitat with a limited size a population dispersing density-dependently can establish a stationary distribution, whereas a population dispersing randomly either goes to extinction or grows explosively.     

The role of density-dependent dispersal in source-sink dynamics

I investigate two aspects of source-sink theory that have hitherto received little attention: density-dependent dispersal and the cost of dispersal to sources. The cost arises because emigration reduces the per capita growth rate of sources, thus predisposing them to extinction. I show that source-sink persistence depends critically on the interplay between these two factors. When the emigration rate increases with abundance at an accelerating rate, dispersal costs to sources is the lowest and risk of source-sink extinction the least. When the emigration rate increases with abundance at a decelerating rate, dispersal costs to sources is the highest and the risk of source-sink extinction the greatest. Density-independent emigration has an intermediate effect. Thus, density-dependent dispersal per se does not increase or decrease source-sink persistence relative to density-independent dispersal. The exact mode of dispersal is crucial. A key point to appreciate is that these effects of dispersal on source-sink extinction arise from the temporal density-dependence that dispersal induces in the per capita growth rates of source and sink populations. Temporal density-dependence due to dispersal is beneficial at low abundances because it rescues sinks from extinction, and detrimental at high abundances because it drives otherwise viable sources to extinction. These results are robust to the nature of population dynamics in the sink, whether exponential or logistic. They provide a means of assessing the relative costs and benefits of preserving sink habitats given three biological parameters.     

Evolution of density-dependent dispersal in a structured metapopulation

We study the evolution of density-dependent dispersal in a structured metapopulation subject to local catastrophes that eradicate local populations. To this end we use the theory of structured metapopulation dynamics and the theory of adaptive dynamics.The set of evolutionarily possible dispersal functions (i.e., emigration rates as a function of the local population density) is derived mechanistically from an underlying resource-consumer model. The local resource dynamics is of a flow-culture type and consumers leave a local population with a constant probability per unit of time κ when searching for resources but not when handling resources (i.e., eating and digesting). The time an individual spends searching (as opposed to handling) depends on the local resource density, which in turn depends on the local consumer density, and so the average per capita emigration rate depends on the local consumer density as well.The derived emigration rates are sigmoid functions of local consumer population density. The parameters of the local resource-consumer dynamics are subject to evolution. In particular, we find that there exists a unique evolutionarily stable and attracting dispersal rate κ^∗ for searching consumers. The κ^∗ increases with local resource productivity and decreases with resource decay rate. The κ^∗ also increases with the survival probability during dispersal, but as a function of the catastrophe rate it reaches a maximum before dropping off to zero again.     

Negative density-dependent dispersal emerges from the joint evolution of density- and body condition-dependent dispersal strategies

Empirical studies have documented both positive and negative density-dependent dispersal, yet most theoretical models predict positive density dependence as a mechanism to avoid competition. Several hypotheses have been proposed to explain the occurrence of negative density-dependent dispersal, but few of these have been formally modeled. Here, we developed an individual-based model of the evolution of density-dependent dispersal. This model is novel in that it considers the effects of density on dispersal directly, and indirectly through effects on individual condition. Body condition is determined mechanistically, by having juveniles compete for resources in their natal patch. We found that the evolved dispersal strategy was a steep, increasing function of both density and condition. Interestingly, although populations evolved a positive density-dependent dispersal strategy, the simulated metapopulations exhibited negative density-dependent dispersal. This occurred because of the negative relationship between density and body condition: high density sites produced low-condition individuals that lacked the resources required for dispersal. Our model, therefore, generates the novel hypothesis that observed negative density-dependent dispersal can occur when high density limits the ability of organisms to disperse. We suggest that future studies consider how phenotype is linked to the environment when investigating the evolution of dispersal.     

The spatial scale of density-dependent growth and implications for dispersal from nests in juvenile Atlantic salmon

By dispersing from localized aggregations of recruits, individuals may obtain energetic benefits due to reduced experienced density. However, this will depend on the spatial scale over which individuals compete. Here, we quantify this scale for juvenile Atlantic salmon (Salmo salar) following emergence and dispersal from nests. A single nest was placed in each of ten replicate streams during winter, and information on the individual positions (±1 m) and the body sizes of the resulting young-of-the-year (YOY) juveniles was obtained by sampling during the summer. In six of the ten streams, model comparisons suggested that individual body size was most closely related to the density within a mean distance of 11 m (range 2–26 m). A link between body size and density on such a restricted spatial scale suggests that dispersal from nests confers energetic benefits that can counterbalance any survival costs. For the four remaining streams, which had a high abundance of trout and older salmon cohorts, no single spatial scale could best describe the relation between YOY density and body size. Energetic benefits of dispersal associated with reduced local density therefore appear to depend on the abundance of competing cohorts or species, which have spatial distributions that are less predictable in terms of distance from nests. Thus, given a trade-off between costs and benefits associated with dispersal, and variation in benefits among environments, we predict an evolving and/or phenotypically plastic growth rate threshold which determines when an individual decides to disperse from areas of high local density.     

