Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments

This paper presents a study of a nonlinear reaction–diffusion population model in fragmented environments. The model is set on , with periodic heterogeneous coefficients obtained using stochastic processes. Using a criterion of species persistence based on the notion of principal eigenvalue of an elliptic operator, we provided a precise numerical analysis of the interactions between habitat fragmentation and species persistence. The obtained results clearly indicated that species persistence strongly tends to decrease with habitat fragmentation. Moreover, comparing two stochastic models of landscape pattern generation, we observed that in addition to local fragmentation, a more global effect of the position of the habitat patches also influenced species persistence.      

Persistence times of populations with large random fluctuations

The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.     

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.     

The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

Phanerozoic marine biodiversity follows a hyperbolic trend

Changes in marine biodiversity through the Phanerozoic correlate much better with hyperbolic model (widely used in demography and macrosociology) than with exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth arises from a second-order positive feedback between the population size and the rate of technological growth. The hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. The similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics.     

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.     

Survival Analysis of Stochastic Competitive Models in a Polluted Environment and Stochastic Competitive Exclusion Principle

Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.     

Metapopulation moments: coupling, stochasticity and persistence

1.  Spatial heterogeneity has long been viewed as a reliable means of increasing persistence. Here, an analytical model is developed to consider the variation and, hence, the persistence of stochastic metapopulations. This model relies on a novel moment closure technique, which is equivalent to assuming log-normal distributions for the population sizes.
2.  Single-species models show the greatest persistence when the mixing between subpopulations is large, so spatial heterogeneity is of no benefit. This result is confirmed by stochastic simulation of the full metapopulation.
3.  In contrast, natural-enemy models exhibit the greatest persistence for intermediate levels of coupling. When the coupling is too low, there are insufficient rescue effects between the subpopulations to sustain the dynamics, whereas when the coupling is too high all spatial heterogeneity is lost.
4.  The difference in behaviour between the one- and two-species models can be attributed to the oscillatory nature of the natural-enemy system.     

A simple stochastic model for complex coextinctions in mutualistic networks: robustness decreases with connectance

Understanding and predicting species extinctions and coextinctions is a major goal of ecological research in the face of a biodiversity crisis. Typically, models based on network topology are used to simulate coextinctions in mutualistic networks. However, such topological models neglect two key biological features of species interactions: variation in the intrinsic dependence of species on the mutualism, and variation in the relative importance of each interacting partner. By incorporating both types of variation, we developed a stochastic coextinction model capable of simulating extinction cascades far more complex than those observed in previous topological models. Using a set of empirical mutualistic networks, we show that the traditional topological model may either underestimate or overestimate the number and likelihood of coextinctions, depending on the intrinsic dependence of species on the mutualism. More importantly, contrary to topological models, our stochastic model predicts extinction cascades to be more likely in highly connected mutualistic communities.     

Projecting impacts of global climate and land‐use scenarios on plant biodiversity using compositional‐turnover modelling

Nations have committed to ambitious conservation targets in response to accelerating rates of global biodiversity loss. Anticipating future impacts is essential to inform policy decisions for achieving these targets, but predictions need to be of sufficiently high spatial resolution to forecast the local effects of global change. As part of the intercomparison of biodiversity and ecosystem services models of the Intergovernmental Science‐Policy Platform on Biodiversity and Ecosystem Services, we present a fine‐resolution assessment of trends in the persistence of global plant biodiversity. We coupled generalized dissimilarity models, fitted to >52 million records of >254 thousand plant species, with the species–area relationship, to estimate the effect of land‐use and climate change on global biodiversity persistence. We estimated that the number of plant species committed to extinction over the long term has increased by 60% globally between 1900 and 2015 (from ~10,000 to ~16,000). This number is projected to decrease slightly by 2050 under the most optimistic scenario of land‐use change and to substantially increase (to ~18,000) under the most pessimistic scenario. This means that, in the absence of climate change, scenarios of sustainable socio‐economic development can potentially bring extinction risk back to pre‐2000 levels. Alarmingly, under all scenarios, the additional impact from climate change might largely surpass that of land‐use change. In this case, the estimated number of species committed to extinction increases by 3.7–4.5 times compared to land‐use‐only projections. African regions (especially central and southern) are expected to suffer some of the highest impacts into the future, while biodiversity decline in Southeast Asia (which has previously been among the highest globally) is projected to slow down. Our results suggest that environmentally sustainable land‐use planning alone might not be sufficient to prevent potentially dramatic biodiversity loss, unless a stabilization of climate to pre‐industrial times is observed.     

