Multiplicative moments and measures of persistence in ecology

Ecologists and epidemiologists have begun focusing on demographic stochasticity and spatial heterogeneity as important biological factors. With high-powered computers simulation of such systems is a common modelling technique; however we lack a detailed understanding of the processes involved. Moment closure approximations provide a simple method which can be used to capture the main features of a wide variety of stochastic models and to gain a more intuitive understanding. In this paper we give an alternative variation based on multiplicative moments which is equivalent to taking a novel third-order cumulant approximation. The differential equations for these multiplicative moments are far more robust than their additive counterparts. We use this technique to consider the behaviour and persistence of finite metapopulations for two common ecological systems.     

Reconciling theoretical approaches to stochastic patch-occupancy metapopulation models

There appear to be two different kinds of theoretical results about stochastic patch-occupancy metapopulation models: those recently proposed by Gyllenberg and Silvestrov about metapopulations including a very stable patch, and those by Darroch and Seneta about more general metapopulations. Based on the spectral theory of linear operators, it is shown that the results by Gyllenberg and Silvestrov are a limiting case of those by Darroch and Seneta. Taking the examples proposed by Gyllenberg and Silvestrov as a case study, the application and relevance of these results are discussed, with a particular stress to their bearing on real metapopulations.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ?(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Probabilistic measures of persistence and extinction in measles (meta)populations

Persistence and extinction are fundamental processes in ecological systems that are difficult to accurately measure due to stochasticity and incomplete observation. Moreover, these processes operate on multiple scales, from individual populations to metapopulations. Here, we examine an extensive new data set of measles case reports and associated demographics in pre‐vaccine era US cities, alongside a classic England & Wales data set. We first infer the per‐population quasi‐continuous distribution of log incidence. We then use stochastic, spatially implicit metapopulation models to explore the frequency of rescue events and apparent extinctions. We show that, unlike critical community size, the inferred distributions account for observational processes, allowing direct comparisons between metapopulations. The inferred distributions scale with population size. We use these scalings to estimate extinction boundary probabilities. We compare these predictions with measurements in individual populations and random aggregates of populations, highlighting the importance of medium‐sized populations in metapopulation persistence.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

The deterministic limit of a stochastic logistic model with individual variation

We consider a Markov chain model similar to the stochastic logistic model except that it allows for variation amongst individuals in the population. We prove that as the population size grows, the process can be approximated by a deterministic process. The equilibrium points of the limiting process and their stability are determined. Applications to modelling epidemics and metapopulations are discussed.     

Regional structuring of genetic variation in short-lived rock pool populations of Branchipodopsis wolfi (Crustacea: Anostraca)

The genetic structure of three metapopulations of the southern African anostracan Branchipodopsis wolfi was compared by analysing allozyme variation at four loci (PGM, GPI, APK, AAT). In total, 17 local populations from three sites (metapopulations) were analysed from rock pools in south-eastern Botswana ranging from 0.2 to 21 m² in surface area. In three populations we found significant deviations from Hardy-Weinberg (H-W) equilibrium at one or more loci due to heterozygote deficiencies. Genetic variability at one site was significantly lower than at the other sites, which may be linked to a greater incidence of extinction and recolonisation, as the basins at this site are shallower and have shorter hydrocycles. Across all local populations, a significant level of population differentiation was revealed. More than 90% of this variation was explained by differentiation among sites (metapopulations), although this differentiation did not correlate with geographic distance, or with environmental variables. Genetic differentiation among populations within metapopulations was low, but significant at all sites. At only one of the sites was a significantly positive association measured between genetic and geographic distance among local populations. Our data suggest that persistent stochastic events and limited effective long-range dispersal appear to dominate genetic differentiation among populations of B. wolfi inhabiting desert rock pools. The lack of association between geographic distance and genetic or ecological differences between rock pool sites is indicative of historical stochastic events. Low heterozygosity, the significant deviations from H-W equilibrium, and the large inter- but low intra-site differentiation are suggestive of the importance of short-range dispersal. Gene flow between metapopulations of B. wolfi appears to be seriously constrained by distances of 2 km or even less. Received: 28 June 1999 / Accepted: 10 January 2000     

