Extinction risk and the 1/f family of noise models

In order to predict extinction risk in the presence of reddened, or correlated, environmental variability, fluctuating parameters may be represented by the family of 1/f noises, a series of stochastic models with different levels of variation acting on different timescales. We compare the process of parameter estimation for three 1/f models (white, pink and brown noise) with each other, and with autoregressive noise models (which are not 1/f noises), using data from a model time-series (length, T) of population. We then calculate the expected increase in variance and the expected extinction risk for each model, and we use these to explore the implication of assuming an incorrect noise model. When parameterising these models, it is necessary to do so in terms of the measured ("sample") parameters rather than fundamental ("population") parameters. This is because these models are non-stationary: their parameters need not stabilize on measurement over long periods of time and are uniquely defined only over a specified "window" of timescales defined by a measurement process. We find that extinction forecasts can differ greatly between models, depending on the length, T, and the coefficient of variability, CV, of the time series used to parameterise the models, and on the length of time into the future which is to be projected. For the simplest possible models, ones with population itself the 1/f noise process, it is possible to predict the extinction risk based on CV of the observed time series. Our predictions, based on explicit formulae and on simulations, indicate that (a) for very short projection times relative to T, brown and pink noise models are usually optimistic relative to equivalent white noise model; (b) for projection timescales equal to and substantially greater than T, an equivalent brown or pink noise model usually predicts a greater extinction risk, unless CV is very large; and (c) except for very small values of CV, for timescales very much greater than T, the brown and pink models present a more optimistic picture than the white noise model. In most cases, a pink noise is intermediate between white and brown models. Thus, while reddening of environmental noise may increase the long-term extinction probability for stationary processes, this is not generally true for non-stationary processes, such as pink or brown noises.     

Trees of unusual size: biased inference of early bursts from large molecular phylogenies

An early burst of speciation followed by a subsequent slowdown in the rate of diversification is commonly inferred from molecular phylogenies. This pattern is consistent with some verbal theory of ecological opportunity and adaptive radiations. One often-overlooked source of bias in these studies is that of sampling at the level of whole clades, as researchers tend to choose large, speciose clades to study. In this paper, we investigate the performance of common methods across the distribution of clade sizes that can be generated by a constant-rate birth-death process. Clades which are larger than expected for a given constant-rate branching process tend to show a pattern of an early burst even when both speciation and extinction rates are constant through time. All methods evaluated were susceptible to detecting this false signature when extinction was low. Under moderate extinction, both the [Formula: see text]-statistic and diversity-dependent models did not detect such a slowdown but only because the signature of a slowdown was masked by subsequent extinction. Some models which estimate time-varying speciation rates are able to detect early bursts under higher extinction rates, but are extremely prone to sampling bias. We suggest that examining clades in isolation may result in spurious inferences that rates of diversification have changed through time.     

Lag time in microbe growth as an age-dependent branching process with two phase types

We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.     

Aging asexual lineages and the evolutionary maintenance of sex

Finite populations of asexual and highly selfing species suffer from a reduced efficacy of selection. Such populations are thought to decline in fitness over time due to accumulating slightly deleterious mutations or failing to adapt to changing conditions. These within‐population processes that lead nonrecombining species to extinction may help maintain sex and outcrossing through species level selection. Although inefficient selection is proposed to elevate extinction rates over time, previous models of species selection for sex assumed constant diversification rates. For sex to persist, classic models require that asexual species diversify at rates lower than sexual species; the validity of this requirement is questionable, both conceptually and empirically. We extend past models by allowing asexual lineages to decline in diversification rates as they age, that is nonrecombining lineages “senesce” in diversification rates. At equilibrium, senescing diversification rates maintain sex even when asexual lineages, at young ages, diversify faster than their sexual progenitors. In such cases, the age distribution of asexual lineages contains a peak at intermediate values rather than showing the exponential decline predicted by the classic model. Coexistence requires only that the average rate of diversification in asexuals be lower than that of sexuals.     

Reversibility and the age of an allele. I. Moran's infinitely many neutral alleles model

A population of constant size is subjected to mutation, such that each mutant is of a new allelic type. For the particular population model studied in this paper, the age of an allele, whose present frequency is known, is a random variable with distribution independent of the frequencies of other alleles. As a consequence of reversibility of the population process, the age of an allele, from the past to the present, has the same distribution as its time to extinction, from the present into the future. This verifies, and re-interprets, certain diffusion approximations found by Kimura and Ohta [Genetics 75, 199–212 (1973)] and Maruyama [Genet. Res. Cambridge 23, 137–143 (1974)].     

Can a More Variable Species Win Interspecific Competition?

