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

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

Impact of Travel Between Patches for Spatial Spread of Disease

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, ℜ₀. Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of ℜ₀ on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches.     

Modeling the Geographic Spread of Rabies in China

In order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and vaccinated subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium is discussed, the basic reproduction number is calculated, and the effect of moving rates of dogs between patches on the basic reproduction number is studied. To investigate the rabies virus clades lineages, the two-patch submodel is used to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. It is found that the basic reproduction number of the two-patch model could be larger than one even if the isolated basic reproduction number of each patch is less than one. This indicates that the immigration of dogs may make the disease endemic even if the disease dies out in each isolated patch when there is no immigration. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog markets and trades needs to be regulated, and transportation of dogs has to be better monitored and under constant surveillance.     

Life on the edge: reproductive mode and rate of invasive <Emphasis Type="Italic">Phragmites australis</Emphasis> patch expansion

The dynamics of plant invasions from initial colonization through patch expansion are driven in part by mode of reproduction, i.e., sexual (seed) and asexual (clonal fragments and expansion) means. Expansion of existing patches—both rate and mode of spread into a matrix of varying conditions—is important for predicting potential invader impacts. In this study, we used fine-scale genetic assessments and remote sensing to describe both the rate and mode of expansion for 20 Phragmites australis patches in flooded and unflooded wetland units on the Great Salt Lake, UT. We found that the majority of Phragmites patch expansion occurred via clonal spread but we also documented instances of (potentially episodic) seedling recruitment. The mode of patch expansion, inferred from patch edge genet richness, was unrelated to flooding in the wetland unit in the preceding growing season. The rate of Phragmites patch expansion varied from 0.09 to 0.35 year^?1 and was unrelated to the mode of spread. In six patches monitored across two years, monoclonal patches stayed monoclonal, whereas patches with higher genet richness had a marked increase in diversity in the second year. The findings of the present study suggest how this partially clonal species can exploit the benefits of both sexual (i.e., genetic recombination, widespread dispersal, colonization of new areas) and asexual reproduction (i.e., stability of established clones suited to local environmental conditions) to become one of the most successful wetland plant invaders. To control this species, both forms of reproduction need to be fully addressed through targeted management actions.     

Threshold dynamics in a time-delayed epidemic model with dispersal

The global dynamics of a time-delayed model with population dispersal between two patches is investigated. For a general class of birth functions, persistence theory is applied to prove that a disease is persistent when the basic reproduction number is greater than one. It is also shown that the disease will die out if the basic reproduction number is less than one, provided that the initial size of the infected population is relatively small. Numerical simulations are presented using some typical birth functions from biological literature to illustrate the main ideas and the relevance of dispersal.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

Body condition dependent dispersal in a heterogeneous environment

We find the evolutionarily stable dispersal behaviour of a population that inhabits a heterogeneous environment where patches differ in safety (the probability that a juvenile individual survives until reproduction) and productivity (the total competitive weight of offspring produced by the local individual), assuming that these characteristics do not change over time. The body condition of clonally produced offspring varies within and between families. Offspring compete for patches in a weighted lottery, and dispersal is driven by kin competition. Survival during dispersal may depend on body condition, and competitive ability increases with increasing body condition. The evolutionarily stable strategy predicts that families abandon patches which are too unsafe or do not produce enough successful dispersers. From families that invest in retaining their natal patches, individuals stay in the patch that are less suitable for dispersal whereas the better dispersers disperse. However, this clear within-family pattern is often not reflected in the population-wide body condition distribution of dispersers or non-dispersers. This may be an explanation why empirical data do not show any general relationship between body condition and dispersal. When all individuals are equally good dispersers, then there exist equivalence classes defined by the competitive weight that remains in a patch. An equivalence class consists of infinitely many dispersal strategies that are selectively neutral. This provides an explanation why very diverse patterns found in body condition dependent dispersal data can all be equally evolutionarily stable.     

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.     

