Equilibrium stability of single-species metapopulations

We investigate the effect of migration between local populations of a single discrete-generation species living in a ring or an array of habitats. The commonly used symmetric dispersal assumption is relaxed to include the biologically more reasonable asymmetric dispersion. It is demonstrated analytically that density independent migration has no effect on the equilibrium stability of individual populations. However, the positive equilibrium may be destabilizing if the migration is density dependent in such a way that it increases with increasing population density at the source patch.     

An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

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 in host-parasitoid assemblages

Most spatial population models assume constant rates of dispersal. However, in a given community, dispersal may not only depend on the density of conspecifics, i.e. density‐dependent dispersal, but also on the density of other species, a phenomenon we term ‘community‐dependent dispersal’. We co‐vary the densities of both the beetle host Callosobruchus chinensis and its parasitoid wasp, Anisopteromalus calandrae, in a laboratory study and record the proportions of each species that disperse within a two‐hour period. The parasitoid in these systems exhibits community‐dependent dispersal – dispersing more frequently when parasitoid density is high and larval host density is low. This supported our prediction that individuals should disperse according to competition for available resources. However, in this study the host's dispersal was independent of density. We suggest that this may be due to less intense selection acting on host dispersal strategies than on the parasitoid. We consider some possible consequences of community‐dependent dispersal for a number of spatial population processes. A well‐known host‐parasitoid metapopulation model is expanded so that it includes a greater range of dispersal functions. When the model is parameterised with the parasitoid community‐dependent dispersal function observed in the empirical study, similar population dynamics are obtained as when fixed‐rate dispersal functions are applied. The importance of dispersal functions for invasions of both competitive and host‐parasitoid systems is also considered. The model results demonstrate that understanding how individuals disperse in response to different species’ population densities is important in determining the rate of spread of an invasion. We suggest that more empirical studies are needed to establish what determines dispersal rate and distance in a range of species, combined with theoretical studies investigating the role of the dispersal function in determining spatial population processes.     

Natal habitat effects drive density—dependent scaling of dispersal decisions

The dispersal behavior of a species is critical for the stability and persistence of its populations across a landscape. How population density affects dispersal decisions is important for predicting these dynamics, as the form of density‐dependent dispersal influences the stability and persistence of populations. Natal habitat experience often has strong impacts on individual dispersal behavior as well, but its influence on density‐dependent dispersal behaviors remains unexplored. Here we address this conceptual gap in two experiments separately examining habitat selection and emigration from recently colonized patches for two species of flour beetle Tribolium sp. We found that interactions between the quality of habitat experienced during natal development and current habitat for dispersal capable adults can strongly affect the form of density dependence, including reversing the direction of nonlinearities (accelerating to decelerating), or even negating the influence of population density for individual dispersal decisions. Across heterogeneous landscapes, where individuals from different populations may experience different natal habitats, this altering of density‐dependent relationships is predicted by theory to fundamentally influence regional population dynamics. Our results indicate that species which occur across heterogeneous environments, such as during conservation reintroductions, or as invasive species spread, have much potential for natal experience to interact with density dependence and influence local and regional population dynamics.     

Density dependent dispersal decisions and the Allee effect

The decision to move between patches in the environment is among the most important life history choices an organism can make. I derive a new density dependent dispersal rule, and examine how dispersal decisions based on avoiding fitness loss associated with an Allee effect or competitive effects impact upon population dynamics in spatially structured populations with qualitatively different dynamics. I also investigate the effects of the number of patches in the system and a limit to the patch sampling time available to dispersers. Dispersing to avoid competitive pressures can destabilise otherwise stable population dynamics, and stabilise chaotic dynamics. Dispersing to avoid an Allee effect does not qualitatively change local population dynamics until eventually driving unstable populations to global extinction with a sufficiently high fitness threshold. A time limit for sampling can stabilise dynamics if dispersal is based on escaping the Allee effect, and rescue populations from global extinction. The results are sensitive to the number of patches available in the environment and suggest that dispersal to avoid an Allee effect will only arise under biologically plausible conditions, i.e. where there is a limit to the number of dispersal attempts that can be made between generations.     