An individual-based approach is used to describe population dynamics. Two kinds of models have been constructed with different distributions illustrating individual variability. In both models, the growth rate of an individual and its final body weight at the end of the growth period, which determines the number of offspring, are functions of the amount of resources assimilated by an individual. In the model with a symmetric distribution, the half saturation constant in the Michaelis–Menten function describing the relationship between the growth of individuals and the amount of resources has a normal distribution. In the model with an asymmetric distribution, resources are not equally partitioned among individuals. The individual who acquired more resources in the past, will acquire more resources in the future. A single population comprising identical individuals has a very short extinction time. If individuals differ in the amount of food assimilated, this time significantly increases irrespectively of the type of model describing population dynamics. Individuals of two populations of competing species use common resources. For larger differences in individual variability, the more variable species will have a longer extinction time and will exclude less variable species. Both populations can also coexist when their variabilities are equal or even when they are slightly different, in the latter case under the condition of high variability of both species. These conclusions have a deterministic nature in the case of the model with the asymmetric distribution—repeated simulations give the same results. In the case of the model with the symmetric distribution, these conclusions are of a statistical nature—if we repeat the simulation many times, then the more variable species will have a longer extinction time more frequently, but some results will happen (although less often) when the less variable species has a longer extinction time. Additionally, in the model with the asymmetric distribution, the result of competition will depend on the way of the introduction of variability into the model. If the higher variability is due to an increase in the proportion of individuals with a low assimilation of resources, it can produce a longer extinction time of the less variable species.

     

To Be Or Not to Be? Variable selection can change the projected fate of a threatened species under future climate

Species distribution models (SDMs) are commonly used to project future changes in the geographic ranges of species, to estimate extinction rates and to plan biodiversity conservation. However, these models can produce a range of results depending on how they are parameterized, and over‐reliance on a single model may lead to overconfidence in maps of future distributions. The choice of predictor variable can have a greater influence on projected future habitat than the range of climate models used. We demonstrate this in the case of the Ptunarra Brown Butterfly, a species listed as vulnerable in Tasmania, Australia. We use the Maxent model to develop future projections for this species based on three variable sets; all 35 commonly used so‐called ‘bioclimatic’ variables, a subset of these based on expert knowledge, and a set of monthly climate variables relevant to the species’ primary activity period. We used a dynamically downscaled regional climate model based on three global climate models. Depending on the choice of variable set, the species is projected either to experience very little contraction of habitat or to come close to extinction by the end of the century due to lack of suitable climate. The different conclusions could have important consequences for conservation planning and management, including the perceived viability of habitat restoration. The output of SDMs should therefore be used to define the range of possible trajectories a species may be on, and ongoing monitoring used to inform management as changes occur.     

Occupancy versus colonization–extinction models for projecting population trends at different spatial scales

Understanding spatiotemporal population trends and their drivers is a key aim in population ecology. We further need to be able to predict how the dynamics and sizes of populations are affected in the long term by changing landscapes and climate. However, predictions of future population trends are sensitive to a range of modeling assumptions. Deadwood‐dependent fungi are an excellent system for testing the performance of different predictive models of sessile species as these species have different rarity and spatial population dynamics, the populations are structured at different spatial scales, and they utilize distinct substrates. We tested how the projected large‐scale occupancies of species with differing landscape‐scale occupancies are affected over the coming century by different modeling assumptions. We compared projections based on occupancy models against colonization–extinction models, conducting the modeling at alternative spatial scales and using fine‐ or coarse‐resolution deadwood data. We also tested effects of key explanatory variables on species occurrence and colonization–extinction dynamics. The hierarchical Bayesian models applied were fitted to an extensive repeated survey of deadwood and fungi at 174 patches. We projected higher occurrence probabilities and more positive trends using the occupancy models compared to the colonization–extinction models, with greater difference for the species with lower occupancy, colonization rate, and colonization:extinction ratio than for the species with higher estimates of these statistics. The magnitude of future increase in occupancy depended strongly on the spatial modeling scale and resource resolution. We encourage using colonization–extinction models over occupancy models, modeling the process at the finest resource‐unit resolution that is utilizable by the species, and conducting projections for the same spatial scale and resource resolution at which the model fitting is conducted. Further, the models applied should include key variables driving the metapopulation dynamics, such as the availability of suitable resource units, habitat quality, and spatial connectivity.     

