Reconstructing Local Population Dynamics in Noisy Metapopulations—The Role of Random Catastrophes and Allee Effects

Reconstructing the dynamics of populations is complicated by the different types of stochasticity experienced by populations, in particular if some forms of stochasticity introduce bias in parameter estimation in addition to error. Identification of systematic biases is critical when determining whether the intrinsic dynamics of populations are stable or unstable and whether or not populations exhibit an Allee effect, i.e., a minimum size below which deterministic extinction should follow. Using a simulation model that allows for Allee effects and a range of intrinsic dynamics, we investigated how three types of stochasticity—demographic, environmental, and random catastrophes— affect our ability to reconstruct the intrinsic dynamics of populations. Demographic stochasticity aside, which is only problematic in small populations, we find that environmental stochasticity—positive and negative environmental fluctuations—caused increased error in parameter estimation, but bias was rarely problematic, except at the highest levels of noise. Random catastrophes, events causing large-scale mortality and likely to be more common than usually recognized, caused immediate bias in parameter estimates, in particular when Allee effects were large. In the latter case, population stability was predicted when endogenous dynamics were actually unstable and the minimum viable population size was overestimated in populations with small or non-existent Allee effects. Catastrophes also generally increased extinction risk, in particular when endogenous Allee effects were large. We propose a method for identifying data points likely resulting from catastrophic events when such events have not been recorded. Using social spider colonies (Anelosimus spp.) as models for populations, we show that after known or suspected catastrophes are accounted for, reconstructed growth parameters are consistent with intrinsic dynamical instability and substantial Allee effects. Our results are applicable to metapopulation or time series data and are relevant for predicting extinction in conservation applications or the management of invasive species.     

Assessing the viability of tri-trophic food chain model in designing a conservation plan: The case of dwindling Quokka population

The extinction and persistence dynamics of quokka (Setonix brachyurus) population at northern jarrah forest of Australia is investigated using mathematical modelling. Predator's management demands a comprehensive understanding of the ecological circumstances associated with predation. Predation by red foxes (Vulpes vulpes) has a significant role in reducing the native animal population. This paper mainly focuses on the extinction dynamics of quokka population and its reduction by red foxes, by qualitative and quantitative analyses of a designed tri-trophic food-chain model composing a prey (quokka), a mesopredator (red fox) and apex predator (dingo). Existence of solution are analysed and shown to be uniformly bounded. We applied the concept of basic reproduction number from epidemiology to the food chain model, to derive a condition for extinction and persistence of predator population. Global stability of the predator-free equilibrium is established by geometric approach. We use Partial Rank Correlation Coefficient (PRCC) for performing global sensitivity analysis to identify most influential model parameter responsible for quokkas prevalence and mortality. Results of numerical simulation for both deterministic and stochastic model confirms the analytical finding and support those of previous studies. The outcome of this work shows that persistence and existence of quokka depend on the demographic impacts of environmental stochasticity on its own population. Based on our results, many conservation strategies are suggested to improve the overall growth of the species. We believe that declination of red fox and the presence of dingoes are important for preserving the uniform occurrence of quokkas.     

Effects on population persistence: the interaction between environmental noise colour,intraspecific competition and space

It is accepted that accurate estimation of risk of population extinction, or persistence time, requires prediction of the effect of fluctuations in the environment on population dynamics. Generally, the greater the magnitude, or variance, of environmental stochasticity, the greater the risk of population extinction. Another characteristic of environmental stochasticity, its colour, has been found to affect population persistence. This is important because real environmental variables, such as temperature, are reddened or positively temporally autocorrelated. However, recent work has disagreed about the effect of reddening environmental stochasticity. Ripa and Lundberg (1996) found increasing temporal autocorrelation (reddening) decreased the risk of extinction, whereas a simple and powerful intuitive argument (Lawton 1988) predicts increased risk of extinction with reddening. This study resolves the apparent contradiction, in two ways, first, by altering the dynamic behaviour of the population models. Overcompensatory dynamics result in persistence times increasing with increased temporal autocorrelation; undercompensatory dynamics result in persistence times decreasing with increased temporal autocorrelation. Secondly, in a spatially subdivided population, with a reasonable degree of spatial heterogeneity in patch quality, increasing temporal autocorrelation in the environment results in decreasing persistence time for both types of competition. Thus, the inclusion of coloured noise into ecological models can have subtle interactions with population dynamics.     

