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.     

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.     

Geographic variation in density-dependent dynamics impacts the synchronizing effect of dispersal and regional stochasticity

Explanations for the ubiquitous presence of spatially synchronous population dynamics have assumed that density-dependent processes governing the dynamics of local populations are identical among disjunct populations, and low levels of dispersal or small amounts of regionalized stochasticity (Moran effect) can act to synchronize populations. In this study we used historical spatially referenced data on gypsy moth (Lymantria dispar) outbreaks to document that density-dependent processes can vary substantially across geographical landscapes. This variation may be due in part to geographical variation in habitat (e.g., variation in forest composition). We then used a second-order log-linear stochastic model to explore how inter-population variation in density-dependent processes affects synchronization via either synchronous stochastic forcing or dispersal. We found that geographical variation in direct density-dependence (first order) greatly diminishes synchrony caused by stochasticity but only slightly decreases synchronization via dispersal. Variation in delayed density-dependence (second order) diluted synchrony caused by regional stochasticity to a lesser extent than first-order variation, but it did not have any influence on synchrony caused by dispersal. In general, synchronization caused by dispersal was primarily dependent upon the instability of populations and only weakly, if at all, affected by similarities in density-dependence among populations. We conclude that studies of synchrony should carefully consider both the nature of the synchronizing agents and the pattern of local density-dependent processes, including how these vary geographically.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Spatial variation and density-dependent dispersal in competitive coexistence

It is well known that dispersal from localities favourable to a species' growth and reproduction (sources) can prevent competitive exclusion in unfavourable localities (sinks). What is perhaps less well known is that too much emigration can undermine the viability of sources and cause regional competitive exclusion. Here, I investigate two biological mechanisms that reduce the cost of dispersal to source communities. The first involves increasing the spatial variation in the strength of competition such that sources can withstand high rates of emigration; the second involves reducing emigration from sources via density-dependent dispersal. I compare how different forms of spatial variation and modes of dispersal influence source viability, and hence source-sink coexistence, under dominance and pre-emptive competition. A key finding is that, while spatial variation substantially reduces dispersal costs under both types of competition, density-dependent dispersal does so only under dominance competition. For instance, when spatial variation in the strength of competition is high, coexistence is possible (regardless of the type of competition) even when sources experience high emigration rates; when spatial variation is low, coexistence is restricted even under low emigration rates. Under dominance competition, density-dependent dispersal has a strong effect on coexistence. For instance, when the emigration rate increases with density at an accelerating rate (Type III density-dependent dispersal), coexistence is possible even when spatial variation is quite low; when the emigration rate increases with density at a decelerating rate (Type II density-dependent dispersal), coexistence is restricted even when spatial variation is quite high. Under pre-emptive competition, density-dependent dispersal has only a marginal effect on coexistence. Thus, the diversity-reducing effects of high dispersal rates persist under pre-emptive competition even when dispersal is density dependent, but can be significantly mitigated under dominance competition if density-dependent dispersal is Type III rather than Type II. These results lead to testable predictions about source-sink coexistence under different regimes of competition, spatial variation and dispersal. They identify situations in which density-independent dispersal provides a reasonable approximation to species' dispersal patterns, and those under which consideration of density-dependent dispersal is crucial to predicting long-term coexistence.     

Effects of local interactions and local migration on stability

Mathematical models can help to resolve the longstanding question of whether more diverse communities are more stable. Here, I focus on how local dispersal and local interactions--hallmarks of spatial communities--affect stability in a spatially implicit model with demographic stochasticity. The results are based on a novel way to analyze moment equations. The main conclusion is that the type and strength of density-dependent factors, such as fecundity and competition, determine whether local dispersal and local interactions increase or decrease stability. Local dispersal has a stabilizing effect when fecundity is high, interspecific competition is either low or high, and the number of species is small. Effects of local migration on stability are amplified when space is explicit.     

Density-dependence as a size-independent regulatory mechanism

The growth function of populations is central in biomathematics. The main dogma is the existence of density-dependence mechanisms, which can be modelled with distinct functional forms that depend on the size of the population. One important class of regulatory functions is the theta-logistic, which generalizes the logistic equation. Using this model as a motivation, this paper introduces a simple dynamical reformulation that generalizes many growth functions. The reformulation consists of two equations, one for population size, and one for the growth rate. Furthermore, the model shows that although population is density-dependent, the dynamics of the growth rate does not depend either on population size, nor on the carrying capacity. Actually, the growth equation is uncoupled from the population size equation, and the model has only two parameters, a Malthusian parameter rho and a competition coefficient theta. Distinct sign combinations of these parameters reproduce not only the family of theta-logistics, but also the van Bertalanffy, Gompertz and Potential Growth equations, among other possibilities. It is also shown that, except for two critical points, there is a general size-scaling relation that includes those appearing in the most important allometric theories, including the recently proposed Metabolic Theory of Ecology. With this model, several issues of general interest are discussed such as the growth of animal population, extinctions, cell growth and allometry, and the effect of environment over a population.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

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.     