How patch configuration affects the impact of disturbances on metapopulation persistence

Disturbances affect metapopulations directly through reductions in population size and indirectly through habitat modification. We consider how metapopulation persistence is affected by different disturbance regimes and the way in which disturbances spread, when metapopulations are compact or elongated, using a stochastic spatially explicit model which includes metapopulation and habitat dynamics. We discover that the risk of population extinction is larger for spatially aggregated disturbances than for spatially random disturbances. By changing the spatial configuration of the patches in the system--leading to different proportions of edge and interior patches--we demonstrate that the probability of metapopulation extinction is smaller when the metapopulation is more compact. Both of these results become more pronounced when colonization connectivity decreases. Our results have important management implication as edge patches, which are invariably considered to be less important, may play an important role as disturbance refugia.     

Metapopulation vicariance explains old endemics on young volcanic islands

Terrestrial plants and animals on oceanic islands occupy zones of volcanism found at intraplate localities and along island arcs at subduction zones. The organisms often survive as metapopulations, or populations of separate sub‐populations connected by dispersal. Although the individual islands and their local subpopulations are ephemeral and unstable, the ecosystem dynamism enables metapopulations to persist in a region, more or less in situ, for periods of up to tens of millions of years. As well as surviving on systems of young volcanic islands, metapopulations can also evolve there; tectonic changes can break up widespread insular metapopulations and produce endemics restricted to fewer islands or even a single island. These processes explain the presence of old endemic clades on young islands, which is often reported in molecular clock studies, and the many distribution patterns in island life that are spatially correlated with tectonic features. Metapopulations can be ruptured by sea floor subsidence, and this occurs with volcanic loading in zones of active volcanism and with sea floor cooling following its production at mid‐ocean ridges. Metapopulation vicariance will also result if an active zone of volcanism is rifted apart. This can be caused by the migration of an arc (by slab rollback) away from a continent or from another subduction zone, by the offset of an arc at transform faults and by sea floor spreading at mid‐ocean ridges. These mechanisms are illustrated with examples from islands in the Caribbean and the Pacific. Endemism on oceanic islands has usually been attributed to chance, long‐distance dispersal, but the processes discussed here will generate endemism on young volcanic islands by vicariance.     

Non-equilibria in small metapopulations: comparing the deterministic Levins model with its stochastic counterpart

In this paper, we examine, for small metapopulations, the stochastic analog of the classical Levins metapopulation model. We study its basic model output, the expected time to metapopulation extinction, for systems which are brought out of equilibrium by imposing sudden changes in patch number and the colonization and extinction parameters. We find that the expected metapopulation extinction time shows different behavior from the relaxation time of the original, deterministic, Levins model. This relaxation time is therefore limited in value for predicting the behavior of the stochastic model. However, predictions about the extinction time for deterministically unviable cases remain qualitatively the same. Our results further suggest that, if we want to counteract the effects of habitat loss or increased dispersal resistance, the optimal conservation strategy is not to restore the original situation, that is, to create habitat or decrease resistance against dispersal. As long as the costs for different management options are not too dissimilar, it is better to improve the quality of the remaining habitat in order to decrease the local extinction rate.     

A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

Stability aware spatial cut of metapopulations ecological networks

Ecological complex networks are common in the study of patched ecological systems where evolving populations interact within and among the patches. The loss of the dispersal connections between patches due to reasons such as erosion of migration corridors and road construction can cause an undesirable partitioning of such networks resulting in instability or negative impact on the metapopulations. A partitioning or spatial cut that is aware of the stability of the dynamics in the resulting daughter sub-networks can be an effective tool in dealing with the situation like proposing road alignment through a metapopulations network. This paper provides some mathematical conditions along with an heuristic graph partitioning algorithm that can help in finding ecologically suitable partitions of the metapopulations networks. Our study noted the crucial role of network connectivity (measured by Fiedler value) in stabilizing the metapopulations. That is, a sufficiently connected metapopulations network along with constrained internal patch dynamics has stable dynamics around its homogeneous co-existential equilibrium solution. With the considered mathematical model in this paper, network partitioning does not alter the internal patch dynamics around its homogeneous equilibrium point, but it can change the connectivity levels in the partitioned subnetworks. Thus, the proposed partitioning problem for an already stable metapopulations network is reduced to finding its subnetworks with desirable connectivity levels.     