Complex dynamics of survival and extinction in simple population models with harvesting

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.     

Life history and population size variability in a relict plant. Different routes towards long-term persistence

A central tenet of conservation biology is that population size affects the persistence of populations. However, many narrow endemic species combine small population ranges and sizes with long persistence, thereby challenging this tenet. I examined the performance of three different-sized populations of Petrocoptis pseudoviscosa (Caryophyllaceae), a palaeoendemic rupicolous herb distributed along a small valley in the Spanish Pyrenees. Reproductive and demographic parameters were recorded over 6 years, and deterministic and stochastic matrix models were constructed to explore population dynamics and extinction risk. Populations differed greatly in structure, fecundity, recruitment, survival rate, and life span. Strong differentiation in life-history parameters and their temporal variability resulted in differential population vulnerability under current conditions and simulated global changes such as habitat fragmentation or higher climatic fluctuations. This study provides insights into the capacity of narrow endemics to survive both at extreme environmental conditions and at small population sizes. When dealing with species conservation, the population size–extinction risk relationship may be too simplistic for ancient, ecologically restricted organisms, and some knowledge of life history may be most important to assess their future.     

Biological control in a disturbed environment

Most ecological and epidemiological models describe systems with continuous uninterrupted interactions between populations. Many systems, though, have ecological disturbances, such as those associated with planting and harvesting of a seasonal crop. In this paper, we introduce host–parasite–hyperparasite systems as models of biological control in a disturbed environment, where the host–parasite interactions are discontinuous. One model is a parasite–hyperparasite system designed to capture the essence of biological control and the other is a host–parasite–hyperparasite system that incorporates many more features of the population dynamics. Two types of discontinuity are included in the models. One corresponds to a pulse of new parasites at harvest and the other reflects the discontinuous presence of the host due to planting and harvesting. Such discontinuities are characteristic of many ecosystems involving parasitism or other interactions with an annual host. The models are tested against data from an experiment investigating the persistent biological control of the fungal plant parasite of lettuce Sclerotinia minor by the fungal hyperparasite Sporidesmium sclerotivorum, over successive crops. Using a combination of mathematical analysis, model fitting and parameter estimation, the factors that contribute the observed persistence of the parasite are examined. Analytical results show that repeated planting and harvesting of the host allows the parasite to persist by maintaining a quantity of host tissue in the system on which the parasite can reproduce. When the host dynamics are not included explicitly in the model, we demonstrate that homogeneous mixing fails to predict the persistence of the parasite population, while incorporating spatial heterogeneity by allowing for heterogeneous mixing prevents fade-out. Including the host''s dynamics lessens the effect of heterogeneous mixing on persistence, though the predicted values for the parasite population are closer to the observed values. An alternative hypothesis for persistence involving a stepped change in rates of infection is also tested and model fitting is used to show that changes in some environmental conditions may contribute to parasite persistence. The importance of disturbances and periodic forcing in models for interacting populations is discussed.     

Demographic consequences of population subdivision on the long-furred woolly mouse opossum (Micoureus paraguayanus) from the Atlantic Forest

Habitat destruction and fragmentation severely affected the Atlantic Forest. Formerly contiguous populations may become subdivided into a larger number of smaller populations, threatening their long-term persistence. The computer package VORTEX was used to simulate the consequences of habitat fragmentation and population subdivision on Micoureus paraguayanus, an endemic arboreal marsupial of the Atlantic Forest. Scenarios simulated hypothetical populations of 100 and 2000 animals being partitioned into 1–10 populations, linked by varying rates of inter-patch dispersal, and also evaluated male-biased dispersal. Results demonstrated that a single population was more stable than an ensemble of populations of equal size, irrespective of dispersal rate. Small populations (10–20 individuals) exhibited high instability due to demographic stochasticity, and were characterized by high rates of extinction, smaller values for metapopulation growth and larger fluctuations in population size and growth rate. Dispersal effects on metapopulation persistence were related to the size of the populations and to the sexes that were capable of dispersing. Male-biased dispersal had no noticeable effects on metapopulation extinction dynamics, whereas scenarios modelling dispersal by both sexes positively affected metapopulation dynamics through higher growth rates, smaller fluctuations in growth rate, larger final metapopulation sizes and lower probabilities of extinction. The present study highlights the complex relationships between metapopulation size, population subdivision, habitat fragmentation, rate of inter-patch dispersal and sex-biased dispersal and indicates the importance of gaining a better understanding of dispersal and its interactions with correlations between disturbance events.     