Density-dependent dispersal and spatial population dynamics

The synchronization of the dynamics of spatially subdivided populations is of both fundamental and applied interest in population biology. Based on theoretical studies, dispersal movements have been inferred to be one of the most general causes of population synchrony, yet no empirical study has mapped distance-dependent estimates of movement rates on the actual pattern of synchrony in species that are known to exhibit population synchrony. Northern vole and lemming species are particularly well-known for their spatially synchronized population dynamics. Here, we use results from an experimental study to demonstrate that tundra vole dispersal movements did not act to synchronize population dynamics in fragmented habitats. In contrast to the constant dispersal rate assumed in earlier theoretical studies, the tundra vole, and many other species, exhibit negative density-dependent dispersal. Simulations of a simple mathematical model, parametrized on the basis of our experimental data, verify the empirical results, namely that the observed negative density-dependent dispersal did not have a significant synchronizing effect.     

Female-biased density-dependent dispersal of a tephritid fly in a fragmented habitat and its implications for population regulation

The aim of this study was to quantify the rate of dispersal as a response to density in the specialist tephritid fly Paroxyna plantaginis (the main seed predator on its patchily distributed host plant, Tripolium vulgare, Asteraceae). Marked flies were released at three different fly densities in artificial host patches. The individual histories of recaptures were recorded as well as migration between patches and invasion by unmarked flies. The loss of marked flies relative to initial density was analysed using maximum likelihood estimation. Females generally had the highest loss rate. When comparing a density-independent model with a density-dependent model of the loss rate, the density-dependent model won four times out of six for the females but not a single time for the males. A stronger immigration rate of females relative to males supported the suggested female-biased dispersal. This indicates a sit-and-wait strategy for the territorial males and a pre-emptive competition strategy for egg-laying substrates for the females. These results may be of general importance for non-frugivorous tephritid systems with unpredictable and almost ephemeral accessibility to host plants and with a dynamics characterised by a high turnover rate and high attack levels. The study presents a method for measuring the propensity of individuals to leave an area as a response to local density. It is further an example of the consequences individual behavioural responses may have on the population dynamics of a patchy population.     

Spatial scaling in a benthic population model with density-dependent disturbance.

This work investigates approaches to simplifying individual-based models in which the rate of disturbance depends on local densities. To this purpose, an individual-based model for a benthic population is developed that is both spatial and stochastic. With this model, three possible ways of approximating the dynamics of mean numbers are examined: a mean-field approximation that ignores space completely, a second-order approximation that represents spatial variation in terms of variances and covariances, and a patch-based approximation that retains information about the age structure of the patch population. Results show that space is important and that a temporal model relying on mean disturbance rates provides a poor approximation to the dynamics of mean numbers. It is possible, however, to represent relevant spatial variation with second-order moments, particularly when recruitment rates are low and/or when disturbances are large and weak. Even better approximations are obtained by retaining patch age information.     

The evolution of slow dispersal rates: a reaction diffusion model

 We consider n phenotypes of a species in a continuous but heterogeneous environment. It is assumed that the phenotypes differ only in their diffusion rates. With haploid genetics and a small rate of mutation, it is shown that the only nontrivial equilibrium is a population dominated by the slowest diffusing phenotype. We also prove that if there are only two possible phenotypes, then this equilibrium is a global attractor and conjecture that this is true in general. Numerical simulations supporting this conjecture and suggesting that this is a robust phenomenon are also discussed. Received: 29 January 1997 / Revised version: 23 September 1997     

Emergence of global behaviour in a host–parasitoid model with density-dependent dispersal in a chain of patches

We present a time discrete spatial host–parasitoid model. The environment is a chain of patches connected by dispersal events. Dispersal of parasitoids is host-density dependent. When the host density is small (resp. high), the proportion of migrant parasitoids is close to unity (resp. to zero). We assume fast patch to patch dispersal with respect to local interactions. Local host–parasitoid interactions are described by the classical Nicholson–Bailey model. By using time scales separation methods (or aggregation methods), we obtain a reduced model that governs the total host and parasitoid densities (obtained by addition over all patches). The aggregated model describes the time evolution of the total number of hosts and parasitoids of the system of patches. This global model is useful to make predictions of emerging behaviour regarding the dynamics of the complete system. We study the effects of number of patches and host density-dependent parasitoid dispersal on the overall stability of the host–parasitoid system. We finally compare our stability results with the CV² > 1 rule.     

