Stochastic von Bertalanffy models, with applications to fish recruitment

We consider three individual-based models describing growth in stochastic environments. Stochastic differential equations (SDEs) with identical von Bertalanffy deterministic parts are formulated, with a stochastic term which decreases, remains constant, or increases with organism size, respectively. Probability density functions for hitting times are evaluated in the context of fish growth and mortality. Solving the hitting time problem analytically or numerically shows that stochasticity can have a large positive impact on fish recruitment probability. It is also demonstrated that the observed mean growth rate of surviving individuals always exceeds the mean population growth rate, which itself exceeds the growth rate of the equivalent deterministic model. The consequences of these results in more general biological situations are discussed.     

Plant Interactions Alter the Predictions of Metabolic Scaling Theory

Metabolic scaling theory (MST) is an attempt to link physiological processes of individual organisms with macroecology. It predicts a power law relationship with an exponent of −4/3 between mean individual biomass and density during density-dependent mortality (self-thinning). Empirical tests have produced variable results, and the validity of MST is intensely debated. MST focuses on organisms’ internal physiological mechanisms but we hypothesize that ecological interactions can be more important in determining plant mass-density relationships induced by density. We employ an individual-based model of plant stand development that includes three elements: a model of individual plant growth based on MST, different modes of local competition (size-symmetric vs. -asymmetric), and different resource levels. Our model is consistent with the observed variation in the slopes of self-thinning trajectories. Slopes were significantly shallower than −4/3 if competition was size-symmetric. We conclude that when the size of survivors is influenced by strong ecological interactions, these can override predictions of MST, whereas when surviving plants are less affected by interactions, individual-level metabolic processes can scale up to the population level. MST, like thermodynamics or biomechanics, sets limits within which organisms can live and function, but there may be stronger limits determined by ecological interactions. In such cases MST will not be predictive.     

Balance between facilitation and resource competition determines biomass-density relationships in plant populations

Theories based on competition for resources predict a monotonic negative relationship between population density and individual biomass in plant populations. They do not consider the role of facilitative interactions, which are known to be important in high stress environments. Using an individual-based 'zone-of-influence' model, we investigated the hypothesis that the balance between facilitative and competitive interactions determines biomass-density relationships. We tested model predictions with a field experiment on the clonal grass Elymus nutans in an alpine meadow. In the model, the relationship between mean individual biomass and density shifted from monotonic to humped as abiotic stress increased. The model results were supported by the field experiment, in which the greatest individual and population biomass were found at intermediate densities in a high-stress alpine habitat. Our results show that facilitation can affect biomass-density relationships.     

Evolution of Size-Dependent Flowering in Onopordum illyricum: A Quantitative Assessment of the Role of Stochastic Selection Pressures

We explore the evolution of delayed, size-dependent reproduction in the monocarpic perennial Onopordum illyricum, using a range of mathematical models, parameterized with long-term field data. Analysis of the long-term data indicated that mortality, flowering, and growth were age and size dependent. Using mixed models, we estimated the variance about each of these relationships and also individual-specific effects. For the field populations, recruitment was the main density-dependent process, although there were weak effects of local density on growth and mortality. Using parameterized growth models, which assume plants grow along a deterministic trajectory, we predict plants should flower at sizes approximately 50% smaller than observed in the field. We then develop a simple criterion, termed the "1-yr look-ahead criterion," based on equating seed production now with that of next year, allowing for mortality and growth, to determine at what size a plant should flower. This model allows the incorporation of variance about the growth function and individual-specific effects. The model predicts flowering at sizes approximately double that observed, indicating that variance about the growth curve selects for larger sizes at flowering. The 1-yr look-ahead approach is approximate because it ignores growth opportunities more than 1 yr ahead. To assess the accuracy of this approach, we develop a more complicated dynamic state variable model. Both models give similar results indicating the utility of the 1-yr look-ahead criterion. To allow for temporal variation in the model parameters, we used an individual-based model with a genetic algorithm. This gave very accurate prediction of the observed flowering strategies. Sensitivity analysis of the model suggested that temporal variation in the parameters of the growth equation made waiting to flower more risky, so selected for smaller sizes at flowering. The models clearly indicate the need to incorporate stochastic variation in life-history analyses.     

