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.     

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.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Mathematical analysis of an age-structured population model with space-limited recruitment

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.     

Breeding migration and population stability

We have modeled habitat shift for reproduction to examine the relationship between the timing of migration and population stability, by modifying Takimoto’s (Am Nat 162:93–109, 2003) consumer–resource model with a consumer’s ontogenetic niche shift. We found that equilibrium was always locally unstable if migration occurs at a fixed time or level of energy storage, whereas it could be stable if the timing of migration was adaptively flexible to maximize reproductive output. The general conditions for stability were safer breeding rather than feeding habitat and abundant resources at the feeding habitat. These results imply that both adopting an adaptive plastic strategy in the timing of migration and choosing to migrate from a rich feeding habitat to a safe breeding habitat can contribute to population stability. We also found that reduced reproductive success with delays in migration, and the survival rate after reproduction, had complicated effects on stability, depending on resource availability at the feeding habitat. The equilibrium was more likely to be stable when reproduction success was only slightly (or greatly) reduced or survival rate was high (or low) if the feeding habitat was rich (or poor). These are significant predictions for ecological study of migrating animals.     

Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution

The effect of dispersal on population size and stability is explored for a population that disperses passively between two discrete habitat patches. Two basic models are considered. In the first model, a single population experiences density-dependent growth in both patches. A graphical construction is presented which allows one to determine the spatial pattern of abundance at equilibrium for most reasonable growth models and rates of dispersal. It is shown under rather general conditions that this equilibrium is unique and globally stable. In the second model, the dispersing population is a food-limited predator that occurs in both a source habitat (which contains a prey population) and a sink habitat (which does not). Passive dispersal between source and sink habitats can stabilize an otherwise unstable predator-prey interaction. The conditions allowing this are explored in some detail. The theory of optimal habitat selection predicts the evolutionarily stable distribution of a population, given that individuals can freely move among habitats so as to maximize individual fitness. This theory is used to develop a heuristic argument for why passive dispersal should always be selectively disadvantageous (ignoring kin effects) in a spatially heterogeneous but temporally constant environment. For both the models considered here, passive dispersal may lead to a greater number of individuals in both habitats combined than if there were no dispersal. This implies that the evolution of an optimal habitat distribution may lead to a reduction in population size; in the case of the predator-prey model, it may have the additional effect of destabilizing the interaction. The paper concludes with a discussion of the disparate effects habitat selection might have on the geographical range occupied by a species.     

Population Dynamics and Spatial Distribution of Coral Reef Fishes: Comparison Between Continuous and Isolated Habitats

Recent studies have shown that there are high degrees of spatial and temporal stability in coral reef fish assemblage structures in a continuous habitat, in contrast to results of observations in isolated habitats. In order to determine the reason for the difference in temporal stability of fish assemblage structures in a continuous habitat site and an isolated habitat site, population dynamics and spatial distributions of coral reef fishes (six species of pomacentrids and two species of apogonids) in the two habitat site were investigated over a 2-year period in an Okinawan coral reef. The population densities of pomacentrid and apogonid species increased in juvenile settlement periods at both sites, but the magnitude of seasonal fluctuation in population density was significantly greater at the isolated habitat site, indicating that the rate of juvenile settlement and mortality rate in the isolated habitat were greater than those in the continuous habitat. The magnitude of aggregation of fishes, which affects density-dependent biological interactions that modify population density such as competition and predation, was also significantly greater at the isolated habitat site, especially in the juvenile settlement season. Most of the fishes at the isolated habitat site exhibited more generalized patterns of microhabitat selection because of less coral coverage and diversity. The seasonal stability in the species composition of fishes was greater at the continuous habitat site than that at the isolated habitat. Our findings suggest that the relative importance of various ecological factors responsible for regulation of the population density of coral reef fishes (e.g., competition, predation, microhabitat selection and post-settlement movement) in a continuous habitat site and the isolated habitat site are different.     

