An extension of the moment closure method

The moment closure method was recently shown in N?sell (Theor. Popul. Biol. 63 (2) (2003a) 159) to give asymptotic approximations of the first few cumulants for the stochastic logistic model. A slight extension of the method is introduced. It is shown to be robust with regard to the specific distributional assumption that is used to achieve moment closure. The phenomenon of spurious solutions is shown to be related to the domain of attraction of the non-spurious critical point.     

Stochastic models of some endemic infections

Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic model. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.     

Extinction times in the subcritical stochastic SIS logistic epidemic

Journal of Mathematical Biology - Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such...     

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

On the interpretation and application of mean times to extinction

As a metric of population viability, conservation biologists routinely predict the mean time to extinction (MTE). Interpretation of MTE depends on the underlying distribution of times to extinction (DTE). Despite claims to the contrary, all information regarding extinction risk can be obtained from this single statistic, the MTE, provided the DTE is exponential. We discuss the proper interpretation of MTE and illustrate how to calculate any population viability statistic when only the MTE is known and the DTE is assumed to be exponential. We also discuss the restrictive assumptions underlying the exponential DTE and the conditions under which alternative models for the DTE are preferable to the conventional (exponential) model. Despite superficial similarities between the exponential and alternative DTEs, several key differences can lead to substantially different interpretations of the MTE.     

Moment closure and the stochastic logistic model

The quasi-stationary distribution of the stochastic logistic model is studied in the parameter region where its body is approximately normal. Improved asymptotic approximations of its first three cumulants are derived. It is shown that the same results can be derived with the aid of the moment closure method. This indicates that the moment closure method leads to expressions for the cumulants that are asymptotic approximations of the cumulants of the quasi-stationary distribution.     

Life history predicts risk of species decline in a stochastic world

Understanding what traits determine the extinction risk of species has been a long-standing challenge. Natural populations increasingly experience reductions in habitat and population size concurrent with increasing novel environmental variation owing to anthropogenic disturbance and climate change. Recent studies show that a species risk of decline towards extinction is often non-random across species with different life histories. We propose that species with life histories in which all stage-specific vital rates are more evenly important to population growth rate may be less likely to decline towards extinction under these pressures. To test our prediction, we modelled declines in population growth rates under simulated stochastic disturbance to the vital rates of 105 species taken from the literature. Populations with more equally important vital rates, determined using elasticity analysis, declined more slowly across a gradient of increasing simulated environmental variation. Furthermore, higher evenness of elasticity was significantly correlated with a reduced chance of listing as Threatened on the International Union for Conservation of Nature Red List. The relative importance of life-history traits of diverse species can help us infer how natural assemblages will be affected by novel anthropogenic and climatic disturbances.     

Relatedness in spatially structured populations with empty sites: An approach based on spatial moment equations

Taking into account the interplay between spatial ecological dynamics and selection is a major challenge in evolutionary ecology. Although inclusive fitness theory has proven to be a very useful tool to unravel the interactions between spatial genetic structuring and selection, applications of the theory usually rely on simplifying demographic assumptions. In this paper, I attempt to bridge the gap between spatial demographic models and kin selection models by providing a method to compute approximations for relatedness coefficients in a spatial model with empty sites. Using spatial moment equations, I provide an approximation of nearest-neighbour relatedness on random regular networks, and show that this approximation performs much better than the ordinary pair approximation. I discuss the connection between the relatedness coefficients I define and those used in population genetics, and sketch some potential extensions of the theory.     

Commentary on Holmes et al. (2007): resolving the debate on when extinction risk is predictable

We reconcile the findings of Holmes et al. ( Ecology Letters , 10 , 2007, 1182) that 95% confidence intervals for quasi-extinction risk were narrow for many vertebrates of conservation concern, with previous theory predicting wide confidence intervals. We extend previous theory, concerning the precision of quasi-extinction estimates as a function of population dynamic parameters, prediction intervals and quasi-extinction thresholds, and provide an approximation that specifies the prediction interval and threshold combinations where quasi-extinction estimates are precise (vs. imprecise). This allows PVA practitioners to define the prediction interval and threshold regions of safety (low risk with high confidence), danger (high risk with high confidence), and uncertainty.     

Understanding foraging behaviour in spatially heterogeneous environments

The role of stochasticity and spatial heterogeneity in foraging systems is investigated. We formulate a spatially explicit model which describes the behaviour of grazing animals in response to local information using simple stochastic rules. In particular the model reflects the biology in that decisions to move to a new location are based on visual assessment of the sward height in a surrounding neighbourhood, whilst the decision to graze the current location is based on the residual sward height and olfactory assessment of local faecal contamination. It is assumed that animals do not interact directly, but do so through modification of, and response to a common environment. Spatial heterogeneity is shown to have significant effects including reducing the equilibrium intake rate and increasing the optimal stocking density, and must therefore be taken into account by resource managers. We demonstrate the relationship between the stochastic spatial model and its non-spatial deterministic counterpart, and in the process derive a moment-closure approximation to the full process, which can be regarded as an intermediate, or pseudo-spatial model. The role of spatial heterogeneity is emphasized, and better understood by comparing the results obtained from each approach. The relative efficiency of random and directed searching behaviour in spatially heterogeneous environments is explored for both clean and contaminated pastures, and the impact of faecal avoidance behaviour assessed.     