Ecological and mathematical considerations on self-thinning in even-aged pure stands

It is emphasized in growth analysis of self-thinning populations that relative mortality rate pertains to the difference between relative growth rates and net assimilation rates, each of which are definable on a mean plant size basis or on a biomass basis. The time trends of the ratio of relative mortality rate to relative growth rates to be expected according to Tadaki's, Shinozaki's and Hozumi's models are compared with that of the eastern white pine population, and a good agreement is exhibited. As an alternative to Hozumi's model, a new model is constructed to unite the logistic theory of plant growth and the 3/2 power law concerning self-thinning, which so far have usually been applied independently to growth analysis. To construct the model the following assumptions are made: the fundamental equation to relate mean plant weight with density in self-thinning population proposed by Shinozaki, and a special population with a specific initial density which follows thew-p trajectory of the 3/2 power law type and has an exponential decrease in its density with biological time. Properties of the model are examined from ecological and mathematical viewpoints.     

An Individual Stand Growth Model for Mean Plant Size Based on the Rule of Self-thinning

A growth model for pure, even-aged stands of plants is asymptoticallybounded above by the self-thinning rule that relates maximumplant size to stand density. The model characterizes accretionin mean size as a deviation from the limiting size. It consistsof a function relating mean size to time and density and a companionsurvival model. The growth model is obtained by substitutingthe survival model for density in the mean size relationship.Model flexibility is demonstrated by fitting it to annual remeasurementsof mean size and number of plants per unit area in a stand ofPinus taeda L. 3/2-power rule, mortality, survival, stand dynamics, plant growth model, loblolly pine     

Individual variability and population regulation: a model of the significance of within-generation density dependence

Most models of theoretical population ecology consider population density as a state variable and thus ignore the fact that populations are composed not of identical average individuals but of individuals which are usually different. However, this individual variability may be important for population regulation. We therefore analysed an individual-based population model which explicitly describes within-generation processes, i.e. individual growth, starvation, and resource dynamics. The results show that if population dynamics are dominated by slow changes in resource level, the population size in the model undergoes wide oscillation, often leading to extinction. If, on the other hand, fast within-generation processes predominate, such as starvation and sudden drops in resource levels, the population fluctuates to a limited extent around an average. Within-generation density dependence may thus be an important mechanism which is largely ignored in classic time-discrete state-variable models. We conclude that the individual-based approach provides important insights into the hierarchical organization of population dynamics, i.e. the relationship between fast processes at the individual level and slower processes at the population level.     

The effects of density, spatial pattern, and competitive symmetry on size variation in simulated plant populations

Patterns of size inequality in crowded plant populations are often taken to be indicative of the degree of size asymmetry of competition, but recent research suggests that some of the patterns attributed to size-asymmetric competition could be due to spatial structure. To investigate the theoretical relationships between plant density, spatial pattern, and competitive size asymmetry in determining size variation in crowded plant populations, we developed a spatially explicit, individual-based plant competition model based on overlapping zones of influence. The zone of influence of each plant is modeled as a circle, growing in two dimensions, and is allometrically related to plant biomass. The area of the circle represents resources potentially available to the plant, and plants compete for resources in areas in which they overlap. The size asymmetry of competition is reflected in the rules for dividing up the overlapping areas. Theoretical plant populations were grown in random and in perfectly uniform spatial patterns at four densities under size-asymmetric and size-symmetric competition. Both spatial pattern and size asymmetry contributed to size variation, but their relative importance varied greatly over density and over time. Early in stand development, spatial pattern was more important than the symmetry of competition in determining the degree of size variation within the population, but after plants grew and competition intensified, the size asymmetry of competition became a much more important source of size variation. Size variability was slightly higher at higher densities when competition was symmetric and plants were distributed nonuniformly in space. In a uniform spatial pattern, size variation increased with density only when competition was size asymmetric. Our results suggest that when competition is size asymmetric and intense, it will be more important in generating size variation than is local variation in density. Our results and the available data are consistent with the hypothesis that high levels of size inequality commonly observed within crowded plant populations are largely due to size-asymmetric competition, not to variation in local density.     

