Logistic growth in the presence of non-white environmental noise

A method is given for studying realistic random fluctuations in the carrying capacity of the logistic population growth model. This method is then applied using an environmental noise based on a Poisson process, and the time-dependent moments of the population probability density calculated. These moments are expressed in terms of a parameter obtained by dividing the correlation time of the environmental fluctuations by the characteristic response time of the population. When this quotient is large (very slow fluctuations tracked by the population) or small (very rapid fluctuations which are averaged), exact solutions are obtained for the probability density itself. It is also shown that at equilibrium, the average population sizes given by these two exact solutions bound all other cases.Numerical simulations confirm these developments and point to a trade-off between population stability and average population size. Additional simulations show that the probability of becoming extinct in a given time is greatest for populations intermediate between tracking and averaging the carrying capacity fluctuations. In addition to specifying when environmental noise can be ignored, these results indicate the direction in which growth parameters evolve in a fluctuating environment.     

Small population instability and island settlement patterns

A model of extinction probability, based on the general theory of island biogeography [MacArthur and Wilson, 1967], is proposed for humans on oceanic islands; extinction probability is determined by island carrying capacity, frequency and amplitude of fluctuations in resources determining carrying capacity, and the net costs of contact and exchange between population units. The model predicts that extinction probability will determine island settlement patterns within an island group resulting in nonsettlement of islands with low carrying capacities and settlement of all islands with high carrying capacities. Data examined from the Marshall Islands tend to support the model. The model is extended to initial atoll colonization patterns. Possible requirements for initial settlement are suggested.Deceased.     

A stochastic logistic diffusion equation

A formal stochastic partial differential equation is introduced as a model for the diffusion of a biological population with a fluctuating birthrate in an environment with a finite local carrying capacity. A unique solution is constructed for a related integral equation, and a perturbation expansion is derived when the fluctuations in the birthrate are multiplied by a small parameter.     

Ecological drivers of guanaco recruitment: variable carrying capacity and density dependence

Ungulates living in predator-free reserves offer the opportunity to study the influence of food limitation on population dynamics without the potentially confounding effects of top-down regulation or livestock competition. We assessed the influence of relative forage availability and population density on guanaco recruitment in two predator-free reserves in eastern Patagonia, with contrasting scenarios of population density. We also explored the relative contribution of the observed recruitment to population growth using a deterministic linear model to test the assumption that the studied populations were closed units. The observed densities increased twice as fast as our theoretical populations, indicating that marked immigration has taken place during the recovery phase experienced by both populations, thus we rejected the closed-population assumption. Regarding the factors driving variation in recruitment, in the low- to medium-density setting, we found a positive linear relationship between recruitment and surrogates of annual primary production, whereas no density dependence was detected. In contrast, in the high-density scenario, both annual primary production and population density showed marked effects, indicating a positive relationship between recruitment and per capita food availability above a food-limitation threshold. Our results support the idea that environmental carrying capacity fluctuates in response to climatic variation, and that these fluctuations have relevant consequences for herbivore dynamics, such as amplifying density dependence in drier years. We conclude that including the coupling between environmental variability in resources and density dependence is crucial to model ungulate population dynamics; to overlook temporal changes in carrying capacity may even mask density dependence as well as other important processes.     

A population dynamics model for annual plants subject to inbreeding depression

Summary We present a population dynamics model for annual plants subject to density dependent competition and a decline in mean individual fitness with inbreeding. An analysis of this model provides three distinct sets of parameter values that define the relative influence of inbreeding depression and density on population growth. First, a population with a relatively high finite rate of increase and a relatively small environmental carrying capacity can persist in spite of low levels of inbreeding depression. These types of population may occur during a bottleneck event that is caused by pure predation (or collecting) pressure rather than loss of habitat. Second, there can exist a minimum viable population size when the finite rate of increase is relatively low and the population is also affected by density: the growth or decline of the population will depend on the initial population size. Third, when the population is small enough to be simultaneously effected by density and by inbreeding depression, there can be no viable population.     

Bayesian Inference on the Effect of Density Dependence and Weather on a Guanaco Population from Chile

Understanding the mechanisms that drive population dynamics is fundamental for management of wild populations. The guanaco (Lama guanicoe) is one of two wild camelid species in South America. We evaluated the effects of density dependence and weather variables on population regulation based on a time series of 36 years of population sampling of guanacos in Tierra del Fuego, Chile. The population density varied between 2.7 and 30.7 guanaco/km², with an apparent monotonic growth during the first 25 years; however, in the last 10 years the population has shown large fluctuations, suggesting that it might have reached its carrying capacity. We used a Bayesian state-space framework and model selection to determine the effect of density and environmental variables on guanaco population dynamics. Our results show that the population is under density dependent regulation and that it is currently fluctuating around an average carrying capacity of 45,000 guanacos. We also found a significant positive effect of previous winter temperature while sheep density has a strong negative effect on the guanaco population growth. We conclude that there are significant density dependent processes and that climate as well as competition with domestic species have important effects determining the population size of guanacos, with important implications for management and conservation.     

