Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics

Infectious diseases provide a particularly clear illustration of the spatiotemporal underpinnings of consumer-resource dynamics. The paradigm is provided by extremely contagious, acute, immunizing childhood infections. Partially synchronized, unstable oscillations are punctuated by local extinctions. This, in turn, can result in spatial differentiation in the timing of epidemics and, depending on the nature of spatial contagion, may result in traveling waves. Measles epidemics are one of a few systems documented well enough to reveal all of these properties and how they are affected by spatiotemporal variations in population structure and demography. On the basis of a gravity coupling model and a time series susceptible-infected-recovered (TSIR) model for local dynamics, we propose a metapopulation model for regional measles dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.     

A diffusive SI model with Allee effect and application to FIV

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.     

Cities and villages: infection hierarchies in a measles metapopulation

An important issue in the dynamics of directly transmitted microparasites is the relationship between infection probability and host density. We use models and extensive spatio-temporal data for the incidence of measles to examine evidence for spatial heterogeneity in transmission probability, in terms of urban–rural hierarchies in infection rate. Pre-vaccination measles data for England and Wales show strong evidence for urban–rural heterogeneities in infection rate – the proportion of urban cases rises significantly before major epidemics. The model shows that this effect is consistent with a higher infection rate in large cities, though small towns have epidemic characteristics intermediate between town and country. Surprisingly, urban and rural areas of the same population size have a similar propensity for local extinction of infection. A spatial map of urban–rural correlations reveals complex regional patterns of synchronization of towns and cities. The hierarchical heterogeneities in infection persist into the vaccine era; their implications for disease persistence and control are discussed.     

Patterns of density dependence in measles dynamics.

An important question in metapopulation dynamics is the influence of external perturbations on the population''s long-term dynamic behaviour. In this paper we address the question of how spatiotemporal variations in demographic parameters affect the dynamics of measles populations in England and Wales. Specifically, we use nonparametric statistical methods to analyse how birth rate and population size modulate the negative density dependence between successive epidemics as well as their periodicity. For the observed spatiotemporal data from 60 cities, and for simulated model data, the demographic variables act as bifurcation parameters on the joint density of the trade-off between successive epidemics. For increasing population size, a transition occurs from an irregular unpredictable pattern in small communities towards a regular, predictable endemic pattern in large places. Variations in the birth rate parameter lead to a bifurcation from annual towards biennial cyclicity in both observed data and model data.     

Effects of experimental population extinction for the spatial population dynamics of the butterfly Parnassius smintheus

While many studies have examined factors potentially impacting the rate of local population extinction, few experimental studies have examined the consequences of extinction for spatial population dynamics. Here we report results from a large‐scale, long‐term experiment examining the effects of local population extinction for the dynamics of surrounding populations. From 2001–2008 we removed all adult butterflies from two large, neighboring populations within a system of 17 subpopulations of the Rocky Mountain Apollo butterfly, Parnassius smintheus. Surrounding populations were monitored using individual, mark–recapture methods. We found that population removal decreased immigration to surrounding populations in proportion to their connectivity to the removed populations. Correspondingly, within‐generation population abundance declined. Despite these effects, we saw little consistent impact between generations. The extinction rates of surrounding populations were unaffected and local population growth was not consistently reduced by the lack of immigration. The broader results show that immigration affects local abundance within generations, but dynamics are mediated by density‐dependence within populations and by broader density‐independent factors acting between generations. The loss of immigrants resulting from extinction has little impact on the persistence of local populations in this system.     

集合种群动态对生境毁坏空间异质性的响应

首次将分形几何(Fractal geometry)与元胞自动机(Cellular automata)相结合,研究了破碎化生境中集合种群的空间分布格局动态,以及集合种群动态对生境毁坏空间异质性的响应。研究发现:(1)各个物种种群在生境中的分布具有很好的分形特征,物种的计盒维数(Box dimension)不仅可以很好地反映种群的空间分布结构,也能很好地反映种群动态。(2)如果将空间因素考虑进来的话,生境毁坏的灭绝债务(Time debt)将大于空间隐含模式所模拟的结果。(3)物种灭绝同时存在强物种灭绝和弱物种灭绝。并且只有在生境随机毁坏下,才与空间隐含的模拟结果比较接近,即强物种中将是最强物种率先灭绝。而在边缘毁坏这种比较集中成块的开发方式下,将是较强的物种灭绝。(4)边缘毁坏相对随机毁坏有利于物种,尤其是弱物种的长期续存。     

Patch quality and context, but not patch number, drive multi-scale colonization dynamics in experimental aquatic landscapes