Stochastic instability and Liapunov stability

Summary The relationship between the deterministic stability of nonlinear ecological models and the properties of the stochastic model obtained by adding weak random perturbations is studied. It is shown that the expected escape time for the stochastic model from a bounded region with nonsingular boundary is determined by a Liapunov function for the nonlinear deterministic model. This connection between stochastic and deterministic models brings together various notions of persistence and vulnerability of ecosystems as defined for deterministically perturbed or randomly perturbed models.     

Scaling biodiversity responses to hydrological regimes

Of all ecosystems, freshwaters support the most dynamic and highly concentrated biodiversity on Earth. These attributes of freshwater biodiversity along with increasing demand for water mean that these systems serve as significant models to understand drivers of global biodiversity change. Freshwater biodiversity changes are often attributed to hydrological alteration by water‐resource development and climate change owing to the role of the hydrological regime of rivers, wetlands and floodplains affecting patterns of biodiversity. However, a major gap remains in conceptualising how the hydrological regime determines patterns in biodiversity's multiple spatial components and facets (taxonomic, functional and phylogenetic). We synthesised primary evidence of freshwater biodiversity responses to natural hydrological regimes to determine how distinct ecohydrological mechanisms affect freshwater biodiversity at local, landscape and regional spatial scales. Hydrological connectivity influences local and landscape biodiversity, yet responses vary depending on spatial scale. Biodiversity at local scales is generally positively associated with increasing connectivity whereas landscape‐scale biodiversity is greater with increasing fragmentation among locations. The effects of hydrological disturbance on freshwater biodiversity are variable at separate spatial scales and depend on disturbance frequency and history and organism characteristics. The role of hydrology in determining habitat for freshwater biodiversity also depends on spatial scaling. At local scales, persistence, stability and size of habitat each contribute to patterns of freshwater biodiversity yet the responses are variable across the organism groups that constitute overall freshwater biodiversity. We present a conceptual model to unite the effects of different ecohydrological mechanisms on freshwater biodiversity across spatial scales, and develop four principles for applying a multi‐scaled understanding of freshwater biodiversity responses to hydrological regimes. The protection and restoration of freshwater biodiversity is both a fundamental justification and a central goal of environmental water allocation worldwide. Clearer integration of concepts of spatial scaling in the context of understanding impacts of hydrological regimes on biodiversity will increase uptake of evidence into environmental flow implementation, identify suitable biodiversity targets responsive to hydrological change or restoration, and identify and manage risks of environmental flows contributing to biodiversity decline.     

Fauna habitat modelling and mapping: A review and case study in the Lower Hunter Central Coast region of NSW

Abstract Habitat models are now broadly used in conservation planning on public lands. If implemented correctly, habitat modelling is a transparent and repeatable technique for describing and mapping biodiversity values, and its application in peri‐urban and agricultural landscape planning is likely to expand rapidly. Conservation planning in such landscapes must be robust to the scrutiny that arises when biodiversity constraints are placed on developers and private landholders. A standardized modelling and model evaluation method based on widely accepted techniques will improve the robustness of conservation plans. We review current habitat modelling and model evaluation methods and provide a habitat modelling case study in the New South Wales central coast region that we hope will serve as a methodological template for conservation planners. We make recommendations on modelling methods that are appropriate when presence‐absence and presence‐only survey data are available and provide methodological details and a website with data and training material for modellers. Our aim is to provide practical guidelines that preserve methodological rigour and result in defendable habitat models and maps. The case study was undertaken in a rapidly developing area with substantial biodiversity values under urbanization pressure. Habitat maps for seven priority fauna species were developed using logistic regression models of species‐habitat relationships and a bootstrapping methodology was used to evaluate model predictions. The modelled species were the koala, tiger quoll, squirrel glider, yellow‐bellied glider, masked owl, powerful owl and sooty owl. Models ranked sites adequately in terms of habitat suitability and provided predictions of sufficient reliability for the purpose of identifying preliminary conservation priority areas. However, they are subject to multiple uncertainties and should not be viewed as a completely accurate representation of the distribution of species habitat. We recommend the use of model prediction in an adaptive framework whereby models are iteratively updated and refined as new data become available.     