Host-parasite population dynamics under combined frequency- and density-dependent transmission

Many host‐parasite models assume that transmission increases linearly with host population density (‘density‐dependent transmission’), but various alternative transmission functions have been proposed in an effort to capture the complexity of real biological systems. The most common alternative (usually applied to sexually transmitted parasites) assumes instead that the rate at which hosts contact one another is independent of population density, leading to ‘frequency‐dependent’ transmission. This straight‐forward distinction generates fundamentally different dynamics (e.g. deterministic, parasite‐driven extinction with frequency‐ but not density‐dependence). Here, we consider the situation where transmission occurs through two different types of contact, one of which is density‐dependent (e.g. social contacts), the other density‐independent (e.g. sexual contacts). Drawing on a range of biological examples, we propose that this type of contact structure may be widespread in natural populations. When our model is characterized mainly by density‐dependent transmission, we find that allowing even small amounts of transmission to occur through density‐independent contacts leads to the possibility of deterministic, parasite‐driven extinction (and lowers the threshold for parasite persistence). Contrastingly, allowing some density‐dependent transmission to occur in a model characterized mainly by density‐independent contacts (i.e. by frequency‐dependent transmission) does not affect the extinction threshold, but does increase the likelihood of parasite persistence. The idea that directly transmitted parasites exploit different types of host contact is not new, but here we show that the impact on dynamics can be fundamental even in the simplest cases. For example, in systems where density‐dependent transmission is normally assumed de facto, we show that parasite‐driven extinction can occur if a small amount of transmission occurs through density‐independent contacts. Many empirical studies are still guided by the traditional density/frequency dichotomy, but our combined transmission function may provide a better model for systems in which both types of transmission occur.     

Community-driven dispersal in an individual-based predator–prey model

We present a spatial, individual-based predator–prey model in which dispersal is dependent on the local community. We determine species suitability to the biotic conditions of their local environment through a time and space varying fitness measure. Dispersal of individuals to nearby communities occurs whenever their fitness falls below a predefined tolerance threshold. The spatiotemporal dynamics of the model is described in terms of this threshold. We compare this dynamics with the one obtained through density-independent dispersal and find marked differences. In the community-driven scenario, the spatial correlations in the population density do not vary in a linear fashion as we increase the tolerance threshold. Instead we find the system to cross different dynamical regimes as the threshold is raised. Spatial patterns evolve from disordered, to scale-free complex patterns, to finally becoming well-organized domains. This model therefore predicts that natural populations, the dispersal strategies of which are likely to be influenced by their local environment, might be subject to complex spatiotemporal dynamics.     

Negative density‐dependent dispersal in tsetse (Glossina spp): red flag or red herring?

A deterministic model of the distribution of tsetse flies (Glossina spp) was used to assess the extent to which the efficacy of control operations would be affected by three different modes of density dependence in per capita adult dispersal: (i) density‐independent dispersal which has been commonly adopted in previous models, (ii) positive density‐dependent dispersal which has occasionally been discussed in the tsetse literature, (iii) negative density‐dependent dispersal (NDDD). The last has recently been suggested, from genetic studies, to change the dispersal rate of tsetse by up to 200‐fold, thereby posing a severe risk for the success of tsetse control operations. Modelling outputs showed that NDDD poses no such risk, provided the mean daily dispersal of tsetse is below about 1 km, which is greater than any rate actually recorded in the field or indicated by the genetic studies. NDDD can be problematic only if tsetse disperse at rates that appear highly unlikely, or even impossible, on energetic grounds. Under some circumstances these high rates would help rather than hinder the control officer. NDDD is not necessary to explain the results of control operations, and not sufficient to explain the results of successful control programmes.     

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.     

On density and extinction in continuous population models

Survival analyses, investigations of extinction and persistence, are executed for populations represented by a nonautonomous differential equation model. The population is assumed governed by density dependent and time varying density independent demographic parameters. While traditional approaches to extinction postulate extinction on an infinite time horizon and at zero abundance level, survival analysis is developed not only for this traditional setting but also on a finite time horizon and at a nonzero threshold level. A main conclusion is that extinction of a temporally stressed population is determined by a totality of density independent and density dependent factors.     