Colonization and extinction are primary drivers of local population dynamics, community structure, and spatial patterns of biological diversity. Existing paradigms of island biogeography, metapopulation biology, and metacommunity ecology, as well as habitat management and conservation biology based on those paradigms, emphasize patch size, number, and isolation as primary characteristics influencing colonization and extinction. Habitat selection theory suggests that patch quality could rival size, number, and isolation in determining rates of colonization and resulting community structure. We used naturally colonized experimental landscapes to address four issues: (a) how do colonizing aquatic beetles respond to variation in patch number, (b) how do they respond to variation in patch quality, (c) does patch context affect colonization dynamics, and (d) at what spatial scales do beetles respond to habitat variation? Increasing patch number had no effect on per patch colonization rates, while patch quality and context were critical in determining colonization rates and resulting patterns of abundance and species richness at multiple spatial scales. We graphically illustrate how variation in immigration rates driven by perceived predation risk (habitat quality) can further modify dynamics of the equilibrium theory of island biogeography beyond predator-driven effects on extinction rates. Our data support the importance of patch quality and context as primary determinants of colonization rate, occupancy, abundance, and resulting patterns of species richness, and reinforce the idea that management of metapopulations for species preservation, and metacommunities for local and regional diversity, should incorporate habitat quality into the predictive equation.     

Similarity in seasonal flow regimes,not regional environmental classifications explain synchrony in brown trout population dynamics in France

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Predators Reduce Extinction Risk in Noisy Metapopulations

Background

Spatial structure across fragmented landscapes can enhance regional population persistence by promoting local “rescue effects.” In small, vulnerable populations, where chance or random events between individuals may have disproportionately large effects on species interactions, such local processes are particularly important. However, existing theory often only describes the dynamics of metapopulations at regional scales, neglecting the role of multispecies population dynamics within habitat patches.

Findings

By coupling analysis across spatial scales we quantified the interaction between local scale population regulation, regional dispersal and noise processes in the dynamics of experimental host-parasitoid metapopulations. We find that increasing community complexity increases negative correlation between local population dynamics. A potential mechanism underpinning this finding was explored using a simple population dynamic model.

Conclusions

Our results suggest a paradox: parasitism, whilst clearly damaging to hosts at the individual level, reduces extinction risk at the population level.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.     

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.     

Spatial dynamics in a metapopulation network: recovery of a rare grasshopper Stauvoderus scalaris from population refuges

A characteristic feature of the spatial distribution of many species is patchiness. This spatial patchiness may be generated by very different processes, e.g. fragmentation, succession and extinction-colonisation dynamics. In this study, we apply a spatial realistic metapopulation model to analyse the occupancy pattern of a rare and endangered grasshopper, Stauroderus scalaris. found in an extensive network of 158 patches. When the study was initiated in 1985 the regional occupancy was 9.3% declining down to 7.1% in 1989. Then there was a spatial expansion of the population and in 1993 as many as 27.3% of the patches were occupied and 32.9% in 1995. During this expansion phase, the dynamics obeyed metapopulation principles: large patches and less isolated ones were more likely to be colonised. In the beginning, local extinction risks were negatively related to patch size and positively influenced by isolation. However, later on neither area nor isolation affected extinction probabilities. Altogether, 20 extinctions and 56 colonisations were observed. The shift in regional occupancy, with a growth of ca 20%, coincides with perturbations to the patch network and the warmest summer in 140 yr. Our results suggest that S. scalaris persists on a dynamic habitat mosaic, where refuges are crucial during adverse periods, and stochastic environmental factors (disturbances and climate), that are correlated over large areas, are generating population dynamic patterns that are hard to predict using current modelling techniques.     

Host and pathogen ecology drive the seasonal dynamics of a fungal disease,white-nose syndrome

Seasonal patterns in pathogen transmission can influence the impact of disease on populations and the speed of spatial spread. Increases in host contact rates or births drive seasonal epidemics in some systems, but other factors may occasionally override these influences. White-nose syndrome, caused by the emerging fungal pathogen Pseudogymnoascus destructans, is spreading across North America and threatens several bat species with extinction. We examined patterns and drivers of seasonal transmission of P. destructans by measuring infection prevalence and pathogen loads in six bat species at 30 sites across the eastern United States. Bats became transiently infected in autumn, and transmission spiked in early winter when bats began hibernating. Nearly all bats in six species became infected by late winter when infection intensity peaked. In summer, despite high contact rates and a birth pulse, most bats cleared infections and prevalence dropped to zero. These data suggest the dominant driver of seasonal transmission dynamics was a change in host physiology, specifically hibernation. Our study is the first, to the best of our knowledge, to describe the seasonality of transmission in this emerging wildlife disease. The timing of infection and fungal growth resulted in maximal population impacts, but only moderate rates of spatial spread.     