Contributions of high- and low-quality patches to a metapopulation with stochastic disturbance

Studies of time-invariant matrix metapopulation models indicate that metapopulation growth rate is usually more sensitive to the vital rates of individuals in high-quality (i.e., good) patches than in low-quality (i.e., bad) patches. This suggests that, given a choice, management efforts should focus on good rather than bad patches. Here, we examine the sensitivity of metapopulation growth rate for a two-patch matrix metapopulation model with and without stochastic disturbance and found cases where managers can more efficiently increase metapopulation growth rate by focusing efforts on the bad patch. In our model, net reproductive rate differs between the two patches so that in the absence of dispersal, one patch is high quality and the other low quality. Disturbance, when present, reduces net reproductive rate with equal frequency and intensity in both patches. The stochastic disturbance model gives qualitatively similar results to the deterministic model. In most cases, metapopulation growth rate was elastic to changes in net reproductive rate of individuals in the good patch than the bad patch. However, when the majority of individuals are located in the bad patch, metapopulation growth rate can be most elastic to net reproductive rate in the bad patch. We expand the model to include two stages and parameterize the patches using data for the softshell clam, Mya arenaria. With a two-stage demographic model, the elasticities of metapopulation growth rate to parameters in the bad patch increase, while elasticities to the same parameters in the good patch decrease. Metapopulation growth rate is most elastic to adult survival in the population of the good patch for all scenarios we examine. If the majority of the metapopulation is located in the bad patch, the elasticity to parameters of that population increase but do not surpass elasticity to parameters in the good patch. This model can be expanded to include additional patches, multiple stages, stochastic dispersal, and complex demography.     

Impact of dispersal on population growth: the role of inter-patch distance

Dispersal can play a key role in the dynamics of patchy populations through patch colonization, and generally this leads to distance-dependent colonization. Less recognised are the roles of dispersal and inter-patch distance on the growth of a population after colonization. We use a laboratory mite model system in which both juveniles and adults can disperse to explore the impact of dispersal, and particularly inter-patch distance, on population dynamics. We examine the dynamics of patches after colonization by manipulating the presence of a dispersal corridor to a source patch at two inter-patch distances. Consistent with many field studies, the results show colonization was slower in more distant patches. Following colonization, the effect of the dispersal corridor on dynamics was dependent on inter-patch distance. In patches near the source, the number of adults tended to increase at a faster rate, and juveniles at a slower rate when connected with a dispersal corridor. In contrast, adult numbers grew slower and juveniles tended to grow faster when connected with a corridor in more distant patches. In the long-term, equilibrium adult numbers were lower in patches connected to the source patch at both distances. These results are likely to be driven by the effects of inter-patch distance on dispersal mortality, and the effects of dispersal on patch abundance and within-patch competition. These results confirm that distance is important for patch colonization and also show that distance can affect population density after colonization. The effects of dispersal and distance on local dynamics could be important in the dynamics of patchy populations in increasingly fragmented landscapes.     

Habitat destruction in mutualistic metacommunities

We investigate a mutualistic metacommunity where the strength of the mutualistic interaction between species is measured by the extent to which the presence of one species on a patch either reduces the extinction rate of the others present on the same patch or increases their ability to colonize other patches. In both cases, a strong enough mutualism enables all species to persist at habitat densities where they would all be extinct in the absence of the interaction. However, a mutualistic interaction that enhances colonization enables the species to persist at lower habitat density than one that suppresses extinction. All species abruptly go extinct (catastrophe) when the habitat density is decreased infinitesimally below a critical value. A comparison of the mean field or spatially implicit case with unrestricted dispersal and colonization to all patches in the system with a spatially explicit case where dispersal is restricted to the immediate neighbours of the original patch leads to the intriguing conclusion that restricted dispersal can be favourable for species that have a beneficial effect on each other when habitat conditions are adverse. When the mutualistic interaction is strong enough, the extinction threshold or critical amount of habitat required for the persistence of all species is lower when the dispersal is locally restricted than when unrestricted ! The persistence advantage for all species created by the mutualistic interaction increases substantially with the number of species in the metacommunity, as does the advantage for restricted dispersal over global dispersal.     