Population structure, habitat features and genetic structure of managed red deer populations

Management of game ungulates alters population structure and habitat features, with potential effects on genetic structure. Here, we study 26 red deer (Cervus elaphus) populations in Spain. We used census data and habitat features as well as genetic information at 11 microsatellite markers from 717 individuals. We found that metapopulations presented a distribution associated with forest interruptions. Within metapopulations, fences did not have a significant effect on red deer genetic structure. The metapopulations we studied presented similar population structure, but they differed in habitat features and genetic structure. The metapopulation with higher resource availability showed a genetic structure pattern in which genetic relatedness between geographically close individuals was high while relatedness between geographically distant individuals was low. Contrarily, the metapopulation with lower resource availability presented a genetic structure pattern in which the genetic relatedness between individuals of different populations was independent of the geographic distance. We discuss the possible connection between resource availability and genetic structure. Finally, we did not find any population or environmental variable related to genetic differentiation within metapopulations.     

How Levins’ dynamics emerges from a Ricker metapopulation model

Understanding the dynamics of metapopulations close to extinction is of vital importance for management. Levins-like models, in which local patches are treated as either occupied or empty, have been used extensively to explore the extinction dynamics of metapopulations, but they ignore the important role of local population dynamics. In this paper, we consider a stochastic metapopulation model where local populations follow a stochastic, density-dependent dynamics (the Ricker model), and use this framework to investigate the behaviour of the metapopulation on the brink of extinction. We determine under which circumstances the metapopulation follows a time evolution consistent with Levins’ dynamics. We derive analytical expressions for the colonisation and extinction rates (c and e) in Levins-type models in terms of reproduction, survival and dispersal parameters of the local populations, providing an avenue to parameterising Levins-like models from the type of information on local demography that is available for a number of species. To facilitate applying our results, we provide a numerical algorithm for computing c and e.     

On the neutrality of molecular genetic markers: pedigree analysis of genetic variation in fragmented populations

Many studies employ molecular markers to infer ecological and evolutionary processes, assuming that variation found at genetic loci offers a reliable representation of stochastic events in natural populations. Increasingly, evidence emerges that molecular markers might not always be selectively neutral. However, only a few studies have analysed how deviations from neutrality could affect estimates of genetic variation, using populations with known genealogy. We monitored changes in allozyme variation over eight generations in captive metapopulations of the butterfly Bicyclus anynana. Population demography was recorded by individually marking 35 000 butterflies and constructing pedigrees. We designed a computer program that simulated the inheritance of founder allozyme alleles in butterfly pedigrees. We thus tested whether the observed transmission of allozyme alleles could be explained by random genetic drift alone, or whether there was evidence for positive or negative selection. This analysis showed that in the smallest metapopulations the loss of allozyme variation exceeded the neutral rate. Possibly, linkage disequilibria between deleterious mutations and marker alleles resulted in background selection and a faster erosion of allozyme variation. In larger metapopulations, one locus (MDH) showed a significant heterozygote excess and smaller than expected loss in heterozygosity, observations consistent with (associative) overdominance. This study demonstrates that the neutrality of molecular markers cannot always be assumed, particularly in small populations with a high mutation load.     

Extinction risks in metapopulations of a beetle inhabiting hollow trees predicted from time series