Incidence and population dynamics of the leaf beetle Gonioctena olivacea in dynamic habitats

In natural as well as in cultural landscapes, disturbance and succession are responsible for the emergence and subsequent disappearance of suitable habitat patches. The dynamics of habitat patches has important consequences for the spatial structure and dynamics of regional populations. However, there are only few studies quantifying both patch dynamics and incidence of insect species in a dynamic landscape over several years. I studied the incidence and population dynamics of the leaf beetle Gonioctena olivacea in a system of dynamic patches of the host plant Scotch broom Cytisus scoparius . The incidence of the beetle was most strongly affected by patch area, whereas connectivity, patch quality, patch age, and landscape context had no or only a minor effect when analysed with logistic regression. The size of local beetle populations was highly fluctuating between the years; however, the population dynamics of the local populations was not synchronous. Adjacent patches did not show higher degrees of synchrony than patches separated by large distances. In the three years of study, local populations became extinct through demographic or environmental stochasticity and patch destruction. Each year >10% of the patches disappeared. The extinction rate of beetles in persistent patches was decreasing with increasing patch area. On the other hand, patches newly emerged and were rapidly colonized by the beetle. The colonization rate depended on patch connectivity. Obviously, Gonioctena olivacea was capable of persisting in this system with high turnover of patches owing to its high dispersal power.     

Effects of landscape and patch-level attributes on regional population persistence

Species responses are influenced by processes operating at multiple scales, yet many conservation studies and management actions are focused on a single scale. Although landscape-level habitat conditions (i.e., habitat amount, fragmentation and landscape quality) are likely to drive the regional persistence of spatially structured populations, patch-level factors (i.e., patch size, isolation, and quality) may also be important. To determine the spatial scales at which habitat factors influence the regional persistence of endangered Ord's kangaroo rats (Dipodomys ordii) in Alberta, Canada, we simulated population dynamics under a range of habitat conditions. Using a spatially-explicit population model, we removed groups of habitat patches based on their characteristics and measured the resulting time to extinction. We used proportional hazards models to rank the influence of landscape and interacting patch-level variables. Landscape quality was the most influential variable followed by patch quality, with both outweighing landscape- and patch-level measures of habitat quantity and fragmentation/proximity. Although habitat conservation and restoration priorities for this population should be in maximizing the overall quality of the landscape, population persistence depends on how this goal is achieved. Patch quality exerted a significant influence on regional persistence, with the removal of low quality road margin patches (sinks) reducing the risk of regional extinction. Strategies for maximizing overall landscape quality that omit patch-level considerations may produce suboptimal or detrimental results for regional population persistence, particularly where complex local population dynamics (e.g., source-sink dynamics) exist. This study contributes to a growing body literature that suggests that the prediction of species responses and future conservation actions may best be assessed with a multi-scale approach that considers habitat quality and that the success of conservation actions may depend on assessing the influences of habitat factors at multiple scales.     

Regional and local scale metapopulation dynamics in the interaction between Callosobruchus maculatus and Anisopteromalus calandrae

Habitat structure increases the persistence of many extinction‐prone resource–consumer interactions. Metapopulation theory is one of the leading approaches currently used to explain why local, ephemeral populations persist at a regional scale. Central to the metapopulation concept is the amount of dispersal occurring between patches, too much or too little can result in regional extinction. In this study, the role of dispersal on the metapopulation dynamics of an over‐exploitative host–parasitoid interaction is assessed. In the absence of the parasitoid the highly vagile bruchid, Callosobruchus maculatus, can maintain a similar population size regardless of the permeability of the inter‐patch matrix and exhibits strong negative density‐dependence. After the introduction of the parasitoid the size of the bruchid population decreases with a corresponding increase in the occurrence of empty patches. In this case, limiting the dispersal of both species decouples the interaction to a greater extent and results in larger regional bruchid populations. Given the disparity between the dispersal rates of the two species, it is proposed that the more dispersive host benefits from the reduction in landscape permeability by increasing the opportunity to colonise empty patches and rescue extinction prone populations. Associated with the introduction of the parasitoid is a shift in the strength of density‐dependence as the population moves from bottom–up towards top–down regulation. The importance of local and regional scale measurements is apparent when the role of individual patches on regional dynamics is considered. By only taking regional dynamics into account the importance of dispersal regime on local dynamics is overlooked. Similarly, when local dynamics were examined, patches were found to have different influences on regional dynamics depending on dispersal regime and patch location.     