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.     

Small-scale demographic variability of the biocolor damselfish, <Emphasis Type="Italic">Stegastes partitus</Emphasis>, in the Florida Keys USA

The demographic responses of reef fish to their environment can be complex and in many cases, quite strong. Growth, mortality, longevity, and even reproductive effort have been demonstrated to vary for the same species of reef fish over scales of 100s to 1,000s of kilometers due to physiological and ecological interactions. Though few studies have explicitly documented it, this sort of habitat-mediated demography can also exist at very local scales. Here we present the results of a 2-year study of the bicolor damselfish, Stegastes partitus, in the Florida Keys, USA. We measured density and distribution, calculated key demographic rates (growth, survival, and fecundity), and characterized the environment (resident fish assemblage, substrate type and complexity, and food availability) of populations living in two adjacent but different habitats, the continuous fore reef and patchy back reef. Fish on the fore reef had an elevated growth rate and asymptotic size, increased mortality, and higher fecundity than fish on the back reef. We identified four potential causative mechanisms for these differences: food availability; competition; intraspecific density-dependent effects; and predation risk. Our data did not support an effect of either food availability or intraspecific density-dependence, but rather suggested that demographic responses are affected by both competition and predation risk.     

Evolution of density- and patch-size-dependent dispersal rates

Based on a marginal value approach, we derive a nonlinear expression for evolutionarily stable (ES) dispersal rates in a metapopulation with global dispersal. For the general case of density-dependent population growth, our analysis shows that individual dispersal rates should decrease with patch capacity and-beyond a certain threshold-increase with population density. We performed a number of spatially explicit, individual-based simulation experiments to test these predictions and to explore further the relevance of variation in the rate of population increase, density dependence, environmental fluctuations and dispersal mortality on the evolution of dispersal rates. They confirm the predictions of our analytical approach. In addition, they show that dispersal rates in metapopulations mostly depend on dispersal mortality and inter-patch variation in population density. The latter is dominantly driven by environmental fluctuations and the rate of population increase. These conclusions are not altered by the introduction of neighbourhood dispersal. With patch capacities in the order of 100 individuals, kin competition seems to be of negligible importance for ES dispersal rates except when overall dispersal rates are low.     

Effects of range position,inter-annual variation and density on demographic transition rates of <Emphasis Type="Italic">Hornungia petraea</Emphasis> populations

The central-marginal model assumes unfavourable and more variable environmental conditions at the periphery of a species’ distribution range to negatively affect demographic transition rates, finally resulting in reduced population sizes and densities. Previous studies on density-dependence as a crucial factor regulating plant population growth have mainly focussed on fecundity and survival. Our objective is to analyse density-dependence in combination with the effect of inter-annual variation and range position on all life stages of an annual plant species, Hornungia petraea, including germination and seed incorporation into the seed bank. As previous studies on H. petraea had revealed a pattern opposite to existing theory with lower population densities at the distribution centre in Italy than at the periphery in Germany, we hypothesised that (1) demographic transition rates are lower, (2) the inter-annual variation in demographic transition rates is higher and (3) the intensity of density-dependence is weaker in Italy than in Germany. To analyse demographic transition rates, we used an autoregressive covariance strategy for repeated measures including density and inter-annual variation. All the three hypotheses were confirmed, but the impact of range position, density-dependence and inter-annual variation differed among the transition steps. All transition rates except fecundity were higher in the German populations than in the Italian populations. Germination rate and incorporation rate into the seed bank were strongly density-dependent. Central populations showed a larger inter-annual variation in fecundity and winter survival rate. Winter survival rate was the only transition step with a stronger density-dependence in peripheral populations. In most cases, these differences between distribution centre and periphery would not have emerged without taking density-dependence and inter-annual variation into account. We conclude that including range position, inter-annual variation and density-dependence in one single statistical model is an important tool for the interpretation of demographic patterns regarding the central-marginal model. Electronic Supplementary Material Supplementary material is available for this article at     

Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities

Although density-dependent dispersal and relative dispersal (the difference in dispersal rates between species) have been documented in natural systems, their effects on the stability of metacommunities are poorly understood. Here we investigate the effects of intra- and interspecific density-dependent dispersal on the regional stability in a predator-prey metacommunity model. We show that, when the dynamics of the populations reach equilibrium, the stability of the metacommunity is not affected by density-dependent dispersal. However, the regional stability, measured as the regional variability or the persistence, can be modified by density-dependent dispersal when local populations fluctuate over time. Moreover these effects depend on the relative dispersal of the predator and the prey. Regional stability is modified through changes in spatial synchrony. Interspecific density-dependent dispersal always desynchronizses local dynamics, whereas intraspecific density-dependent dispersal may either synchronize or desynchronize it depending on dispersal rates. Moreover, intra- and interspecific density-dependent dispersal strengthen the top-down control of the prey by the predator at intermediate dispersal rates. As a consequence the regional stability of the metacommunity is increased at intermediate dispersal rates. Our results show that density-dependent dispersal and relative dispersal of species are keys to understanding the response of ecosystems to fragmentation.     