Effects of asymmetric dispersal and environmental gradients on the stability of host–parasitoid systems

We consider host–parasitoid systems spatially distributed on a row of patches connected by dispersal. We analyze the effects of dispersal frequency, dispersal asymmetry, number of patches and environmental gradients on the stability of the host–parasitoid interactions. To take into account dispersal frequency, the hosts and parasitoids are allowed to move from one patch to a neighboring patch a certain number of times within a generation. When this number is high, aggregation methods can be used to simplify the proposed initial model into an aggregated model describing the dynamics of both the total host and parasitoid populations. We show that as the number of patches increases less asymmetric parasitoid dispersal rates are required for stability. We found that the 'CV²>1 rule' is a valid approximation for stability if host growth rate is low, otherwise the general condition of stability we establish should be preferred. Environmental variability along the row of patches is introduced as gradients on host growth rate and parasitoid searching efficiency. We show that stability is more likely when parasitoids move preferentially towards patches where they have high searching efficiency or when hosts go mainly to patches where they have a low growth rate.     

Does colonization asymmetry matter in metapopulations?

Despite the considerable evidence showing that dispersal between habitat patches is often asymmetric, most of the metapopulation models assume symmetric dispersal. In this paper, we develop a Monte Carlo simulation model to quantify the effect of asymmetric dispersal on metapopulation persistence. Our results suggest that metapopulation extinctions are more likely when dispersal is asymmetric. Metapopulation viability in systems with symmetric dispersal mirrors results from a mean field approximation, where the system persists if the expected per patch colonization probability exceeds the expected per patch local extinction rate. For asymmetric cases, the mean field approximation underestimates the number of patches necessary for maintaining population persistence. If we use a model assuming symmetric dispersal when dispersal is actually asymmetric, the estimation of metapopulation persistence is wrong in more than 50% of the cases. Metapopulation viability depends on patch connectivity in symmetric systems, whereas in the asymmetric case the number of patches is more important. These results have important implications for managing spatially structured populations, when asymmetric dispersal may occur. Future metapopulation models should account for asymmetric dispersal, while empirical work is needed to quantify the patterns and the consequences of asymmetric dispersal in natural metapopulations.     

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator–prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator–prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.     

A graph-based approach to investigating the influence of the landscape on population spread processes

Modelling of landscape connectivity is a key point in the study of the movement of populations within a given landscape. For studies focused on the preservation of biodiversity, graph-based methods provide an interesting framework to investigate the landscape influence on population spread processes. Such an approach is described here, based on the mapping of landscape categories in habitat patches, including a diachronic data set describing the population spread within the habitat patches. A minimum planar graph was built by computing spatial distances between all pairs of neighbouring patches. From this structure, two types of analysis are proposed: one focused on the links of the graph and consists in correlating spatial distances and gap indicators computed from the diachronic data. The other was based on the correlations between population data and connectivity metrics at the patch level. As an example, this approach was applied to the spread of the fossorial water vole on the Jura plateau (France), with annual population data covering eleven years from 1989 to 2000. Link analysis allowed to find an optimal set of resistance values used in the least-cost distances computations, and thus to build a relevant graph. From this graph, patch analysis displayed a cyclic correlation between a metric based on potential dispersal flux and the population density, outlining the strong role of landscape connectivity in the population spread. The present study clearly shows that landscape modelling and graph-based approach can produce parameters which are consistent with field observations and thus pave the way to simulating the effect of landscape modification on population dynamics.     