Bird behaviour and environmental planning: approaches in the study of wader populations

Predicting the effect of changes in land-use on bird populations requires a degree of understanding of their population dynamics which is seldom available. Such knowledge is especially difficult to acquire if the birds occupy a variety of habitats of differing quality and they are also migratory. The effects of habitat loss at a particular time of year on the year-round population dynamics are then difficult to predict. The problem is discussed using wintering waders as examples. Unless a species can extend its present range, the initial effect of habitat loss is to increase bird density. Whether this affects population size will depend on whether bird density affects either winter survival or subsequent breeding success. Measuring such density-dependent relationships is in practice extremely difficult in migratory populations. However, behavioural studies help in testing the key hypothesis that birth and death rates are affected by bird density, and may even allow the form of any density-dependent functions to be deduced. Simulation modelling then allows the effects of habitat loss on the overall population dynamics to be explored. The general point is that behavioural studies play an important part in Environmental Impact Studies because behaviour is the main way in which birds respond to environmental changes and compete for limited resources, such as diminishing habitat. They are therefore likely to provide insights when making predictions about the responses of birds and populations to habitat loss.     

Density-dependent migration and synchronism in metapopulations

A spatially explicit metapopulation model with density-dependent dispersal is proposed in order to study the stability of synchronous dynamics. A stability criterion is obtained based on the computation of the transversal Liapunov number of attractors on the synchronous invariant manifold. We examine in detail a special case of density-dependent dispersal rule where migration does not occur if the patch density is below a certain critical density, while the fraction of individuals that migrate to other patches is kept constant if the patch density is above the threshold level. Comparisons with density-independent migration models indicate that this simple density-dependent dispersal mechanism reduces the stability of synchronous dynamics. We were able to quantify exactly this loss of stability through the frequency that synchronous trajectories are above the critical density.     

Competition for space in a heterogeneous environment

We consider effects of competition for space in a heterogeneous environment, making use of nonlinear interaction-diffusion equations. Competition for space is assumed to mean mutual repulsive interactions that force other individuals to disperse from a crowded region. In other words, we are concerned with density-dependent dispersal forced by population pressures. Spatial heterogeneity is incorporated in the growth rates, and the environment is assumed to have a favorable habitat for two populations surrounded by largely hostile regions. Space-independent migration rates are assumed. We ignore the usual density-dependence in the growth rates to focus our attention on density-dependence in the migration rates. Our main conclusion is that two populations can coexist if the interspecific repulsive forces are weaker than the intraspecific ones. It is also emphasized that density-dependent dispersal in a heterogeneous environment is not always a stabilizing agent, and that either of two populations may become extinct by competition for space. Finally, the resemblance of our results to those from Lotka-Volterra competition equations is suggested.     

Competition and coexistence in flowing habitats with a hydraulic storage zone

This paper examines a model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. In this habitat, one or two microbial populations grow while consuming a single nutrient resource. Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, the theory of bifurcation and the theory of monotone dynamical systems. A single population persists if it can invade the trivial steady state of an empty habitat. Under some conditions, persistence occurs in the presence of a hydraulic storage zone when it would not in an otherwise equivalent flowing habitat without such a zone. Coexistence of two competing species occurs if each can invade the semi-trivial steady state established by the other species. Numerical work shows that both coexistence and enhanced persistence due to a storage zone occur for biologically reasonable parameters.     

Dispersal among habitats varying in fitness: reciprocating migration through ideal habitat selection

Current evolutionary models of dispersal set the ends of a continuum where the number of individuals emigrating from a habitat either equals the number of individuals immigrating (balanced dispersal) or where emigrants flow from a source habitat to a corresponding sink. Theories of habitat selection suggest a more sophisticated conditional strategy where individuals disperse from habitats where they have the greatest impact on fitness to habitats where their per capita impact is lower. Asymmetries between periods of population growth and decline result in a reciprocating dispersal strategy where the direction of migration is reversed as populations wax and wane. Thus, for example, if net migration of individuals flows from high- to low-density habitats during periods of population growth, net migration will flow in the opposite direction during population decline. Stochastic simulations and analytical models of reciprocating dispersal demonstrate that fitness, carrying capacity, stochastic dynamics, and interference from dominants interact to determine whether dispersal is balanced between habitats, or whether one habitat or the other acts as a net donor of dispersing individuals. While the pattern of dispersal may vary, each is consistent with an underlying strategy of density-dependent habitat selection.     