Cooperation‐mediated plasticity in dispersal and colonization

Kin selection theory predicts that costly cooperative behaviors evolve most readily when directed toward kin. Dispersal plays a controversial role in the evolution of cooperation: dispersal decreases local population relatedness and thus opposes the evolution of cooperation, but limited dispersal increases kin competition and can negate the benefits of cooperation. Theoretical work has suggested that plasticity of dispersal, where individuals can adjust their dispersal decisions according to the social context, might help resolve this paradox and promote the evolution of cooperation. Here, we experimentally tested the hypothesis that conditional dispersal decisions are mediated by a cooperative strategy: we quantified the density‐dependent dispersal decisions and subsequent colonization efficiency from single cells or groups of cells among six genetic strains of the unicellular Tetrahymena thermophila that differ in their aggregation level (high, medium, and low), a behavior associated with cooperation strategy. We found that the plastic reaction norms of dispersal rate relative to density differed according to aggregation level: highly aggregative genotypes showed negative density‐dependent dispersal, whereas low‐aggregation genotypes showed maximum dispersal rates at intermediate density, and medium‐aggregation genotypes showed density‐independent dispersal with intermediate dispersal rate. Dispersers from highly aggregative genotypes had specialized long‐distance dispersal phenotypes, contrary to low‐aggregation genotypes; medium‐aggregation genotypes showing intermediate dispersal phenotype. Moreover, highly aggregation genotypes showed evidence for beneficial kin‐cooperation during dispersal. Our experimental results should help to resolve the evolutionary conflict between cooperation and dispersal: cooperative individuals are expected to avoid kin‐competition by dispersing long distances, but maintain the benefits of cooperation by dispersing in small groups.     

Density‐dependent dispersal and the formation of range borders.

Knowledge about the mechanisms of range formation is crucial for scientifically based species conservation strategies in the face of ongoing global climate change. In recent years an increasing amount of studies have focused on the influences of density‐dependent dispersal on demographic and biogeographical patterns. However, it still remains unclear, to what extent and in what ways this strategy would affect the range formation of species. In order to fill this gap, we present a study using individual‐based simulations of a species with discrete generations living along a dispersal mortality gradient. We compare the evolution of range sizes for species following density‐dependent and density‐independent emigration. Furthermore we assess the influence of environmental stochasticity and Allee effects on range formation, as both processes are known to play an important role for dispersal evolution. We find that density‐dependent dispersal always results in much wider ranges than unconditional dispersal. Increasing environmental stochasticity, a predicted consequence of climate change, can remarkably expand the ranges of species living in such connectivity gradients if dispersal decisions are based on local population density. A strong Allee effect causes range contraction for both strategies, but the effect is considerably less dramatic under density‐dependent compared to density‐independent emigration. We strongly recommend accounting for these findings in future attempts to model species’ range shifts due to climate change.     

Testing the interaction between environmental variation and dispersal strategy on population dynamics using a soil mite experimental system

Dispersal can play an important role in both the local and regional dynamics of populations. Empirical studies have shown that the proportion of individuals dispersing is often density dependent, which may have implications for the effect of dispersal on populations. In this study, we manipulate the dispersal strategy of adults within two-patch laboratory populations of soil mites and compare the consequences of fixed (density-independent) and density-dependent dispersal in environments of constant and temporally varying resource availability. Effects of dispersal on population dynamics were dependent on the presence of environmental variation. Both dispersal strategies tended to spatially homogenize the population abundance of adults in a variable environment. However, the effect of environmental variation on mean adult abundance was greater with density-dependent dispersal than with fixed dispersal. Adult dispersal did not affect juvenile or egg abundance. This study demonstrates the potential significance of density-dependent dispersal for population dynamics, but emphasizes the role of the environmental context.     