Extinction models for cancer stem cell therapy

Cells with stem cell-like properties are now viewed as initiating and sustaining many cancers. This suggests that cancer can be cured by driving these cancer stem cells to extinction. The problem with this strategy is that ordinary stem cells are apt to be killed in the process. This paper sets bounds on the killing differential (difference between death rates of cancer stem cells and normal stem cells) that must exist for the survival of an adequate number of normal stem cells. Our main tools are birth-death Markov chains in continuous time. In this framework, we investigate the extinction times of cancer stem cells and normal stem cells. Application of extreme value theory from mathematical statistics yields an accurate asymptotic distribution and corresponding moments for both extinction times. We compare these distributions for the two cell populations as a function of the killing rates. Perhaps a more telling comparison involves the number of normal stem cells N_H at the extinction time of the cancer stem cells. Conditioning on the asymptotic time to extinction of the cancer stem cells allows us to calculate the asymptotic mean and variance of N_H. The full distribution of N_H can be retrieved by the finite Fourier transform and, in some parameter regimes, by an eigenfunction expansion. Finally, we discuss the impact of quiescence (the resting state) on stem cell dynamics. Quiescence can act as a sanctuary for cancer stem cells and imperils the proposed therapy. We approach the complication of quiescence via multitype branching process models and stochastic simulation. Improvements to the τ-leaping method of stochastic simulation make it a versatile tool in this context. We conclude that the proposed therapy must target quiescent cancer stem cells as well as actively dividing cancer stem cells. The current cancer models demonstrate the virtue of attacking the same quantitative questions from a variety of modeling, mathematical, and computational perspectives.     

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.     

Colonization and persistence ability explain the extent to which plant species fill their potential range

Aim How species traits and environmental conditions affect biogeographical dynamics is poorly understood. Here we test whether estimates of a species’ evolutionary age, colonization and persistence ability can explain its current ‘range filling’ (the ratio between realized and potential range size). Location Fynbos biome (Cape Floristic Region, South Africa). Methods For 37 species of woody plants (Proteaceae), we estimate range filling using atlas data and distribution models, evolutionary age using molecular phylogenies, and persistence ability using estimates of individual longevity (which determines the probability of extinction of local populations). Colonization ability is estimated from validated process‐based seed dispersal models, the arrangement of potential habitat, and data on local abundance. To relate interspecific variation in range filling to evolutionary age, colonization and persistence ability, we use two complementary model types: phenomenological linear models and the process‐based metapopulation model of Levins. Results Linear model analyses show that range filling increases with a species’ colonization and persistence ability but is not affected by species age. Moreover, colonization ability is a better predictor of range filling than its component variables (local abundance and dispersal ability). The phylogenetically independent interaction between colonization and persistence ability is significant (P < 0.05) for 97% of 180 alternative phylogenies. While the selected linear model explains 42% of the variance in arcsine transformed range filling, the Levins model performs more poorly. It overestimates range filling for realistic parameter values and produces unrealistic parameter estimates when fitted statistically. Main conclusions Colonization and local extinction seem to shape Proteaceae range dynamics on ecological rather than macroevolutionary time‐scales. Our results suggest that the positive abundance–range size relationship in this group is due primarily to the effect of abundance on colonization. In summary, this study contributes to a process‐based understanding of range dynamics and highlights the importance of colonization for the future survival of Fynbos Proteaceae.     

The expected extinction time of a population within a system of interacting biological populations

The quasi-stationary distribution of a population within a system of interacting populations is approximated by a stochastic logistic process. The parameters of this process can be expressed in the parameters of the full system. Using the diffusion approximation, an expression for the expected extinction time is derived from this logistic process. Since the expected extinction time is expressed in the parameters of the full system, the effect of these parameters on the extinction risk can be easily evaluated, which may be of use for studies in ecology, conservation biology and epidemiology. The outcome is compared with simulation results for the case of a prey-predator system.     

Distribution of phylogenetic diversity under random extinction

Phylogenetic diversity is a measure for describing how much of an evolutionary tree is spanned by a subset of species. If one applies this to the unknown subset of current species that will still be present at some future time, then this ‘future phylogenetic diversity’ provides a measure of the impact of various extinction scenarios in biodiversity conservation. In this paper, we study the distribution of future phylogenetic diversity under a simple model of extinction (a generalized ‘field of bullets’ model). We show that the distribution of future phylogenetic diversity converges to a normal distribution as the number of species grows, under mild conditions, which are necessary. We also describe an algorithm to compute the distribution efficiently, provided the edge lengths are integral, and briefly outline the significance of our findings for biodiversity conservation.     

