Threshold parameters and metapopulation persistence

A method is presented to estimate the minimum viable metapopulation size based on the basic reproductive number R ₀ and the expected time to extinction τ _E for epidemiological models. We exemplify our approach with two simple deterministic metapopulation models of the patch occupancy type and then proceed to stochastic versions that permit the estimation of the minimum viable metapopulation size.     

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.     

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.     

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.     

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.     

Effects of population subdivision and catastrophes on the persistence of a land snail metapopulation

We modeled the dynamics of a metapopulation of the land snail Arianta arbustorum in north-eastern Switzerland to investigate the effect of population subdivision on the persistence of a land snail metapopulation and to analyze the interaction between spatial factors, population subdivision, and catastrophes. We developed a spatially structured, stochastic, age-structured metapopulation model with field data from previous studies on the metapopulation in Switzerland and experimental and meteorological data. The model incorporated distance-dependent dispersal through stream banks, correlated environmental fluctuations, and catastrophes resulting from heavy rains. The results point to various complex interactions among factors involved in metapopulation dynamics and suggest that in some cases population subdivision may act to decrease threats from environmental fluctuations and catastrophes.     

Effects of population size and metapopulation dynamics on a mating-system polymorphism

The evolutionary dynamics of neutral alleles under the Wright-Fisher model are well understood. Similarly, the effect of population turnover on neutral genetic diversity in a metapopulation has attracted recent attention in theoretical studies. Here we present the results of computer simulations of a simple model that considers the effects of finite population size and metapopulation dynamics on a mating-system polymorphism involving selfing and outcrossing morphs. The details of the model are based on empirical data from dimorphic populations of the annual plant Eichhornia paniculata, but the results are also of relevance to species with density-dependent selfing rates in general. In our model, the prior selfing rate is determined by two alleles segregating at a single diploid locus. After prior selfing occurs, some remaining ovules are selfed through competing self-fertilisation in finite populations as a result of random mating among gametes. Fitness differences between the mating-system morphs were determined by inbreeding depression and pollen discounting in a context-dependent manner. Simulation results showed evidence of frequency dependence in the action of pollen discounting and inbreeding depression in finite populations. In particular, as a result of selfing in outcrossers through random mating among gametes, selfers experienced a "fixation bias" through drift, even when the mating-system locus was selectively neutral. In a metapopulation, high colony turnover generally favoured the fixation of the outcrossing morph, because inbreeding depression reduced opportunities for colony establishment by selfers through seed dispersal. Our results thus demonstrate that population size and metapopulation processes can lead to evolutionary dynamics involving pollen and seed dispersal that are not predicted for large populations with stable demography.     

The effect of phenotypic variation on metapopulation persistence

Demographic stochasticity (due to the probabilistic nature of the birth–death process) and demographic heterogeneity (between-individual differences in demographic parameters) have long been seen as factors affecting extinction risk. While demographic stochasticity can be independent of underlying species traits, demographic heterogeneity may strongly depend on phenotypic variation. However, how phenotypic variation can affect extinction risk is largely unknown. Here, I develop a stochastic metapopulation model that takes into account the effects of demographic stochasticity and phenotypic variation in the traits controlling colonization rates to assess what the effect of phenotypic variation may be on the persistence of the metapopulation. Although phenotypic variation can lead to a decrease in metapopulation persistence under some conditions, it also may lead to an increase in persistence whenever phenotypic mismatch—or the distance between the optimal trait value and the population mean—is large. This mismatch can in turn arise from a variety of ecological and evolutionary reasons, including weak selection or a recent history of invasion. Last, the effect of phenotypic variation has a deterministic component on colonization rates, and a stochastic component on persistence through colonization rates, but both are important to understand the overall effect. These results have important implications for the conservation of threatened species and management practices that may historically have overlooked phenotypic variation as unimportant noise around mean values of interest.     

Effects of oscillatory electric fields on internal membranes: an analytical model

We derive an analytical model of the potential differences induced across plasma and internal organelle membranes in suspended cells exposed to oscillatory electric fields. Multiple shells are modeled using iterative applications of the single-shell calculation with mobile charges. This work is motivated, in part, by recent results suggesting the ability to use alternating current (ac) fields to noninvasively monitor enzyme activity within internal membranes, particularly the mitochondrial electron transport chain. Previous work, on induced transmembrane voltages in cells subjected to ac fields, has mainly been limited to oscillatory potentials across the plasma membrane. Here we first develop a three-membrane model, consisting of a plasma membrane surrounding inner and outer membranes representing an internal organelle, such as a mitochondrion. Frequency-dependent transmembrane potentials are modeled for spherical, weakly conducting membrane shells enclosing a conductive cytoplasm surrounding an idealized internal organelle. We then use a two-shell model to simulate induced ac membrane potentials of a suspended isolated mitochondrion in which the outer membrane is usually much more permeable than the inner membrane.     