cdpop: A spatially explicit cost distance population genetics program

Spatially explicit simulation of gene flow in complex landscapes is essential to explain observed population responses and provide a foundation for landscape genetics. To address this need, we wrote a spatially explicit, individual-based population genetics model (cdpop). The model implements individual-based population modelling with Mendelian inheritance and k-allele mutation on a resistant landscape. The model simulates changes in population and genotypes through time as functions of individual based movement, reproduction, mortality and dispersal on a continuous cost surface. This model will be a valuable tool for the study of landscape genetics by increasing our understanding about the effects of life history, vagility and differential models of landscape resistance on the genetic structure of populations in complex landscapes.     

A New Stochastic Individual-Based Model for Pattern Formation and its Application to Predator–Prey Systems

Reaction–diffusion theory has played a very important role in the study of pattern formation in biology. However, a group of individuals is described by a single state variable representing population density in reaction–diffusion models, and interaction between individuals can be included only phenomenologically. In this paper, we propose a new scheme that seamlessly combines individual-based models with elements of reaction–diffusion theory and apply it to predator–prey systems as a test of our scheme. In the model, starvation periods and the time to reproductive maturity are modeled for individual predators. Similarly, the life cycle and time to reproductive maturity of an individual prey are modeled. Furthermore, both predators and prey migrate through a two-dimensional space. To include animal migration in the model, we use a relationship between the diffusion and the random numbers generated according to a two-dimensional bivariate normal distribution. Despite the simplicity of this model, our scheme successfully produces logistic patterns and oscillations in the population size of both predator and prey. The peak for the predator population oscillation lags slightly behind the prey peak. The simplicity of this scheme will aid additional study of spatially distributed negative-feedback systems.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.     

Effects of Density and Extinction Coefficient on Size Variability in Plant Populations

The effects of density and extinction coefficient on size variability,as measured by the coefficient of variation of plant weightin even-aged monocultures, were investigated theoretically usinga diffusion model of growth and size distribution and a canopyphotosynthesis model over the range of densities at which self-thinning(size-dependent mortality) does not occur. Size inequality (thecoefficient of variation of plant weight) increases with increasingdensity or leaf area index at each growth stage. Plants witherect leaves are prone to lower size inequality than plantswith horizontal leaves. These results agree well with existingobservations on even-aged plant monocultures and suggest thatcompetition between plants is mainly one-sided (competitionfor light). One sided competition affects size variability througha G(t, x) function (mean growth of plants of size x at timet per unit time). Two-sided competition (including competitionfor nutrients) affects size variability through a D(t, x) function(variance of growth of plants of size x at time t per unit time).In this case, size inequality decreases with increasing density.The importance of studying size variability is emphasized. Helianthus annus L., size variability, size inequality, coefficient of variation, competition, density effect, extinction coefficient, diffusion model, canopy photosynthesis model     

Evolutionary dynamics as a component of stage-structured matrix models: an example using Trillium grandiflorum

Evolution by natural selection improves fitness and may therefore influence population trajectories. Demographic matrix models are often employed in conservation studies to project population dynamics, but such analyses have not incorporated evolutionary dynamics. We project evolutionarily informed population trajectories for a population of the perennial plant Trillium grandiflorum, which is declining due to high levels of herbivory by white-tailed deer. Individuals with later flowering times are less often consumed, so there is selection on this trait. We first incorporated selection analyses into a deterministic matrix model in three ways (corresponding to different methods that have been used for analyzing evolution in structured populations). Because it is not clear which of these methods works best for stage-structured models, we compared each with a more realistic, individual-based model. Deterministic models using fitness averaged over the phenotypic distribution gave trajectories that were similar to those of the individual-based model, whereas the deterministic model using fitness at the mean phenotype gave a much faster rate of evolution than that which was observed. This illustrates that subtle differences in the way in which one splices evolution into demographic models can have a large effect on expected outcomes. This study demonstrates that, by combining demographic and selection analyses, one can gauge the potential relevance of evolution to population dynamics and persistence.     

