Spatial network structure and metapopulation persistence

We explore the relationship between network structure and dynamics by relating the topology of spatial networks with its underlying metapopulation abundance. Metapopulation abundance is largely affected by the architecture of the spatial network, although this effect depends on demographic parameters here represented by the extinction-to-colonization ratio (e/c). Thus, for moderate to large e/c-values, regional abundance grows with the heterogeneity of the network, with uniform or random networks having the lowest regional abundances, and scale-free networks having the largest abundance. However, the ranking is reversed for low extinction probabilities, with heterogeneous networks showing the lowest relative abundance. We further explore the mechanisms underlying such results by relating a node's incidence (average number of time steps the node is occupied) with its degree, and with the average degree of the nodes it interacts with. These results demonstrate the importance of spatial network structure to understanding metapopulation abundance, and serve to determine under what circumstances information on network structure should be complemented with information on the species life-history traits to understand persistence in heterogeneous environments.     

Effects of demographic parameters on metapopulation size and persistence: an analytical stochastic model

An analytical stochastic metapopulation model is developed. It describes how individuals will be distributed among patches as a function of density-dependent birth, death and emigration rates, and the probability of successful dispersal. The model includes demographic stochasticity, but not catastrophes, environmental stochasticity or variation in patch size and suitability. All patches are equally likely to be colonized by migrants. The model predicts: (a) mean and variance of the number of individuals per patch; (b) probability distribution of individuals per patch; (c) mean number of individuals in transit; and (d) turn-over rate and expected persistence time of a single patch. The model shows that (a) dispersal rates must be intermediate in order to ensure metapopulation persistence; (b) the mean number of individuals per patch is often well below the carrying capacity; (c) long transit times and/or high mortality during dispersal reduce the mean number of individuals per patch; (d) density-dependent emigration responses will usually increase metapopulation size and persistence compared with density-independent dispersal; (e) an increase in the per capita net growth rate can both increase and decrease metapopulation size and persistence depending on whether dispersal rates are high or low; (f) density-independent birth, death, and emigration rates lead to a spatial pattern described by the negative binomial distribution.     

Effects of colonization asymmetries on metapopulation persistence

Ocean currents, prevailing winds, and the hierarchical structures of river networks are known to create asymmetries in re-colonization between habitat patches. The impacts of such asymmetries on metapopulation persistence are seldom considered, especially rarely in theoretical studies. Considering three classical models (the island, the stepping stone and the distance-dependent model), we explore how metapopulation persistence is affected by (i) asymmetry in dispersal strength, in which the colonization rate between two patches differs in direction, and (ii) asymmetry in connectivity, in which the overall colonization pattern displays asymmetry (circulating or dendritic networks). Viability can be drastically reduced when directional bias in dispersal strength is higher than 25%. Re-colonization patterns that allow for strong local connectivity provide the highest persistence compared to systems that allow circulation. Finally, asymmetry has relatively weak effects when metapopulations maintain strong general connectivity.     

A metapopulation paradox: partial improvement of habitat may reduce metapopulation persistence

The adverse influence of habitat degradation on the survival of populations may sometimes be amplified by rapid evolution over ecological timescales. This phenomenon of evolutionary suicide has been described in theoretical as well as empirical studies. However, no studies have suggested that habitat improvement could possibly also trigger an evolutionary response that would result in a decline in population size. We use individual-based simulations to demonstrate the potential for such a paradoxical response. An increase in the quality, size, or stability of only a fraction of the habitat patches in a metapopulation may result in an evolutionary decline in the dispersal propensity of individuals, followed by a decrease in recolonization, a reduction in the number of patches occupied, a decline in overall population size, and even extinction. Thus, well-intended conservation efforts that ignore potential evolutionary consequences of habitat management may increase the extinction risk of populations.     

On optimal size and number of reserves for metapopulation persistence

Habitat fragmentation is generally considered to be detrimental to the persistence of natural populations. In nature management, one therefore tends to prefer few large nature reserves over many small nature reserves having equal total area. This paper examines whether this preference is warranted in a metapopulation framework with circular reserves (patches) by formulating the dependence of metapopulation persistence on the size and number of reserves, both of which depend on reserve radius if the total area is kept constant. Two measures of metapopulation persistence are used: R(0), the number of patches colonized by an occupied patch during its lifetime as an occupied patch, and T(e), the expected time to extinction. These two measures are functions of the extinction and colonization rates of the metapopulation. Several mechanisms for the extinction and colonization processes are formulated from which the dependence of these rates on reserve radius is calculated. It turns out that T(e)generally increases with reserve radius for all mechanisms, which supports the preference of few large reserves. However, R(0)supports this preference only in the case of some special, rather unrealistic, mechanisms. In many other, more realistic, cases an intermediate reserve size exists for which metapopulation persistence measured by R(0)is optimal.     