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.     

Carrying capacity and demographic stochasticity: scaling behavior of the stochastic logistic model

The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R-1)(2) N/R(R+1), where N is the habitat size and R is the basic reproductive number.     

On the role of body size in a tri-trophic metapopulation model

 A particular tri-trophic (resource, prey, predator) metapopulation model with dispersal of preys and predators is considered in this paper. The analysis is carried out numerically, by finding the bifurcations of the equilibria and of the limit cycles with respect to prey and predator body sizes. Two routes to chaos are identified. One is characterized by an intriguing cascade of flip and tangent bifurcations of limit cycles, while the other corresponds to the crisis of a strange attractor. The results are summarized by partitioning the space of body sizes in eight subregions, each one of which is associated to a different asymptotic behavior of the system. Emphasis is put on the possibility of having different modes of coexistence (stationary, cyclic, and chaotic) and/or extinction of the predator population. Received 1 August 1995; received in revised form 8 January     

Effects of dispersal in a tri-trophic metapopulation model

The aim of this paper is to understand how dispersal in a patchy environment influences the stability properties of tri-trophic metapopulations. Differential equation models for tri-trophic metapopulations are formulated and analysed. The patchy nature of the metapopulations is incorporated through dispersal phases. Two variants are studied: one with a dispersal phase for the top and one with a dispersal phase for the middle level. A complete characterisation of stable and unstable equilibria is given and the possibility of invasion in these food chains is studied. A dispersal phase for the middle level can destabilize the bottom level-middle level interaction, because of the delay that dispersal causes in the reaction to the resource. When the middle level is not efficiently controlled by the top level, the unstable bottom level-middle level pair can destabilize the entire food chain. Dispersal for the top level can destabilize in the same way. A characterisation of the long term behaviour of the models is given. Bistability with a stable three species equilibrium and a stable limit cycle is one of the possibilities.     

Network theory and metapopulation persistence: incorporating node self‐connections

Network analysis is gaining increasing importance in conservation planning. However, which network metrics are the best predictors of metapopulation persistence is still unresolved. Here, we identify a critical limitation of graph theory‐derived network metrics that have been proposed for this purpose: their omission of node self‐connections. We resolve this by presenting modifications of existing network metrics, and developing entirely new metrics, that account for node self‐connections. Then, we illustrate the performance of these new and modified metrics with an age‐structured metapopulation model for a real‐world marine reserve network case study, and we evaluate the robustness of our findings by systematically varying particular features of that network. Our new and modified metrics predict metapopulation persistence much better than existing metrics do, even when self‐connections are weak. Existing metrics become good predictors of persistence only when self‐connections are entirely absent, an unrealistic scenario in the overwhelming majority of metapopulation applications. Our study provides a set of novel tools that can substantially enhance the extent to which network metrics can be employed to understand, and manage for, metapopulation persistence.     

Balanced input allows optimal encoding in a stochastic binary neural network model: an analytical study

Recent neurophysiological experiments have demonstrated a remarkable effect of attention on the underlying neural activity that suggests for the first time that information encoding is indeed actively influenced by attention. Single cell recordings show that attention reduces both the neural variability and correlations in the attended condition with respect to the non-attended one. This reduction of variability and redundancy enhances the information associated with the detection and further processing of the attended stimulus. Beyond the attentional paradigm, the local activity in a neural circuit can be modulated in a number of ways, leading to the general question of understanding how the activity of such circuits is sensitive to these relatively small modulations. Here, using an analytically tractable neural network model, we demonstrate how this enhancement of information emerges when excitatory and inhibitory synaptic currents are balanced. In particular, we show that the network encoding sensitivity--as measured by the Fisher information--is maximized at the exact balance. Furthermore, we find a similar result for a more realistic spiking neural network model. As the regime of balanced inputs has been experimentally observed, these results suggest that this regime is functionally important from an information encoding standpoint.     

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 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.     

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.     

生境破坏的模式对集合种群动态和续存的影响

构建了空间关联的集合种群模型,该模型不但包含了种群的空间结构信息,而且引入了破坏生境的全局密度和局部密度两个指标,它们描述了破坏生境的模式.模型揭示了破坏生境的空间分布格局复杂地影响了集合种群的动态和续存,破坏和未破坏生境斑块的均匀混合不利于集合种群的增长和续存,而生境类型聚集分布可以促进集合种群的快速增长和长期续存;对于两种斑块类型相对均匀混合的生境来说,均匀场假设可能会高估集合种群的续存,对于相对斑块类型高度聚集的生境,均匀场假设可能会低估集合种群的续存;物种的迁移范围也会影响集合种群的续存,迁移范围越大的物种越容易抵御生境的破坏而免遭灭绝.这意味着在生物保护中不能仅仅考虑生境的恢复和斑块质量的改善,生境结构的构建也是很重要的,加强生境斑块之间的连通性也有利于物种的长期续存.