Considering threats to population viability of the endangered Korean long-tailed goral (Naemorhedus caudatus) using VORTEX

Population viability analysis (PVA) has frequently been used in conservation biology to predict extinction rates for threatened or endangered species. In this study, we used VORTEX to model Korean long-tailed goral (Naemorhedus caudatus) using previously collected ecological data. We focused on modelling population extinction, mean population size and heterozygosity. The minimum viable population size was found to be at least 50 gorals for 100 years, regardless of carrying capacity. However, populations with fewer than 50 gorals could not remain successful in the model. Inbreeding depression, catastrophes and supplementation also affected patterns of population extinction, mean population size and heterozygosity. Supplementation with new individuals had the strongest effect on extinction, mean population size and heterozygosity, followed by initial population size, inbreeding, catastrophes and carrying capacity. These results suggest that a supplementation by extra goral individuals from goral proliferation facilities would be the most helpful means for the restoration programme. More Korean goral-specific information regarding demographic and habitat parameters is needed for further PVA of the species.     

Forms of density regulation and (quasi-) stationary distributions of population sizes in birds

The theta-logistic model of density regulation is an especially flexible class of density regulation models where different forms of non-linear density regulation can be expressed by only one parameter, θ. Estimating the parameters of the theta-logistic model is, however, challenging. This is mainly due to the need for information concerning population growth at low densities as well as data on fluctuations around the carrying capacity K in order to estimate the strength of density regulation. Here we estimate parameters of the theta-logistic model for 28 populations of three species of birds that have grown from very small population sizes followed by a period of fluctuations around K. We then use these parameters to estimate the quasi-stationary distribution of population size. There were often large uncertainties in these parameters specifying the form of density regulation that were generally independent of the duration of the study period. In contrast, precision in the estimates of environmental variance increased with the length of the time series. In most of the populations, a large proportion of the probability density of the (quasi-) stationary distribution of population sizes was located at intermediate population sizes relative to K. Thus, we suggest that the (quasi-) stationary distribution of population sizes represents a useful summary statistic that in many cases provides a more robust characterisation of basic population dynamics (e.g. range of variation in population fluctuations or proportion of time spent close to K) than can be obtained from analyses of single model parameters.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.     

Contingency and statistical laws in replicate microbial closed ecosystems

Contingency, the persistent influence of past random events, pervades biology. To what extent, then, is each course of ecological or evolutionary dynamics unique, and to what extent are these dynamics subject to a common statistical structure? Addressing this question requires replicate measurements to search for emergent statistical laws. We establish a readily replicated microbial closed ecosystem (CES), sustaining its three species for years. We precisely measure the local population density of each species in many CES replicates, started from the same initial conditions and kept under constant light and temperature. The covariation among replicates of the three species densities acquires a stable structure, which could be decomposed into discrete eigenvectors, or "ecomodes." The largest ecomode dominates population density fluctuations around the replicate-average dynamics. These fluctuations follow simple power laws consistent with a geometric random walk. Thus, variability in ecological dynamics can be studied with CES replicates and described by simple statistical laws.     

Structure and function of a non-interactive, reactive insect-plant system

Rhagoletis alternata is a common tephritid fly in central Europe, whose larvae feed on the hypanthium of rose hips. The resource-consumer system is “non-interactive”, i.e. the insect has little or no impact on host plant fitness and therefore is not able to influence the rate at which larval food resources are renewed. The system is “reactive”, since fluctuations in the carrying capacity (hip density) of the host plant are important for determining year-to-year fluctuations in the insect's population size. Insect fluctuations exceed those of its carrying capacity. The insect's efficient exploitation strategy, maximizing its fitness at high as well as low resource supply, must be attributed to the variable and unpredictable relationship between resource availability and consumer density. The only regulatory mechanism is contest competition when larval densities exceed the carrying capacity. Due to the low impact of the insect, its exploitation strategy is apparently not opposed by mechanisms selecting for defence in the host plant. This lack of defence and the efficient exploitation strategy may be important factors for the frequently observed high degree of the resource utilization by the insect. Received: 3 November 1997 / Accepted: 22 January 1998     

Environmental Fluctuations and Level of Density-Compensation Strongly Affects the Probability of Fixation and Fixation Times

The probability of, and time to, fixation of a mutation in a population has traditionally been studied by the classic Wright–Fisher model where population size is constant. Recent theoretical expansions have covered fluctuating populations in various ways but have not incorporated models of how the environment fluctuates in combination with different levels of density-compensation affecting fecundity. We tested the hypothesis that the probability of, and time to, fixation of neutral, advantageous and deleterious mutations is dependent on how the environment fluctuates over time, and on the level of density-compensation. We found that fixation probabilities and times were dependent on the pattern of autocorrelation of carrying capacity over time and interacted with density-compensation. The pattern found was most pronounced at small population sizes. The patterns differed greatly depending on whether the mutation was neutral, advantageous, or disadvantageous. The results indicate that the degree of mismatch between carrying capacity and population size is a key factor, rather than population size per se, and that effective population sizes can be very low also when the census population size is far above the carrying capacity. This study highlights the need for explicit population dynamic models and models for environmental fluctuations for the understanding of the dynamics of genes in populations.     