On the evolution of dispersal via heterogeneity in spatial connectivity

Dispersal has long been recognized as a mechanism that shapes many observed ecological and evolutionary processes. Thus, understanding the factors that promote its evolution remains a major goal in evolutionary ecology. Landscape connectivity may mediate the trade-off between the forces in favour of dispersal propensity (e.g. kin-competition, local extinction probability) and those against it (e.g. energetic or survival costs of dispersal). It remains, however, an open question how differing degrees of landscape connectivity may select for different dispersal strategies. We implemented an individual-based model to study the evolution of dispersal on landscapes that differed in the variance of connectivity across patches ranging from networks with all patches equally connected to highly heterogeneous networks. The parthenogenetic individuals dispersed based on a flexible logistic function of local abundance. Our results suggest, all else being equal, that landscapes differing in their connectivity patterns will select for different dispersal strategies and that these strategies confer a long-term fitness advantage to individuals at the regional scale. The strength of the selection will, however, vary across network types, being stronger on heterogeneous landscapes compared with the ones where all patches have equal connectivity. Our findings highlight how landscape connectivity can determine the evolution of dispersal strategies, which in turn affects how we think about important ecological dynamics such as metapopulation persistence and range expansion.     

Effects of environmental synchrony and density-dependent dispersal on temporal and spatial slopes of Taylor's law

Taylor's law (TL) is an empirical rule that describes an approximate relationship between the variance and mean of population density: log₁₀(variance) ≈ log₁₀(a) + b × log₁₀(mean). Population synchrony is another prevailing feature observed in empirical populations. This study investigated the effects of environmental synchrony and density-dependent dispersal on the temporal (b_T) and spatial (b_S) slopes of TL, using an empirical dataset of gray-sided vole populations and simulation analyses based on the second-order autoregressive (AR) model. Eighty-five empirical populations satisfied the temporal and spatial TLs with b_T = 1.943 (±SE 0.143) and b_S = 1.579 (±SE 0.136). The pairwise synchrony of population was 0.377 ± 0.199 (mean ± SD). Most simulated populations that obeyed the AR model satisfied the form of the temporal and spatial TLs without being affected by the environmental synchrony and density-dependent dispersal; however, those simulated slopes were too steep. The incorporation of environmental synchrony resulted in reduced simulated slopes, but those slopes, too, were still unrealistically steep. By incorporating density-dependent dispersal, simulated slopes decreased and fell within a realistic range. However, the simulated populations without environmental synchrony did not exhibit an adequate degree of density synchrony. In simulations that included both environmental synchrony and density-dependent dispersal, 92.7% of the simulated datasets provided realistic values for b_T, b_S and population synchrony. Because the two slopes were more sensitive to the variation of density-dependent dispersal than that of environmental synchrony, density-dependent dispersal may be the key to the determination of b_T and b_S.     

On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible–infectious–susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction–diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.     

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     

Evolution of positive and negative density-dependent dispersal

Understanding the evolution of density-dependent dispersal strategies has been a major challenge for evolutionary ecologists. Some existing models suggest that selection should favour positive and others negative density-dependence in dispersal. Here, we develop a general model that shows how and why selection may shift from positive to negative density-dependence in response to key ecological factors, in particular the temporal stability of the environment. We find that in temporally stable environments, particularly with low dispersal costs and large group sizes, habitat heterogeneity selects for negative density-dependent dispersal, whereas in temporally variable environments, particularly with high dispersal costs and small group sizes, habitat heterogeneity selects for positive density-dependent dispersal. This shift reflects the changing balance between the greater competition for breeding opportunities in more productive patches, versus the greater long-term value of offspring that establish themselves there, the latter being very sensitive to the temporal stability of the environment. In general, dispersal of individuals out of low-density patches is much more sensitive to habitat heterogeneity than is dispersal out of high-density patches.