Ancient and dead trees are declining habitats harbouring many threatened species. These habitats are naturally patchy, and inhabiting species might exhibit metapopulation dynamics at a small spatial scale. In this study, the demography and metapopulation dynamics was analysed for Osmoderma eremita , which is an endangered beetle species associated with tree hollows in Europe. Extinction risks of O. eremita populations were predicted using Monte Carlo simulations based on time series of population assessments. Predicted occurrence patterns were consistent with field observations from an area with many small stands in which the populations are believed to have been more or less isolated from each other during the last 150–200 yr. Population growth was found to be density dependent. Carrying capacity was proportional to the volume of wood mould (i.e. loose material of dead wood in the tree hollows), which varied widely between hollow trees. This generates large differences in local extinction risks between hollow trees. The predicted metapopulation extinction risk was much higher if the habitat dynamics (formation, gradual increase and deterioration of tree hollows) were taken into consideration than in predictions yielded by models in which the amount of wood mould was assumed to be constant over time. Thus, this system has features from both mainland-island metapopulations and habitat-tracking metapopulations, and is rather far from a classic metapopulation. For the long-term persistence of the species in hollow trees, the habitat dynamics seem to be more important than demographic processes. Since the formation and deterioration of suitable tree are partly stochastic processes, there is a considerable extinction risk for many O. eremita populations, because they mainly rely on only one or a few trees with large amounts of wood mould.     

Stabilizing Spatially-Structured Populations through Adaptive Limiter Control

Stabilizing the dynamics of complex, non-linear systems is a major concern across several scientific disciplines including ecology and conservation biology. Unfortunately, most methods proposed to reduce the fluctuations in chaotic systems are not applicable to real, biological populations. This is because such methods typically require detailed knowledge of system specific parameters and the ability to manipulate them in real time; conditions often not met by most real populations. Moreover, real populations are often noisy and extinction-prone, which can sometimes render such methods ineffective. Here, we investigate a control strategy, which works by perturbing the population size, and is robust to reasonable amounts of noise and extinction probability. This strategy, called the Adaptive Limiter Control (ALC), has been previously shown to increase constancy and persistence of laboratory populations and metapopulations of Drosophila melanogaster. Here, we present a detailed numerical investigation of the effects of ALC on the fluctuations and persistence of metapopulations. We show that at high migration rates, application of ALC does not require a priori information about the population growth rates. We also show that ALC can stabilize metapopulations even when applied to as low as one-tenth of the total number of subpopulations. Moreover, ALC is effective even when the subpopulations have high extinction rates: conditions under which another control algorithm had previously failed to attain stability. Importantly, ALC not only reduces the fluctuation in metapopulation sizes, but also the global extinction probability. Finally, the method is robust to moderate levels of noise in the dynamics and the carrying capacity of the environment. These results, coupled with our earlier empirical findings, establish ALC to be a strong candidate for stabilizing real biological metapopulations.     

Habitat structure mediates biodiversity effects on ecosystem properties

Much of what we know about the role of biodiversity in mediating ecosystem processes and function stems from manipulative experiments, which have largely been performed in isolated, homogeneous environments that do not incorporate habitat structure or allow natural community dynamics to develop. Here, we use a range of habitat configurations in a model marine benthic system to investigate the effects of species composition, resource heterogeneity and patch connectivity on ecosystem properties at both the patch (bioturbation intensity) and multi-patch (nutrient concentration) scale. We show that allowing fauna to move and preferentially select patches alters local species composition and density distributions, which has negative effects on ecosystem processes (bioturbation intensity) at the patch scale, but overall positive effects on ecosystem functioning (nutrient concentration) at the multi-patch scale. Our findings provide important evidence that community dynamics alter in response to localized resource heterogeneity and that these small-scale variations in habitat structure influence species contributions to ecosystem properties at larger scales. We conclude that habitat complexity forms an important buffer against disturbance and that contemporary estimates of the level of biodiversity required for maintaining future multi-functional systems may need to be revised.     

Host fragmentation and helminth parasites: Hedging your bets against extinction

We consider the probability of parasite extinction due to anthropogenic fragmentation of host populations and in the absence of host extinction. We conclude that extinction at infrapopulation and infracommunity levels is both common and trivial. Extinction may occur in communities at higher levels but only if metapopulations or suprapopulations become extinct. Suprapopulations are highly complex and unlikely to become extinct in the face of simple host fragmentation. We acknowledge parasite metapopulations as being the most likely to become extinct, but only locally. Our reasoning for this is that, in the absence of complete host extinction, populations of the parasite in other fragments are likely to serve as sources for reinvasion (e.g. a rescue effect). We identify a number of features that may act as hedges against extinction for many parasites and conclude by attempting to identify what form an extinction might take.