The role of landscape-dependent disturbance and dispersal in metapopulation persistence

The fundamental processes that influence metapopulation dynamics (extinction and recolonization) will often depend on landscape structure. Disturbances that increase patch extinction rates will frequently be landscape dependent such that they are spatially aggregated and have an increased likelihood of occurring in some areas. Similarly, landscape structure can influence organism movement, producing asymmetric dispersal between patches. Using a stochastic, spatially explicit model, we examine how landscape-dependent correlations between dispersal and disturbance rates influence metapopulation dynamics. Habitat patches that are situated in areas where the likelihood of disturbance is low will experience lower extinction rates and will function as partial refuges. We discovered that the presence of partial refuges increases metapopulation viability and that the value of partial refuges was contingent on whether dispersal was also landscape dependent. Somewhat counterintuitively, metapopulation viability was reduced when individuals had a preponderance to disperse away from refuges and was highest when there was biased dispersal toward refuges. Our work demonstrates that landscape structure needs to be incorporated into metapopulation models when there is either empirical data or ecological rationale for extinction and/or dispersal rates being landscape dependent.     

Extinction dynamics and the regional persistence of a tree frog metapopulation

The concept of a metapopulation acknowledges local extinctions as a natural part of the dynamics of a patchily distributed population. However, if extinctions are not balanced by recolonizations or if there is a high degree of spatial synchrony of local extinctions, this poses a threat to and will reduce the metapopulation persistence time. Here we show that, in a metapopulation network of 378 pond patches used by the tree frog (Hyla arborea), even though extinctions are frequent (mean extinction probability p(e) = 0.24) they pose no threat to the metapopulation as they are balanced by recolonizations (p(c) = 0.33). In any one year there was a pattern of large populations tending to persist while small populations became extinct. The total number of individuals belonging to populations that went extinct was small (< 5%) compared with those populations that persisted. A spatial autocorrelation analysis indicated no clustering of local extinctions. The tree frog metapopulation studied consisted of a set of larger, persistent populations mixed with smaller populations characterized by high turnover dynamics.     

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.     

The effects of spatially correlated perturbations and habitat configuration on metapopulation persistence

All natural populations are confronted with the temporal variability of their environment, and most of them occur in fragmented habitats. I use spatially explicit modeling to examine the interactive effects of habitat configuration, habitat deterioration and spatially correlated environmental perturbations on the viability of fragmented populations. When considering a fixed amount of habitat, viability is maximized for an intermediate (optimal) density of habitat patches, allowing effective dispersal without strong environmental correlation among patches. Starting from this optimal density, I show that the scale of habitat change (density reduction vs range contraction) interacts with the direction of change (reduction vs improvement of habitat availability). An improvement in habitat availability is always more beneficial if occurring through an increase of the species range, while a reduction in habitat availability is always more critical if occurring through a reduction of patch density (even in the presence of environmental correlation). In the context of the 'single large or several small' debate in reserve design, results indicate that a large number of small patches may be optimal to long-term species persistence if the species range increases with the number of patches.     

The effects of connectivity on metapopulation persistence: network symmetry and degree correlations

A spatial metapopulation is a mosaic of interconnected patch populations. The complex routes of colonization between the patches are governed by the metapopulation''s dispersal network. Over the past two decades, there has been considerable interest in uncovering the effects of dispersal network topology and its symmetry on metapopulation persistence. While most studies find that the level of symmetry in dispersal pattern enhances persistence, some have reached the conclusion that symmetry has at most a minor effect. In this work, we present a new perspective on the debate. We study properties of the in- and out-degree distribution of patches in the metapopulation which define the number of dispersal routes into and out of a particular patch, respectively. By analysing the spectral radius of the dispersal matrices, we confirm that a higher level of symmetry has only a marginal impact on persistence. We continue to analyse different properties of the in–out degree distribution, namely the ‘in–out degree correlation’ (IODC) and degree heterogeneity, and find their relationship to metapopulation persistence. Our analysis shows that, in contrast to symmetry, the in–out degree distribution and particularly, the IODC are dominant factors controlling persistence.     