Single species models with logistic growth and dissymmetric impulse dispersal

In this paper, two classes of single-species models with logistic growth and impulse dispersal (or migration) are studied: one model class describes dissymmetric impulsive bi-directional dispersal between two heterogeneous patches; and the other presents a new way of characterizing the aggregate migration of a natural population between two heterogeneous habitat patches, which alternates in direction periodically. In this theoretical study, some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness and global stability of positive periodic solutions are established by using analysis based on the theory of discrete dynamical systems. From this study, we observe that the dynamical behavior of populations with impulsive dispersal differs greatly from the behavior of models with continuous dispersal. Unlike models where the dispersal is continuous in time, in which the travel losses associated with dispersal make it difficult for such dispersal to evolve e.g., [25], [26], [28], in the present study it was relatively easy for impulsive dispersal to positively affect populations when realistic parameter values were used, and a rich variety of behaviors were possible. From our results, we found impulsive dispersal seems to more nicely model natural dispersal behavior of populations and may be more relevant to the investigation of such behavior in real ecological systems.     

Quantile regression estimates of animal population trends

Ecologists often estimate population trends of animals in time series of counts using linear regression to estimate parameters in a linear transformation of multiplicative growth models, where logarithms of rates of change in counts in time intervals are used as response variables. We present quantile regression estimates for the median (0.50) and interquartile (0.25, 0.75) relationships as an alternative to mean regression estimates for common density-dependent and density-independent population growth models. We demonstrate that the quantile regression estimates are more robust to outliers and require fewer distributional assumptions than conventional mean regression estimates and can provide information on heterogeneous rates of change ignored by mean regression. We provide quantile regression trend estimates for 2 populations of greater sage-grouse (Centrocercus urophasianus) in Wyoming, USA, and for the Crawford population of Gunnison sage-grouse (Centrocercus minimus) in southwestern Colorado, USA. Our selected Gompertz models of density dependence for both populations of greater sage-grouse had smaller negative estimates of density-dependence terms and less variation in corresponding predicted growth rates (λ) for quantile than mean regression models. In contrast, our selected Gompertz models of density dependence with piecewise linear effects of years for the Crawford population of Gunnison sage-grouse had predicted changes in λ across years from quantile regressions that varied more than those from mean regression because of heterogeneity in estimated λs that were both less and greater than mean estimates. Our results add to literature establishing that quantile regression provides better behaved estimates than mean regression when there are outlying growth rates, including those induced by adjustments for zeros in the time series of counts. The 0.25 and 0.75 quantiles bracketing the median provide robust estimates of population changes (λ) for the central 50% of time series data and provide a 50% prediction interval for a single new prediction without making parametric distributional assumptions or assuming homogeneous λs. Compared to mean estimates, our quantile regression trend estimates for greater sage-grouse indicated less variation in density-dependent λs by minimizing sensitivity to outlying values, and for Gunnison sage-grouse indicated greater variation in density-dependent λs associated with heterogeneity among quantiles.     

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     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

Positive and negative density-dependence and boom-bust dynamics in enemy-victim populations: a mountain pine beetle case study

Negative density-dependent population regulation in exploitative species is well studied. Positive density-dependence can arise if exploiters must cooperate to obtain access to well-defended resources. Most studies, however, focus on the first type of density-dependence at the expense of the other. Using a parasitoid-host model, we explored how positive density-dependence driven by host defenses in combination with negative density-dependence due to competition for resources impact transient population dynamics. Inspired by interactions between the mountain pine beetle and its pine hosts, we formulated a model of enemy-victim interactions in discrete-time in which the victim is capable of deadly self-defense against exploitation. We fitted the model to data and then analyzed its non-equilibrium dynamics to determine what conditions promote boom-bust dynamics. When present together, strong Allee effects and overcompensating competition for resources among exploiters can cause their populations to irrupt and then crash even though many exploitable resources remain. Accelerating population irruptions followed by precipitous collapse occur for realistic parameter values of our model of mountain pine beetle dynamics. Insect dynamics are often dominated by sudden irruptions and collapses on short time scales. Population crashes in exploitative species often happen enigmatically even when exploitable resources are not depleted. Herein, we argue that strong Allee effects in combination with overcompensation provide a plausible explanation for these boom-bust dynamics in some species.