扩散对破碎化景观上宿主-寄生种群动态的影响

景观破碎化和扩散是空间种群模型的重要因素,对生物入侵存在着深远的影响。本章将基于偶对近似模型,探讨由局部和全局宿主-寄生相互作用共同决定的扩散模式对破坏性景观上疾病入侵与传播的影响。其中,生境破坏由生境丧失量与生境破碎化程度来描述。模拟结果显示,宿主和病毒的全局扩散对疾病的入侵与种群密度产生不对称效应：病毒的全局扩散对系统产生的影响较宿主的全局扩散更为显著。不同扩散模式下,生境丧失越高或破碎化程度越低,均将越有害于寄生病毒的入侵;同时,生境的破坏程度也显著地影响了入侵阈值对扩散模式的响应机制。本文研究结果暗示,景观破碎化的空间分布格局以及病毒扩散的限制均可作为物种保护与管理中有效的疾病控制策略。该研究结果在一定意义上丰富和发展了寄生感染理论,为物种保护提供了生态学理论依据。     

How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionarily stable dispersal strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches. In addition, we introduce a more easily calculated quantity Rm that relates to fitness in the same manner as R0 relates to fitness in ordinary population dynamics: the Rm of a mutant is only defined when the resident population dynamics converges to a point equilibrium and Rm is larger (smaller) than 1 if and only if mutant fitness is positive (negative). Rm corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. Efficient algorithms for calculating its numerical value are provided. As an example of the usefulness of these concepts we calculate the evolutionarily stable conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions. Below a threshold density x, at which staying and leaving are equality profitable, everybody should stay and above x everybody should leave, where profitability is measured as the mean number of dispersers produced through lines of descent consisting of non-dispersers.     

Local density and group size interacts with age and sex to determine direction and rate of social dispersal in a polygynous mammal

Movement away from an area or social group in response to increasing density (density‐dependent dispersal) is known for most species; why it evolves is fundamental to our understanding of ecology and evolution. However, we have yet to fully appreciate how individuals of varying conditions (e.g., age and sex) might differently consider effects of density (quorum) when deciding to disperse or not, and scale dependence in their sense of quorum. We tracked movements of all individuals of a naturalized population of feral horses (Equus ferus caballus; Sable Island National Park Reserve, Nova Scotia, Canada) during a period of rapid population growth (N increased from 375 to 484 horses from 2008 to 2010). Permanent dispersal from breeding groups (bands) was positively density dependent for all age and sex categories with respect to local density (horses/km², bounded by the 99th percentile of individual movements [8000 m]), but was negatively and positively density dependent for males and females, respectively, in relation to group (band) size. Dispersal was generally female biased, with the exception of foals which moved with their mothers (no sex effect), and for yearlings and subadults when band sizes were smaller than average, in which case males dispersed at higher rates than females. Dispersal distance was positively related to local density. We conclude that dispersal rate can be both positively and negatively density dependent for feral horses, contingent on the state of individuals and the scale at which quorum with respect to choosing to disperse or not is assessed. Scale effects and interactions of density‐dependent and sex‐ and age‐biased dispersal may have both ecological and evolutionary consequences through effects on resource and mate competition.     

Evolution of Complex Density-Dependent Dispersal Strategies

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.     

On fitness in structured metapopulations

In this paper we derive a general expression measuring fitness in general structured metapopulation models. We apply the theory to a model structured by local population size and in which local dynamics is explicitly modelled. In particular, we calculate the evolutionarily stable dispersal strategy for individuals that can assess the local population density in the case where only dispersal is subject to evolutionary control but all other model ingredients are assumed fixed. We show that there exists a threshold size such that at ESS everyone should stay as long as the population size is below the threshold and everyone should disperse immediately as the population size reaches the threshold. Received: 13 August 1999 / Revised version: 2 May 2001 / Published online: 12 October 2001     

Fitness-dependent dispersal in metapopulations and its consequences for persistence and synchrony

1. We present a novel metapopulation model where dispersal is fitness dependent: the strength of migration from a site is dependent on the expected reproductive fitness of individuals there. Furthermore, individuals continue to migrate until they reach a suitable habitat where their expected fitness is above a threshold value.
2. Fitness-dependent dispersal has a very strong stabilizing effect on population dynamics, even when the intrinsic dynamics of populations in the absence of dispersal exhibit complex high-amplitude oscillations. This stabilizing effect is much stronger than that of the density-independent dispersal normally considered in metapopulation models.
3. Even when fitness-dependent dispersal does not stabilize the dynamics in a formal sense, it generally leads to simplification, with complex or even chaotic fluctuations being reduced to simple cycles.
4. This form of dispersal also has a strong tendency to synchronize local population dynamics across the spatial extent of the metapopulation.
5. These conclusions are robust to the addition of strong stochasticity in the migration threshold.