Effects of metapopulation processes on measures of genetic diversity

Many species persist as a metapopulation under a balance between the local extinction of subpopulations or demes and their recolonization through dispersal from occupied patches. Here we review the growing body of literature dealing with the genetic consequences of such population turnover. We focus our attention principally on theoretical studies of a classical metapopulation with a 'finite-island' model of population structure, rather than on 'continent-island' models or 'source-sink' models. In particular, we concern ourselves with the subset of geographically subdivided population models in which it is assumed that all demes are liable to extinction from time to time and that all demes receive immigrants. Early studies of the genetic effects of population turnover focused on population differentiation, such as measured by F(ST). A key advantage of F(ST) over absolute measures of diversity is its relative independence of the mutation process, so that different genes in the same species may be compared. Another advantage is that F(ST) will usually equilibrate more quickly following perturbations than will absolute levels of diversity. However, because F(ST) is a ratio of between-population differentiation to total diversity, the genetic effects of metapopulation processes may be difficult to interpret in terms of F(ST) on its own, so that the analysis of absolute measures of diversity in addition is likely to be informative. While population turnover may either increase or decrease F(ST), depending on the mode of colonization, recurrent extinction and recolonization is expected always to reduce levels of both within-population and species-wide diversity (piS and piT, respectively). One corollary of this is that piS cannot be used as an unbiased estimate of the scaled mutation rate, theta, as it can, with some assumptions about the migration process, in species whose demes do not fluctuate in size. The reduction of piT in response to population turnover reflects shortened mean coalescent times, although the distribution of coalescence times under extinction colonization equilibrium is not yet known. Finally, we review current understanding of the effect of metapopulation dynamics on the effective population size.     

Modelling the unpredictability of future biodiversity in ecological networks

We consider the question of how accurately we can hope to predict future biodiversity in a world in which many interacting species are at risk of extinction. Simple models assuming that species’ extinctions occur independently are easily analysed, but do not account for the fact that many species depend on or otherwise interact with each other. In this paper we evaluate the effect of explicitly incorporating ecological dependencies on the predictive ability of models of extinction. In particular, we compare a model in which species’ extinction rates increase because of the extinction of their prey to a model in which the same average rate increase takes place, but in which extinctions occur independently from species to species. One might expect that including this ecological information would make the prediction of future biodiversity more accurate, but instead we find that accounting for food web dependencies reveals greater uncertainty. The expected loss of biodiversity over time is similar between the two models, but the variance in future biodiversity is considerably higher in the model that includes species interactions. This increased uncertainty is because of the non-independence of species—the tendency of two species to respond similarly to the loss of a species on which both depend. We use simulations to show that this increase in variance is robust to many variations of the model, and that its magnitude should be largest in food webs that are highly dependent on a few basal species. Our results should hold whenever ecological dependencies cause most species’ extinction risks to covary positively, and illustrate how more information does not necessarily improve our ability to predict future biodiversity loss.     

Potential tree species extinction,colonization and recruitment in Afromontane forest relicts

Tree species regeneration determines future forest structure and composition, but is often severely hampered in small forest relicts. To study succession, long-term field observations or simulation models are used but data, knowledge or resources to run such models are often scarce in tropical areas. We propose and implement a species accounting equation, which includes the co-occurring events extinction, colonization and recruitment and which can be solved by using data from a single inventory. We solved this species accounting equation for the 12 remaining Afromontane cloud forest relicts in Taita Hills, Kenya by comparing the tree species present among the seedling, sapling and mature tree layer in 82 plots. A simultaneous ordination of the seedling, sapling and mature tree layer data revealed that potential species extinctions, colonizations and recruitments may induce future species shifts. On landscape level, the potential extinction debt amounted to 9% (7 species) of the regional species pool. On forest relict level, the smallest relicts harbored an important proportion of the tree species diversity in the regeneration layer. The average potential recruitment credit, defined as species only present as seedling or sapling, was 3 and 6 species for large and small forest relicts, while the average potential extinction debt was 12 and 4 species, respectively. In total, both large and small relicts are expected to lose approximately 20% of their current local tree species pool. The species accounting equations provide a time and resource effective tool and give an improved understanding of the conservation status and possible future succession dynamics of forest relicts, which can be particularly useful in a context of participatory monitoring.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

常见生物生长模型的时差性分析及其应用

生长曲线是估计生物年龄的重要方法之一.在实际工作中,有时会出现对生物年龄的计算起点时间存在着一定差异的情形.例如在一些有关哺乳动物生长的研究中,年龄有出生年龄和受精年龄的区分.这种年龄计算时间上的差异可能导致一些生物生长模型出现不同的拟合结果.本文分析了4种常见的三参数生长模型（Spillman、Logistic、Gompertz和Bertalanffy）的时差性特征.结果表明,这4个方程均具有时差不变性,即无论时间(年龄)起点如何,它们对生物生长数据的拟合结果都一致.文中还引用了小毛足鼠体质量生长数据,采用两种年龄进行了实例比较.     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.