Patterns of synchrony in natterjack toad breeding activity and reproductive success at local and regional scales

Empirical studies of the spatiotemporal dynamics of populations are required to better understand natural fluctuations in abundance and reproductive success, and to better target conservation and monitoring programmes. In particular, spatial synchrony in amphibian populations remains little studied. We used data from a comprehensive three year study of natterjack toad Bufo calamita populations breeding at 36 ponds to assess whether there was spatial synchrony in the toad breeding activity (start and length of breeding season, total number of egg strings) and reproductive success (premetamorphic survival and production of metamorphs). We defined a novel approach to assess the importance of short‐term synchrony at both local and regional scales. The approach employs similarity indices and quantifies the interaction between the temporal and spatial components of populations using mixed effects models. There was no synchrony in the toad breeding activity and reproductive success at the local scale, suggesting that populations function as individual clusters independent of each other. Regional synchrony was apparent in the commencement and duration of the breeding season and in the number of egg strings laid (indicative of female population size). Regional synchrony in both rainfall and temperature are likely to explain the patterns observed (e.g. Moran effect). There was no evidence supporting regional synchrony in reproductive success, most likely due to spatial variability in the environmental conditions at the breeding ponds, and to differences in local population fitness (e.g. fecundity). The small scale asynchronous dynamics and regional synchronous dynamics in the number of breeding females indicate that it is best to monitor several populations within a subset of regions. Importantly, variations in the toad breeding activity and reproductive success are not synchronous, and it is thus important to consider them both when assessing the conservation status of pond‐breeding amphibians.     

Lotka's Game in Predator–Prey Theory: Linking Populations to Individuals

This paper further examines an individual-based model of a spatially distributed predator–prey population that demonstrates strong spatial structuring in contrast with predictions from its representative analytic formulation. Examination of a small, localized population reveals that extinctions due to demographic stochasticity dominate the dynamics. Local extinction dynamics produce wave pulses and the interactions of these wave pulses constitute global dynamics. The results motivate a population-level cell-based model with each cell representing a local population and parameterized by local extinction probabilities, rather than individual-based interaction rates. A detailed comparison of spatiotemporal plots from the two modelling frameworks shows that the population-level model captures the broad range of dynamics exhibited by the individual-based model. The agreement between these two complementary theoretical frameworks, one formulated at the level of individuals, the other at the level of populations, provides a mechanistic understanding of the dynamics.     

Spatial ecology of large herbivore populations

Models of the dynamics of large herbivore populations represent density feedbacks on the population growth rate either directly or indirectly through interactions with vegetation resources. Neither approach incorporates the spatial heterogeneity that is an essential feature of most natural environments, and modifies the population dynamics generated. This is especially true for large herbivores exploiting food resources that are rooted in space but temporally variable in quantity and quality both seasonally and annually. In this review I explore how environmental variation at different spatiotemporal scales influences the abundance of herbivore populations controlled via resources, predators or social mechanisms. Changes in abundance can be spatially disparate and dependent on different resource components at different stages of the seasonal cycle, including buffer resources restricting population crashes in extremely adverse years. GPS telemetry enables movement responses generating spatial patterns to be documented in fine spatiotemporal detail, including migration and dispersal. Models incorporating spatial heterogeneity either implicitly or explicitly are outlined, exemplifying how herbivores cope with temporal variability by exploiting spatial variability in resources and conditions. Global human dominance is generating widened climatic variation while opportunities for herbivore movements are becoming constricted. Theoretical population ecologists need to shift their focus from the workings of demographic structure towards effects of changing environmental contexts, in order to project the likely trajectories of large herbivore populations through the Anthropocene.     

Migration, coherence and persistence in a fragmented landscape

The chance of local extinction is high during periods of small population size. Accordingly, a metapopulation made of local communities that support internal population cycling may face the threat of regional extinction if the local dynamics is coherent (synchronized). These systems achieve maximum sustainability at an intermediate level of migration that allows recolonization but prevents synchronization. Here we implement an individual-based simulation technique to examine the maximum persistence condition for a system of patch habitats connected by passive migration. The models discussed in this paper take into consideration realistic elements of metapopulations, such as migration cost, disordered spatial structure, frustration and environmental noise. It turns out that the state with maximum anti-correlation between neighboring patches is the most sustainable one, even in the presence of these complications. The results suggest, at least for small systems, a model independent conservation strategy: coherence between neighboring local communities has, in general, a negative impact, and population will benefit from intervention that increases anti-correlations.