Fitting parameters of stochastic birth-death models to metapopulation data

Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth-death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth-death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models.     

Threshold conditions for infection persistence in complex host-vectors interactions

As classically defined by Macdonald in the early 1950s, for the case of diseases with one vector and one host, the Basic Reproduction Number, R0, is defined as the number of secondary infections caused by a single infective of the same type (vector or host) during its infectiousness period in an entirely susceptible population. In the case of a disease which has one vector and one host, it is easy to show that R0 coincides with the threshold for the establishment of an endemic state: if R0 > 1 (< 1), the disease can invade (cannot invade) the host population. In this paper we examine various epidemic situations in which there are more than one vector and/or host. We show that in those more complex systems it is not possible to deduce a single R0 but rather a threshold for infection persistence which is a composite of several quantities closely related to the classical expression of R0. Another definition of R0 given by Diekmann, Heesterbeek and Metz, and denoted in this paper R0NGO is discussed and applied as an alternative to calculate the thresholds for infection establishment.     

The equilibrium assumption in estimating the parameters of metapopulation models

1.  The construction of a predictive metapopulation model includes three steps: the choice of factors affecting metapopulation dynamics, the choice of model structure, and finally parameter estimation and model testing.
2.  Unless the assumption is made that the metapopulation is at stochastic quasi-equilibrium and unless the method of parameter estimation of model parameters uses that assumption, estimates from a limited amount of data will usually predict a trend in metapopulation size.
3.  This implicit estimation of a trend occurs because extinction-colonization stochasticity, possibly amplified by regional stochasticity, leads to unequal numbers of observed extinction and colonization events during a short study period.
4.  Metapopulation models, such as those based on the logistic regression model, that rely on observed population turnover events in parameter estimation are sensitive to the implicit estimation of a trend.
5.  A new parameter estimation method, based on Monte Carlo inference for statistically implicit models, allows an explicit decision about whether metapopulation quasi-stability is assumed or not.
6. Our confidence in metapopulation model parameter estimates that have been produced from only a few years of data is decreased by the need to know before parameter estimation whether the metapopulation is in quasi-stable state or not.
7. The choice of whether metapopulation stability is assumed or not in parameter estimation should be done consciously. Typical data sets cover only a few years and rarely allow a statistical test of a possible trend. While making the decision about stability one should consider any information about the landscape history and species and metapopulation characteristics.     

Conserving extirpated sites: using habitat quality to manage unoccupied patches for metapopulation persistence

Conservation of metapopulations requires managing extirpated sites, particularly with current threats of increased fragmentation and displacement from global warming. Determining the habitat requirements of threatened species and how they relate to defining characteristics of occupied and unoccupied sites is key to managing suitable habitat in extirpated patches. Due to habitat destruction and degradation, the endangered Ohlone tiger beetle (Cicindela ohlone) is found in only five sites of a once more extensive metapopulation in Santa Cruz County, California. To determine the role of habitat quality in classifying sites, I measured vegetation and ground cover as well as plant and soil composition in sites in which C. ohlone are present, extirpated, and absent. I used conditional inference trees to determine what habitat factors significantly predicted the different sites types. I also analyzed habitat characteristics within present sites to determine factors that predicted egg-laying habitat. As isolation has been shown to be an important driver of metapopulation patch extirpation, I tested the spatial autocorrelation of C. ohlone occupancy to determine if extirpated patches were significantly isolated. Habitat characteristics successfully differentiated nearly 90 % of extirpated plots, which were not isolated from occupied sites. Sites in which C. ohlone are currently present were classified as having at least 10 % cover of bare ground, high forb cover, low litter cover and depth, and high soil bulk density, characteristics that extirpated sites lacked. I illustrate how the defining characteristics could be used to manage habitat in extirpated and absents sites for potential recolonization or translocation, which is vital for metapopulation persistence.     