Comparison of reintroduction and enhancement effects on metapopulation viability

Metapopulation viability depends upon a balance of extinction and colonization of local habitats by a species. Mechanisms that can affect this balance include physical characteristics related to natural processes (e.g. succession) as well as anthropogenic actions. Plant restorations can help to produce favorable metapopulation dynamics and consequently increase viability; however, to date no studies confirm this is true. Population viability analysis (PVA) allows for the use of empirical data to generate theoretical future projections in the form of median time to extinction and probability of extinction. In turn, PVAs can inform and aid the development of conservation, recovery, and management plans. Pitcher's thistle (Cirsium pitcheri) is a dune endemic that exhibited metapopulation dynamics. We projected viability of three natural and two restored populations with demographic data spanning 15–23 years to determine the degree the addition of reintroduced population affects metapopulation viability. The models were validated by comparing observed and projected abundances and adjusting parameters associated with demographic and environmental stochasticity to improve model performance. Our chosen model correctly predicted yearly population abundance for 60% of the population‐years. Using that model, 50‐year projections showed that the addition of reintroductions increases metapopulation viability. The reintroduction that simulated population performance in early‐successional habitats had the maximum benefit. In situ enhancements of existing populations proved to be equally effective. This study shows that restorations can facilitate and improve metapopulation viability of species dependent on metapopulation dynamics for survival with long‐term persistence of C. pitcheri in Indiana likely to depend on continued active management.     

The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations

Analysing the persistence and viability of small populations is a key issue in extinction theory and population viability analysis. However, there is still no consensus on how to quantify persistence and viability. We present an approach to evaluate any simulation model concerned with extinction. The approach is devised from general Markov models of stochastic population dynamics. From these models, we distil insights into the general mathematical structure of the risk of extinction by time t, P₀(t). From this mathematical structure, we devise a simple but effective protocol – the ln(1−P₀)-plot – which is applicable for situations including environmental noise or catastrophes. This plot delivers two quantities which are fundamental to the assessment of persistence and viability: the intrinsic mean time to extinction, T_m, and the probability c₁ of the population reaching the established phase. The established phase is characterized by typical fluctuations of the population's state variable which can be described by quasi-stationary probability distributions. The risk of extinction in the established phase is constant and given by 1/T_m. We show that T_m is the basic currency for the assessment of persistence and viability because T_m is independent of initial conditions and allows the risk of extinction to be calculated for any time horizon. For situations where initial conditions are important, additionally c₁ has to be considered.     

Multiple attractors in a discrete competition model

The existence of multiple attractors in a competition model implies that the question of coexistence vs. extinction can depend on initial conditions. A discrete stage-structured model of two competing species is derived from a well-tested single-species model of insect populations, and is shown to exhibit multiple attractors for parameter values similar to those used in laboratory experiments which demonstrated chaos in population dynamics. The corresponding basins of attraction are investigated and shown to have very complex structures, and the initial stage structure of the populations is shown to have a significant impact on final outcomes.     

Asynchronization of local population dynamics and persistence of a metapopulation: a lesson from an endangered composite plant, Aster kantoensis

Fragmentation of a large habitat makes local populations less linked to others, and a whole population structure changes to a metapopulation. The smaller a local population is, the more strengthened extinction factors become. Then, frequent extinctions of local populations threaten persistence of the metapopulation unless recolonizations occur rapidly enough after local extinctions. Spatially structured models have been more widely used for predicting future population dynamics and for assessing the extinction risk of a metapopulation. In this article, we first review such spatially structured models that have been applied to conservation biology, focusing on effects of asynchronization among local population dynamics on persistence of the whole metapopulation. Second, we introduce our ongoing project on extinction risk assessment of an endangered composite biennial plant, Aster kantoensis, in the riverside habitat, based on a lattice model for describing its spatiotemporal population dynamics. The model predicted that the extinction risk of A. kantoensis depends on both the frequency of flood occurrence and the time to coverage of a local habitat by other competitively stronger perennials. Finally, we present a measure (Hassell and Pacala's CV ²) for quantifying the effect of asynchronization among local population dynamics on the persistence of a whole metapopulation in conservation ecology. Received: January 12, 2000 / Accepted: February 8, 2000     