The Effect of Multiple Paternity on Genetic Diversity of Small Populations during and after Colonisation

Genetic variation within and among populations is influenced by the genetic content of the founders and the migrants following establishment. This is particularly true if populations are small, migration rate low and habitats arranged in a stepping-stone fashion. Under these circumstances the level of multiple paternity is critical since multiply mated females bring more genetic variation into founder groups than single mated females. One such example is the marine snail Littorina saxatilis that during postglacial times has invaded mainland refuge areas and thereafter small islands emerging due to isostatic uplift by occasional rafting of multiply mated females. We modelled effects of varying degrees of multiple paternity on the genetic variation of island populations colonised by the founders spreading from the mainland, by quantifying the population heterozygosity during both the transient colonisation process, and after a steady state (with migration) has been reached. During colonisation, multiple mating by males increased the heterozygosity by in comparison with single paternity, while in the steady state the increase was compared with single paternity. In the steady state the increase of heterozygosity due to multiple paternity is determined by a corresponding increase in effective population size. During colonisation, by contrast, the increase in heterozygosity is larger and it cannot be explained in terms of the effective population size alone. During the steady-state phase bursts of high genetic variation spread through the system, and far from the mainland this led to short periods of high diversity separated by long periods of low diversity. The size of these fluctuations was boosted by multiple paternity. We conclude that following glacial periods of extirpation, recolonization of isolated habitats by this species has been supported by its high level of multiple paternity.     

Age- and density-dependent population dynamics in static and variable environments

We consider a general model of a single-species population with age- and density-dependent per capita birth and death rates. In a static environment we show that if the per capita death rate is independent of age, then the local stability of any stationary state is guaranteed by the requirement that, in the region of the steady state, the density dependence of the birth rate should be negative and that of the death rate positive. In a variable environment we show that, provided the system is locally stable, small environmental fluctuations will give rise to small age structure and population fluctuations which are related to the driving environmental fluctuations by a simple “transfer function.” We illustrate our general theory by examining a model with a per capita death rate which is age and density independent and a per capita birth rate which is zero up to some threshold age a₀, adopts a finite density-dependent value up to a maximum age a_o + α, and is zero thereafter. We conclude from this model that resonance due specifically to single-species age-structure effects will only be of practical importance in populations whose members have a life cycle consisting of a long immature phase followed by a short burst of intense reproductive effort (α a_o).     

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.     

Density-dependent migration and human population structure in historical Massachusetts

Studies of population structure often focus on the effects of population size and migration rates on genetic variation. Few studies, however, have investigated the relationship between these two factors. The purpose of this paper is to determine the extent to which migration (and gene flow) is density-dependent (that is, affected by population size) for populations in historical Massachusetts. Data from 4,859 marriage records were analyzed from four populations in north-central Massachusetts during the time period 1741 to 1849. These data were placed into 29 samples defined in terms of population and time cohort. Within each cohort the overall exogamy rate was computed along with three estimates of gene flow based on marital migration: local migration (k), long-distance migration (m), and effective migration rate (me). Three samples show unusually low rates that reflect the history of settlement. Regression analyses were used with the remaining samples, and they show nonlinear density-dependent migration that is unrelated to temporal trends. Migration is highest in samples with small population sizes (less than 800) and large population sizes (greater than 1,600). Migration is lowest in medium-sized populations. Two processes are suggested to explain this curvilinear relationship of migration and population size. In small populations, the lack of suitable potential mates and/or availability of settled land leads to an increase in migration into the population. As population size increases, this migration decreases. After populations reach a certain size, migration increases again, most likely reflecting the economic pull of larger populations. These patterns could act to enhance, or counter, genetic drift, depending on the direction of density dependence.     