Spatially structured metapopulation models: global and local assessment of metapopulation capacity

We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence capacity lambda(M) defines the threshold condition for long-term metapopulation persistence as lambda(M)>delta, where delta is defined by the extinction and colonization rate parameters of the focal species. Metapopulation invasion capacity lambda(I) sets the condition for successful invasion of an empty network from one small local population as lambda(I)>delta. The metapopulation capacities lambda(M) and lambda(I) are defined as the leading eigenvalue or a comparable quantity of an appropriate "landscape" matrix. Based on these definitions, we present a classification of a very general class of deterministic, continuous-time and discrete-time metapopulation models. Two specific models are analyzed in greater detail: a spatially realistic version of the continuous-time Levins model and the discrete-time incidence function model with propagule size-dependent colonization rate and a rescue effect. In both models we assume that the extinction rate increases with decreasing patch area and that the colonization rate increases with patch connectivity. In the spatially realistic Levins model, the two types of metapopulation capacities coincide, whereas the incidence function model possesses a strong Allee effect characterized by lambda(I)=0. For these two models, we show that the metapopulation capacities can be considered as simple sums of contributions from individual habitat patches, given by the elements of the leading eigenvector or comparable quantities. We may therefore assess the significance of particular habitat patches, including new patches that might be added to the network, for the metapopulation capacities of the network as a whole. We derive useful approximations for both the threshold conditions and the equilibrium states in the two models. The metapopulation capacities and the measures of the dynamic significance of particular patches can be calculated for real patch networks for applications in metapopulation ecology, landscape ecology, and conservation biology.     

A metapopulation model for the spread and persistence of contagious bovine pleuropneumonia (CBPP) in African sedentary mixed crop-livestock systems

Contagious bovine pleuropneumonia (CBPP) is endemic in several developing countries. Our objective is to evaluate the regional CBPP spread and persistence in a mixed crop-livestock system in Africa. A stochastic compartmental model in metapopulation is used, in which between-herd animal movements and the within-herd infection dynamics are explicitly represented. Hundred herds of varying size are modelled, each sending animals to n other herds (network degree). Animals are susceptible, latent, infectious, chronic carrier or resistant. The role of chronic carriers in CBPP spread being still debated, several chronic periods and infectiousness are tested. A sensitivity analysis is performed to evaluate the influence on model outputs of these parameters and of pathogen virulence, between-herd movement rate, network degree, and calves recruitment. Model outputs are the probability that individual- and group-level reproductive numbers R₀ and R_* are above one, the metapopulation infection duration, the probability of CBPP endemicity (when CBPP persists over 5 years), and the epidemic size in infected herds and infected animals. The most influential parameters are related to chronic carriers (infectiousness and chronic period), pathogen virulence, and recruitment rate. When assuming no CBPP re-introduction in the region, endemicity is only probable if chronic carriers are assumed infectious for at least 1 year and to shed the pathogen in not too low an amount. It becomes highly probable when assuming high pathogen virulence and high recruitment rate.     

Aggregate statistical measures and metapopulation dynamics

There are two main types of metapopulation models. Spatially implicit models are analytically tractable but neglect spatial heterogeneities. Spatially explicit models are more realistic but too complex. In this paper, I build a bridge between both approximations. I derive a new metapopulation model using a well-known technique in population genetics. Spatial heterogeneities are captured by an aggregate statistical measure of spatial correlation. When this correlation is zero, i.e., space is homogeneous, the model becomes the well-known Levins' model. As spatial correlation increases, equilibrium patch occupancy decreases from what would be expected under the spatially homogeneous assumption. I proceed by testing how well spatial complexities from a spatially explicit simulation can be encapsulated by such an aggregate statistical measure.     

Optimal patch use and metapopulation dynamics

Summary We compared the metapopulation dynamics of predator—prey systems with (1) adaptive global dispersal, (2) adaptive local dispersal, (3) fixed global dispersal and (4) fixed local dispersal by predators. Adaptive dispersal was modelled using the marginal value theorem, such that predators departed patches when the instantaneous rate of prey capture was less than the long-term rate of prey capture averaged over all patches, scaled to the movement time between patches. Adaptive dispersal tended to stabilize metapopulation dynamics in a similar manner to conventional fixed dispersal models, but the temporal dynamics of adaptive dispersal models were more unpredictable than the smooth oscillations of fixed dispersal models. Moreover, fixed and adaptive dispersal models responded differently to spatial variation in patch productivity and the degree of compartmentalization of the system. For both adaptive dispersal and fixed dispersal models, localized (stepping-stone) dispersal was more strongly stabilizing than global (island) dispersal. Variation among predators in the probability of dispersal in relation to local prey density had a strong stabilizing influence on both within-patch and metapopulation dynamics. These results suggest that adaptive space use strategies by predators could have important implications for the dynamics of spatially heterogeneous trophic systems.