The evolution of juvenile-adult interactions in populations structured in age and space

We study the evolution of a spatially structured population with two age classes using spatial moment equations. In the model, adults can either help juveniles by increasing their survival, or adopt a cannibalistic behaviour and consume juveniles. While cannibalism is the sole evolutionary outcome when the population is well-mixed, both cannibalism and parental care can be evolutionarily stable if the population is viscous. Our analysis allows us to make two main technical points. First, we present a method to define invasion fitness in class-structured viscous populations, which allows us to apply adaptive dynamics methodology. Second, we show that ordinary pair approximation introduces an important quantitative bias in the evolutionary model, even on random networks. We propose a correction to the ordinary pair approximation that yields quantitative accuracy, and discuss how the bias associated with this approach is precisely what allows us to identify subtle aspects associated with the evolutionary dynamics of spatially structured populations.     

A mathematical synthesis of niche and neutral theories in community ecology

The debate between niche-based and neutral community theories centers around the question of which forces shape predominantly ecological communities. Niche theory attributes a central role to niche differences between species, which generate a difference between the strength of intra- and interspecific interactions. Neutral theory attributes a central role to migration processes and demographic stochasticity. One possibility to bridge these two theories is to combine them in a common mathematical framework. Here we propose a mathematical model that integrates the two perspectives. From a niche-based perspective, our model can be interpreted as a Lotka-Volterra model with symmetric interactions in which we introduce immigration and demographic stochasticity. From a neutral perspective, it can be interpreted as Hubbell's local community model in which we introduce a difference between intra- and interspecific interactions. We investigate the stationary species abundance distribution and other community properties as functions of the interaction coefficient, the immigration rate and the strength of demographic stochasticity.     

Extinction times and size of the surviving species in a two-species competition process

We investigate a stochastic model for the competition between two species. Based on percentiles of the maximum number of individuals in the ecosystem, we present an approximating model for which the extinction time can be thought of as a phase-type random variable. We determine formulae for the probabilities of extinction and the moments of the extinction time. We discuss the use of several quasi-stationary assumptions. We include a comparative study between existing asymptotic results, results obtained from a simulation of the process, and our solution.     

Allee effects, mating systems and the extinction risk in populations with two sexes

The Allee effect is one of the population consequences of sexual reproduction that has received increased attention in recent years. Due to its impact on small population dynamics, it is commonly accepted that Allee effects should render populations more extinction prone. In particular, monogamous species are considered more susceptible to the Allee effect and hence, more extinction prone, than polygamous species. Although this hypothesis has received theoretical support, there is little empirical evidence. In this study, we investigate (1) how variation in tertiary sex ratio affects the presence and intensity of the Allee effect induced by mating system, as well as (2) how this effect contributes to extinction risk. In contrast with previous predictions, we show that all mating systems are likely to experience a strong Allee effect when the operational sex ratio (OSR) is balanced. This strong Allee effect does not imply being exceptionally extinction prone because it is associated with an OSR that result in a relatively small extinction risk. As a consequence, the impact of Allee effects on overall extinction risk is buffered. Moreover, the OSR of natural populations appears to be often male biased, thus making it unlikely that they will suffer from an Allee effect induced by mating system.     

Extinction and endemism in the New Zealand avifauna

Aim Species belonging to higher taxa endemic to islands are more likely to go extinct following human arrival. This selectivity may occur because more highly endemic island species possess features that make them uniquely vulnerable to impacts associated with human arrival, specifically: (1) restricted distribution (2) reduced predator escape response, including loss of flight, and (3) life history traits, such as large body mass, associated with greater susceptibility to hunting or habitat loss. This study aims to identify which of these features can explain the selective extinction of more highly endemic bird species in New Zealand. Location North and South Island, New Zealand. Methods Bird species breeding in New Zealand prior to human arrival were classified according to whether they became extinct or not during two periods of human settlement, prehistoric (post‐Maori but pre‐European arrival) and historic (post‐European arrival). We modelled the relationships between extinction probability, level of endemism and life history traits in both periods. Results The prehistoric extinction–endemism relationship can be explained entirely by the selective extinction of large‐bodied species, whereas the historic extinction–endemism relationship appears due to increased susceptibility to introduced predators resulting from the loss of predator escape responses, including loss of flight. Conclusions These features may explain extinction–endemism relationships more generally, given that human hunting and predator introductions are major impacts associated with human arrival on islands.