Cross-cultural analysis of migration rates: effects of geographic distance and population size.

A model is developed that treats migration rates among populations as a function of the geographic distance between them and the size of both sources and recipient population. Specifically, mij/mjj = a(Ni/Nj)pe-bd, where mij/mjj is the relative migration rate into population j from population i, Ni is the size of the source population, Nj is the size of the recipient population, d is the geographic distance between populations i and j, p is a measure of differential density-dependence, b is a measure of distance decay, and a is an adjustment parameter with little demographic meaning. Methods of parameter estimation and hypothesis testing using maximum likelihood are outlined. These methods are applied to migration matrix data from 13 samples obtained from the literature representing a wide range of ecological settings. All samples show a significant effect of geographic distance on migration, and all but one show a significant effect of differential population size. All but one sample show an overall tendency for migration to be negative density-dependent; that is, the relative migration rate is greater from larger populations to smaller populations than the reverse.     

Asymmetry in the Presence of Migration Stabilizes Multistrain Disease Outbreaks

We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model.     

Density-dependent processes in the life history of fishes: evidence from laboratory populations of zebrafish Danio rerio

Population regulation is fundamental to the long-term persistence of populations and their responses to harvesting, habitat modification, and exposure to toxic chemicals. In fish and other organisms with complex life histories, regulation may involve density dependence in different life-stages and vital rates. We studied density dependence in body growth and mortality through the life-cycle of laboratory populations of zebrafish Danio rerio. When feed input was held constant at population-level (leading to resource limitation), body growth was strongly density-dependent in the late juvenile and adult phases of the life-cycle. Density dependence in mortality was strong during the early juvenile phase but declined thereafter and virtually ceased prior to maturation. Provision of feed in proportion to individual requirements (easing resource limitation) removed density dependence in growth and substantially reduced density dependence in mortality, thus indicating that 'bottom-up' effects act on growth as well as mortality, but most strongly on growth. Both growth and mortality played an important role in population regulation, with density-dependent growth having the greater impact on population biomass while mortality had the greatest impact on numbers. We demonstrate a clear ontogenic pattern of change in density-dependent processes within populations of a very small (maximum length 5 mm) fish, maintained in constant homogeneous laboratory conditions. The patterns are consistent with those distilled from studies on wild fish populations, indicating the presence of broad ontogenic patterns in density-dependent processes that are invariant to maximum body size and hold in homogeneous laboratory, as well as complex natural environments.     

Temporal quota corrections based on timing of harvest in a small game species

Theoretical models have shown that the effect of removing a given proportion of the population can be profoundly different if the harvest takes place late in the season compared to early. We explore the effect of these differences using theoretical models based on the concept of demographic value and empirical data on seasonal patterns of natural mortality risk in two contrasting populations of willow ptarmigan in Norway. Based on the theoretical models, we found that changes in the timing of harvest have a much stronger effect in populations with relatively low annual survival compared to populations characterized by longevity typical for species with slow life histories. Also, the timing of harvest is more influential in cases with constant mortality hazards compared to a situation with density-dependent natural mortality. Empirical data from two study populations of willow ptarmigan showed large deviations from the theoretical predictions of models with both constant and density-dependent mortality hazards. There were also large differences in both the temporal pattern and magnitude of annual survival between the two ptarmigan populations (54 vs 26% annual survival). Site differences illustrate the importance of knowledge of both the magnitude and temporal pattern of natural mortality hazard to be able to correctly predict the effect of changing the timing of harvest in a population. In the two ptarmigan populations, we show how harvest quotas can be adjusted in accordance to the empirical estimates of natural mortality risk and how this determines the effects of shifting from harvesting early to late in the annual cycle.