Correlated extinctions, colonizations and population fluctuations in a highly connected ringlet butterfly metapopulation

 The persistence of metapopulations is likely to be highly dependent on whether population dynamics are correlated among habitat patches as a result of migration between patches and spatially-correlated environmental stochasticity (weather effects). We examined whether population dynamics of the ringlet butterfly, Aphantopus hyperantus, were synchronous in an area of approximately 0.5 km², with respect to extinction, colonization and population fluctuations. Monks Wood Butterfly Monitoring Scheme transect count data from 1973 to 1995, revealed (A) a major environmental perturbation, the drought of 1976, which caused synchronized extinctions of A. hyperantus in subsequent years, (B) synchronized recolonization in years following the large number of apparent extinctions, and (C) population changes by A. hyperantus were highly correlated in many of the 14 sections of the transect, presumably reflecting similar responses to environmental stochasticity, and the exchange of individuals among sections. However, extinction and population synchrony depended on habitat type. Following the 1976 drought, A. hyperantus apparently became extinct from the most open and most shady habitats it occupied, with some persistence in habitats of intermediate shading, thus showing retraction to core populations in central parts of an environmental gradient, albeit with an average shift to relatively open habitat. Populations at extreme ends of the environmental gradient occupied by A. hyperantus fluctuated least synchronously, suggesting a potential buffering effect of habitat heterogeneity, but this was not crucial to survival after the 1976 drought. Thus, not all habitats are equally important to persistence. Correlated temporal dynamics, variation in habitat quality and the interaction between habitat quality and temporal environmental stochasticity are important determinants of metapopulation persistence and should be incorporated in metapopulation models. Received: 26 April 1996 / Accepted: 17 July 1996     

Regional dynamics and dispersal in the vascular plant Silene viscosa (L.) inhabiting small islands

We investigated and compared the patch occupancy and colonisation and extinction over a 40-yr period in the vascular plant Silene viscosa in two archipelagos off the Swedish east coast. We also assessed the importance of regional vs local dynamics for regional persistence by examining factors affecting colonisation and extinction processes.
We found an effect of isolation on the patch occupancy in 1957, but this effect disappeared 1997. The effect of isolation was not detected when analysing colonisations and extinctions during the 40-yr period separately, which suggests that the colonisations and extinctions since 1957 have changed the patch occupancy of S. viscosa . The change in the patch occupancy is attributed to an increased colonisation, presumably mediated by the increase of the greylag goose Anser anser . The large amplitude of population size change, the high persistence of populations during a 40-yr period, and the long term persistence for some populations all suggest that the distribution pattern approximates a distribution that is expected to be generated from an Island-Mainland model, where a number of large populations are persistent and a number of smaller populations are more prone to extinction due to catastrophes (nesting cormorants) and environmental stochasticity (summer drought). Most evidence suggests that regional processes are of less importance than local processes for regional persistence on a time scale of 40 yr for S. viscosa .     

Stabilizing Spatially-Structured Populations through Adaptive Limiter Control

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

Scaling of the mean and variance of population dynamics under fluctuating regimes

Theoretical ecologists have long sought to understand how the persistence of populations depends on the interactions between exogenous (biotic and abiotic) and endogenous (e.g., demographic and genetic) drivers of population dynamics. Recent work focuses on the autocorrelation structure of environmental perturbations and its effects on the persistence of populations. Accurate estimation of extinction times and especially determination of the mechanisms affecting extinction times is important for biodiversity conservation. Here we examine the interaction between environmental fluctuations and the scaling effect of the mean population size with its variance. We investigate how interactions between environmental and demographic stochasticity can affect the mean time to extinction, change optimal patch size dynamics, and how it can alter the often-assumed linear relationship between the census size and the effective population size. The importance of the correlation between environmental and demographic variation depends on the relative importance of the two types of variation. We found the correlation to be important when the two types of variation were approximately equal; however, the importance of the correlation diminishes as one source of variation dominates. The implications of these findings are discussed from a conservation and eco-evolutionary point of view.     