The natural variability of vital rates and associated statistics

The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

The Roles of Spatial Pattern and Size Variation in Shaping Height Inequality of Plant Population

Game-theoretic models predict that there is an ESS height for the plant population to which all individual plants should converge. To attain this conclusion, the neighborhood factors were assumed to be equal for all the individual plants, and the spatial pattern and size variation of population were left without consideration, which is clearly not right for the scenario of plant competition. We constructed a spatially-explicit, individual-based model to explore the impacts of spatial structure and size variation on individual plant’s height and population’s height hierarchies under the light competition. The monomorphic equilibrium of height that all the individual plants will converge to only exists for a population growing in a strictly uniform spatial pattern with no size variation. When the spatial pattern of the population is non-uniform or there’s size variation among individual plants, the critical heights that individual plants will finally reach are different from each other, and the height inequality at the end of population growth will increase when the population’s spatial pattern’s degree of deviation from uniform and population’s size variation increase. Our results argue strongly for the importance of spatial pattern and neighborhood effects in generating the diversity of population’s height growth pattern.     

松嫩平原碱化草甸天然翅碱蓬种群的密度制约模型

翅碱蓬是一年生藜科植物,耐碱性极强。本文根据在单—种群落随机取样的调查数据,从种群水平分析了松嫩平原碱化草甸天然翅碱蓬种群的密度制约特征。结果表明,翅碱蓬种群在不同生长期及不同数量性状的密度制约模型均可由多种函数形式同时较好地表达出来。但本文仅以相关性最好的拟合方程作为种群某一性状密度制约特征的模型。孕蕾期的平均植株重和单位面积籽实重的密度制约表现为变形双曲线函数y=a+b/x形式;籽实成熟期的平均植株重、平均植株籽实重、平均植株种子数和单位面积种子数均表现为幂函数y=ax-b形式;地上生物量在孕蕾期为变形双曲线函数y=1/(a+bx)形式,在籽实成熟期为对数函数y=a-blnx形式。     

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator–prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator–prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator–prey model.      

Effects of competitive asymmetry on a local density model of plant interference

Although competition between plants is usually asymmetric (i.e. larger plants have a disproportionate effect on smaller plants) almost all models of plant competition at the local level have assumed symmetric competition. We add a simple version of competitive asymmetry to the local density neighborhood models of plant interference and population dynamics developed by Pacala & Silander (1985, Am. Nat. 125, 385-411; 1987, Oikos 48, 217-224) by assuming that plants within a neighborhood can be put in a linear dominance hierarchy based upon their initial size. The size of a focal plant is a function of the number of dominant and the number of subordinate neighbors within its neighborhood, with subordinate neighbors having less of an effect than dominant ones. Asymmetry prevents precipitous changes in focal plant size with changes in local density, making the relationship between focal plant size and local density hyperbolic, even if the symmetric model is not hyperbolic. Thus, asymmetry makes the model conform to the law of constant final yield, irrespective of the form of the relationship between plant size and local crowding. Asymmetry also prevents population dynamic oscillations in the model in cases in which it would occur in the absence of asymmetry. The results show that asymmetry has major effects on a model of local interference in plants, and point to the importance of including it in such models.     

林窗模型及其在全球气候变化研究中的应用

林窗模型是基于个体的广泛应用于森林长期动态变化的模拟与预测的模型,是研究森林生态系统对气候变化响应的有效工具。本文把林窗模型的发展与演变过程概括为3个阶段:萌芽阶段、飞速发展阶段和提高阶段;展望了林窗模型的未来发展趋势;简要阐述了在全球气候变化背景下应用模型研究森林与气候间关系的可行性与必要性;对国际上相关的研究热点和前沿问题进行了探讨;综述了国内的研究现状,指出国内林窗模型的预测研究应以改进现有模型、构建新模型、耦合多模型作为未来的发展方向。