Carrying Capacity and Potential Production of Ungulates for Human Use in a Mexican Tropical Dry Forest

Data are provided on the carrying capacity and potential production for sustainable human use of white-tailed deer ( Odocoileus virginianus ) and collared peccary ( Pecari tajacu ) in a protected tropical dry forest at Chamela on the Pacific coast of Mexico. In this paper, the carrying capacity was defined as the equilibrium density plus the number of animals removed by predators. The equilibrium point was estimated from the density dependent relationship between the finite population growth rate and the current density according to a logistic model. Annual density was estimated using the line transect method. Carrying capacity estimates were 16.5 to 17.2 deer/km² and 9.3–9.5 peccaries/km², representing a combined biomass of 841–874 kg/km². A potential production for human use of 2.1 deer/km² and 4.4 peccaries/km² was estimated employing the model of Robinson and Redford (1991) . The data suggest that, in the protected tropical dry forest of Chamela, the density and biomass of wild ungulates can maintain a similar or greater density and biomass than other Neotropical forests. To obtain an accurate estimation of the maximum sustainable yield ( MSY ), it is necessary to consider predation. From a management point of view, it is important to consider that carrying capacity varies as a function of the rainfall pattern.     

Postcatastrophe Population Dynamics and Density Dependence of an Endemic Island Duck

ABSTRACT Laysan ducks (Anas laysanensis) are restricted to approximately 9 km² in the Northwestern Hawaiian Islands, USA. To evaluate the importance of density dependence for Laysan ducks, we conducted a Bayesian analysis to estimate the parameters of a Gompertz model and the magnitude of process variation and observation error based on the fluctuations in Laysan duck abundance on Laysan Island from 1994 to 2007. This model described a stationary distribution for the population at carrying capacity that fluctuates around a long-term mean of 456 ducks and is between 316 to 636 ducks 95% of the time. This range of expected variability can be used to identify changes in population size that warn of catastrophic events. Density-dependent population dynamics may explain the recovery of Laysan duck from catastrophic declines and allow managers to identify population monitoring thresholds.     

Spatial heterogeneity in climate change effects decouples the long‐term dynamics of wild reindeer populations in the high Arctic

The ‘Moran effect’ predicts that dynamics of populations of a species are synchronized over similar distances as their environmental drivers. Strong population synchrony reduces species viability, but spatial heterogeneity in density dependence, the environment, or its ecological responses may decouple dynamics in space, preventing extinctions. How such heterogeneity buffers impacts of global change on large‐scale population dynamics is not well studied. Here, we show that spatially autocorrelated fluctuations in annual winter weather synchronize wild reindeer dynamics across high‐Arctic Svalbard, while, paradoxically, spatial variation in winter climate trends contribute to diverging local population trajectories. Warmer summers have improved the carrying capacity and apparently led to increased total reindeer abundance. However, fluctuations in population size seem mainly driven by negative effects of stochastic winter rain‐on‐snow (ROS) events causing icing, with strongest effects at high densities. Count data for 10 reindeer populations 8–324 km apart suggested that density‐dependent ROS effects contributed to synchrony in population dynamics, mainly through spatially autocorrelated mortality. By comparing one coastal and one ‘continental’ reindeer population over four decades, we show that locally contrasting abundance trends can arise from spatial differences in climate change and responses to weather. The coastal population experienced a larger increase in ROS, and a stronger density‐dependent ROS effect on population growth rates, than the continental population. In contrast, the latter experienced stronger summer warming and showed the strongest positive response to summer temperatures. Accordingly, contrasting net effects of a recent climate regime shift—with increased ROS and harsher winters, yet higher summer temperatures and improved carrying capacity—led to negative and positive abundance trends in the coastal and continental population respectively. Thus, synchronized population fluctuations by climatic drivers can be buffered by spatial heterogeneity in the same drivers, as well as in the ecological responses, averaging out climate change effects at larger spatial scales.     