Density-dependent demographic variation determines extinction rate of experimental populations

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of Daphnia magna to test hypothesized effects of environmental variation on population persistence. Here, half of those data were used to select and fit a stochastic model of population growth to predict extinctions of populations in the other half. When density-dependent demographic stochasticity was detected and incorporated in simple stochastic models, rates of population extinction were accurately predicted or only slightly biased. However, when density-dependent demographic stochasticity was not accounted for, as is usual when forecasting extinction of threatened and endangered species, predicted extinction rates were severely biased. Thus, an experimental demonstration shows that reliable estimates of extinction risk may be obtained for populations in variable environments if high-quality data are available for model selection and if density-dependent demographic stochasticity is accounted for. These results suggest that further consideration of density-dependent demographic stochasticity is required if predicted extinction rates are to be relied upon for conservation planning.     

High local substrate availability stabilizes a cooperative trait

Cooperative behavior is widely spread in microbial populations. An example is the expression of an extracellular protease by the lactic acid bacterium Lactococcus lactis, which degrades milk proteins into free utilizable peptides that are essential to allow growth to high cell densities in milk. Cheating, protease-negative strains can invade the population and drive the protease-positive strain to extinction. By using multiple experimental approaches, as well as modeling population dynamics, we demonstrate that the persistence of the proteolytic trait is determined by the fraction of the generated peptides that can be captured by the cell before diffusing away from it. The mechanism described is likely to be relevant for the evolutionary stability of many extracellular substrate-degrading enzymes.     

Co‐adaptation impacts the robustness of predator–prey dynamics against perturbations

Global change threatens the maintenance of ecosystem functions that are shaped by the persistence and dynamics of populations. It has been shown that the persistence of species increases if they possess larger trait adaptability. Here, we investigate whether trait adaptability also affects the robustness of population dynamics of interacting species and thereby shapes the reliability of ecosystem functions that are driven by these dynamics. We model co‐adaptation in a predator–prey system as changes to predator offense and prey defense due to evolution or phenotypic plasticity. We investigate how trait adaptation affects the robustness of population dynamics against press perturbations to environmental parameters and against pulse perturbations targeting species abundances and their trait values. Robustness of population dynamics is characterized by resilience, elasticity, and resistance. In addition to employing established measures for resilience and elasticity against pulse perturbations (extinction probability and return time), we propose the warping distance as a new measure for resistance against press perturbations, which compares the shapes and amplitudes of pre‐ and post‐perturbation population dynamics. As expected, we find that the robustness of population dynamics depends on the speed of adaptation, but in nontrivial ways. Elasticity increases with speed of adaptation as the system returns more rapidly to the pre‐perturbation state. Resilience, in turn, is enhanced by intermediate speeds of adaptation, as here trait adaptation dampens biomass oscillations. The resistance of population dynamics strongly depends on the target of the press perturbation, preventing a simple relationship with the adaptation speed. In general, we find that low robustness often coincides with high amplitudes of population dynamics. Hence, amplitudes may indicate the robustness against perturbations also in other natural systems with similar dynamics. Our findings show that besides counteracting extinctions, trait adaptation indeed strongly affects the robustness of population dynamics against press and pulse perturbations.     

Metapopulation dynamics: Does it help to have more of the same?

With the interest in conservation biology shifting towards processes from patterns, and to populations from communities, the theory of metapopulation dynamics is replacing the equilibrium theory of island biogeography as the population ecology paradigm in conservation biology. The simplest models of metapopulation dynamics make predictions about the effects of habitat fragmentation - size and isolation of habitat patches - on metapopulation persistence. The simple models may be enriched by considerations of the effects of demographic and environmental stochasticity on the size and extinction probability of local populations. Environmental stochasticity affects populations at two levels: it makes local extinctions more probable, and it also decreases metapopulation persistence time by increasing the correlation of extinction events across populations. Some controversy has arisen over the significance of correlated extinctions, and how they may affect the optimal subdivision of metapopulations to maximize their persistence time.