Permanence of delayed population model with dispersal loss

Permanence of a dispersal single-species population model where environment is partitioned into several patches is considered. The species not only requires some time to disperse between the patches but also has some possibility to die during its dispersion. The model is described by delay differential equations. The existence of 'super' food-rich patch is proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence implies permanence if each food-poor patch is chained to the super food-rich patch. Furthermore, it is proven that partial persistence is ensured if there exist food-rich patches and the dispersion of the species among the patches are small. When the dispersion is large, the partial persistence is realized under relatively small dispersion time.     

How do birds search for breeding areas at the landscape level? Interpatch movements of male ortolan buntings

Animal movements at large spatial scales are of great importance in population ecology, yet little is known due to practical problems following individuals across landscapes. We studied the whole Norwegian population of a small songbird (ortolan bunting, Emberiza hortulana ) occupying habitat patches dispersed over nearly 500 km². Movements of colour-ringed males were monitored during ten years, and extensive long-distance dispersal was recorded. More than half of all cases of breeding dispersal took place within one breeding season, and males moved up to 43 km between singing territories, using 1–22 d. Natal dispersal was usually to a habitat patch close to the natal patch, or within the natal patch if it was large. Breeding dispersal movements were often long-distance, beyond neighbouring patches, and up to 11–19 patches were overflown. Movements of at least 6–9 km across areas of unsuitable habitat occurred regularly. The number of patches visited was low (1–4) even though search costs in terms of time spent moving from one site to another were relatively low (often only a few days even for distances >10 km). Most males seemed to use a threshold tactic when choosing a patch, but returns to previously visited patches were recorded, including some cases of commuting. In conclusion, male ortolan buntings have a surprising ability to move quickly at the landscape level, and this resulted in a high connectivity of patches. We discuss our results in relation to optimal searching strategies, in particular the use of within-breeding season versus post-breeding season search, conspecific attraction and adaptive late arrival of young birds.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

The effect of travel loss on evolutionarily stable distributions of populations in space

A key assumption of the ideal free distribution (IFD) is that there are no costs in moving between habitat patches. However, because many populations exhibit more or less continuous population movement between patches and traveling cost is a frequent factor, it is important to determine the effects of costs on expected population movement patterns and spatial distributions. We consider a food chain (tritrophic or bitrophic) in which one species moves between patches, with energy cost or mortality risk in movement. In the two-patch case, assuming forced movement in one direction, an evolutionarily stable strategy requires bidirectional movement, even if costs during movement are high. In the N-patch case, assuming that at least one patch is linked bidirectionally to all other patches, optimal movement rates can lead to source-sink dynamics where patches with negative growth rates are maintained by other patches with positive growth rates. As well, dispersal between patches is not balanced (even in the two-patch case), leading to a deviation from the IFD. Our results indicate that cost-associated forced movement can have important consequences for spatial metapopulation dynamics. Relevance to marine reserve design and the study of stream communities subject to drift is discussed.     

Evolutionary suicide and evolution of dispersal in structured metapopulations

 We study the evolution of dispersal in a structured metapopulation model. The metapopulation consists of a large (infinite) number of local populations living in patches of habitable environment. Dispersal between patches is modelled by a disperser pool and individuals in transit between patches are exposed to a risk of mortality. Occasionally, local catastrophes eradicate a local population: all individuals in the affected patch die, yet the patch remains habitable. We prove that, in the absence of catastrophes, the strategy not to migrate is evolutionarily stable. Under a given set of environmental conditions, a metapopulation may be viable and yet selection may favor dispersal rates that drive the metapopulation to extinction. This phenomenon is known as evolutionary suicide. We show that in our model evolutionary suicide can occur for catastrophe rates that increase with decreasing local population size. Evolutionary suicide can also happen for constant catastrophe rates, if local growth within patches shows an Allee effect. We study the evolutionary bifurcation towards evolutionary suicide and show that a discontinuous transition to extinction is a necessary condition for evolutionary suicide to occur. In other words, if population size smoothly approaches zero at a boundary of viability in parameter space, this boundary is evolutionarily repelling and no suicide can occur. Received: 10 November 2000 / Revised version: 13 February 2002 / Published online: 17 July 2002