EVOLUTION AND EXTINCTION IN A CHANGING ENVIRONMENT: A QUANTITATIVE-GENETIC ANALYSIS

Because of the ubiquity of genetic variation for quantitative traits, virtually all populations have some capacity to respond evolutionarily to selective challenges. However, natural selection imposes demographic costs on a population, and if these costs are sufficiently large, the likelihood of extinction will be high. We consider how the mean time to extinction depends on selective pressures (rate and stochasticity of environmental change, and strength of selection), population parameters (carrying capacity, and reproductive capacity), and genetics (rate of polygenic mutation). We assume that in a randomly mating, finite population subject to density-dependent population growth, individual fitness is determined by a single quantitative-genetic character under Gaussian stabilizing selection with the optimum phenotype exhibiting directional change, or random fluctuations, or both. The quantitative trait is determined by a finite number of freely recombining, mutationally equivalent, additive loci. The dynamics of evolution and extinction are investigated, assuming that the population is initially under mutation-selection-drift balance. Under this model, in a directionally changing environment, the mean phenotype lags behind the optimum, but on the average evolves parallel to it. The magnitude of the lag determines the vulnerability to extinction. In finite populations, stochastic variation in the genetic variance can be quite pronounced, and bottlenecks in the genetic variance temporarily can impair the population's adaptive capacity enough to cause extinction when it would otherwise be unlikely in an effectively infinite population. We find that maximum sustainable rates of evolution or, equivalently, critical rates of environmental change, may be considerably less than 10% of a phenotypic standard deviation per generation.     

The evolution of self-fertilization in density-regulated populations

The evolution of selfing in hermaphrodites has been studied to reveal the demographic conditions that lead to intermediate selfing rates. Using a demographic model based on Ricker-type density regulation, we assume first that, independent of population density, inbred individuals survive less well than outbred individuals and second, that inbred and outbred individuals differ in their competitive abilities in density-regulated populations. The evolution of selfing, driven by inbreeding depression and the cost of outcrossing, is then analysed for three fundamentally different demographic scenarios: stable population densities, deterministically varying population densities (resulting from cyclical or chaotic population dynamics) and stochastic fluctuations of carrying capacities (resulting from environmental noise). We show that even under stable demographic conditions evolutionary outcomes are not confined to either complete selfing or full outcrossing. Instead, intermediate selfing rates arise under a wide range of conditions, depending on the nature of competitive interactions between inbred and outbred individuals. We also explore the evolution of selfing under deterministic and stochastic density fluctuations to demonstrate that such environmental conditions can evolutionarily stabilize intermediate selfing rates. This is the first study, to our knowledge, to consider in detail the effect of density regulation on the evolution of selfing rates.     

A general analysis of resource allocation by competing individuals.

A linear model for population dynamics in a stationary stochastic environment is introduced based on linearizing the N-species Lotka-Volterra competition equations in discrete time. Iteration of the linear model shows the sequence of population sizes to be formed from a simple linear operation on the sequence of carrying capacities. The transfer function for this operation is calculated and the spectral properties of time series data on population size follow directly.The above approach is illustrated with a symmetrical two-species competition system assuming white noise variation in the carrying capacities. The results are interpreted in detail with the following ideas. (1) The intrinsic rate of increase governs the “responsiveness” of the population to changes in the carrying capacity; (2) one effect of competition is to reduce the “effective rate of increase” of the population. Increasing competition can produce effects identical to that of lowering the intrinsic rate of increase; (3) the other effect of competition is to communicate the stochastic variation in one species' carrying capacity to its competitors. The end result of this communication depends critically on the cross-correlation scheme among the carrying capacities of the competing species.     

完达山东部地区东北虎主要猎物种群的时空动态模拟

基于能体现直接与间接人为干扰的不同意外死亡率和环境容纳量情景,使用景观尺度的动物种群模型（LAPS）模拟了1990—2009年完达山东部地区东北虎主要猎物种群的时空动态,研究了意外死亡率和环境容纳量对种群动态的影响,并直观展现了研究区内动物集群的时空分布状况,比较了不同生境斑块类型中个体密度的差异.结果表明：意外死亡率对研究区动物种群动态的影响较环境容纳量大;灌丛中动物种群的密度高于阔叶林中的密度.研究结果为有效进行东北虎主要猎物的保护与管理提供了科学依据,但相关的定量验证还需深入研究.     

THE EVOLUTIONARY DYNAMICS OF SPITE IN FINITE POPULATIONS

Spite, the shady relative of altruism, involves paying a fitness cost to inflict a cost on some recipient. Here, we investigate a density dependent dynamic model for the evolution of spite in populations of changing size. We extend the model by introducing a dynamic carrying capacity. Our analysis shows that it is possible for unconditionally spiteful behavior to evolve without population structure in any finite population. In some circumstances spiteful behavior can contribute to its own stability by limiting population growth. We use the model to show that there are differences between spite and altruism, and to refine Hamilton’s original argument about the insignificance of spite in the wild. We also discuss the importance of fixing the measure of fitness to classify behaviors as selfish or spiteful.