Range size in mid-domain models of species diversity

Geographical patterns of species diversity have been examined using mid-domain null models, in which the ranges of individual species are simulated by randomly arranging them on a bounded one- or two-dimensional continent. These models have shown that structured patterns in the geographical distribution of biodiversity can arise even under a fully stochastic procedure. In particular, mid-domain models have demonstrated that the random generation of ranges of different sizes and locations can produce a gradient of species diversity similar to the one found in real assemblages, with a peak at the middle of a continent. A less explored feature of mid-domain models is the pattern of range-size frequency distribution. Numerical simulations have provided some insights about the geographic pattern of average range size, but no exploration of the shape of range-size frequency distributions has been carried out. Here I present analytical and numerical models that generate explicit predictions for patterns of range size under the assumptions of mid-domain models of species diversity. Some generalizations include: (1) Mid-domain models predict no geographic gradient of average range size; the mean range size of species occurring at any point on a continent is constant (0.5 of the extent of the continent in the one-dimensional model, 0.25 of the area of the continent in the two-dimensional case); (2) Variance in range size is lowest at the middle of a continent and highest near the corners of a square-shaped continent; (3) The range-size frequency distribution is highly right-skewed at any point of a continent, but the skewness is highest near the corners. Despite their alleged weaknesses, mid-domain models are adequate null models against which real-world patterns can be contrasted.     

Predicting extinction risks under climate change: coupling stochastic population models with dynamic bioclimatic habitat models

Species responses to climate change may be influenced by changes in available habitat, as well as population processes, species interactions and interactions between demographic and landscape dynamics. Current methods for assessing these responses fail to provide an integrated view of these influences because they deal with habitat change or population dynamics, but rarely both. In this study, we linked a time series of habitat suitability models with spatially explicit stochastic population models to explore factors that influence the viability of plant species populations under stable and changing climate scenarios in South African fynbos, a global biodiversity hot spot. Results indicate that complex interactions between life history, disturbance regime and distribution pattern mediate species extinction risks under climate change. Our novel mechanistic approach allows more complete and direct appraisal of future biotic responses than do static bioclimatic habitat modelling approaches, and will ultimately support development of more effective conservation strategies to mitigate biodiversity losses due to climate change.     

Niche partitioning and stochastic processes shape community structure following whitefly invasions

One of the most detrimental impacts of invasive species is the exclusion of native species, which reduces biodiversity and can alter community structure. Coexistence between invaders and native species across large scales, however, might be promoted by niche partitioning and/or stochastic processes, even when one species is excluded in some habitats. Here, we examined the effects of species traits, stochastic processes, and niche partitioning on coexistence of two morphocryptic whitefly species in the Bemisia tabaci complex: the invasive Mediterranean (MED) species and the native Middle East-Asia Minor 1 (MEAM1) species. These species engage in intense reproductive interference, which can result in the exclusion of one species or the other in shared habitats. Both species, however, have coexisted in sympatry in Israel for many years, where MED is invasive and MEAM1 is native. Using a spatially explicit model, we show that both stochastic processes and niche partitioning can promote coexistence between MEAM1 and MED, although predicted community structure differs drastically in each scenario. Comparison of field observations with model results indicated that variation in habitat use leading to niche partitioning was a primary factor driving coexistence between MEAM1 and MED across landscapes, although stochastic processes affected the establishment of rare species within habitats. In many systems, combining models with field surveys can be used to isolate and test mechanisms underlying patterns of community structure following invasions.     

The population biology of bacterial plasmids: a hidden Markov model approach

Horizontal plasmid transfer plays a key role in bacterial adaptation. In harsh environments, bacterial populations adapt by sampling genetic material from a horizontal gene pool through self-transmissible plasmids, and that allows persistence of these mobile genetic elements. In the absence of selection for plasmid-encoded traits it is not well understood if and how plasmids persist in bacterial communities. Here we present three models of the dynamics of plasmid persistence in the absence of selection. The models consider plasmid loss (segregation), plasmid cost, conjugative plasmid transfer, and observation error. Also, we present a stochastic model in which the relative fitness of the plasmid-free cells was modeled as a random variable affected by an environmental process using a hidden Markov model (HMM). Extensive simulations showed that the estimates from the proposed model are nearly unbiased. Likelihood-ratio tests showed that the dynamics of plasmid persistence are strongly dependent on the host type. Accounting for stochasticity was necessary to explain four of seven time-series data sets, thus confirming that plasmid persistence needs to be understood as a stochastic process. This work can be viewed as a conceptual starting point under which new plasmid persistence hypotheses can be tested.     

Metapopulation theory for fragmented landscapes

We review recent developments in spatially realistic metapopulation theory, which leads to quantitative models of the dynamics of species inhabiting highly fragmented landscapes. Our emphasis is in stochastic patch occupancy models, which describe the presence or absence of the focal species in habitat patches. We discuss a number of ecologically important quantities that can be derived from the full stochastic models and their deterministic approximations, with a particular aim of characterizing the respective roles of the structure of the landscape and the properties of the species. These quantities include the threshold condition for persistence, the contributions that individual habitat patches make to metapopulation dynamics and persistence, the time to metapopulation extinction, and the effective size of a metapopulation living in a heterogeneous patch network.