Fitting and Calibrating a Multilevel Mixed-Effects Stem Taper Model for Maritime Pine in NW Spain

Stem taper data are usually hierarchical (several measurements per tree, and several trees per plot), making application of a multilevel mixed-effects modelling approach essential. However, correlation between trees in the same plot/stand has often been ignored in previous studies. Fitting and calibration of a variable-exponent stem taper function were conducted using data from 420 trees felled in even-aged maritime pine (Pinus pinaster Ait.) stands in NW Spain. In the fitting step, the tree level explained much more variability than the plot level, and therefore calibration at plot level was omitted. Several stem heights were evaluated for measurement of the additional diameter needed for calibration at tree level. Calibration with an additional diameter measured at between 40 and 60% of total tree height showed the greatest improvement in volume and diameter predictions. If additional diameter measurement is not available, the fixed-effects model fitted by the ordinary least squares technique should be used. Finally, we also evaluated how the expansion of parameters with random effects affects the stem taper prediction, as we consider this a key question when applying the mixed-effects modelling approach to taper equations. The results showed that correlation between random effects should be taken into account when assessing the influence of random effects in stem taper prediction.     

Development of a Compatible Taper Function and Stand-Level Merchantable Volume Model for Chinese Fir Plantations

Chinese fir (Cunninghamia lanceolata [Lamb.] Hook) is one of the most important plantation tree species in China with good timber quality and fast growth. It covers an area of 8.54 million hectare, which corresponds to 21% of the total plantation area and 32% of total plantation volume in China. With the increasing market demand, an accurate estimation and prediction of merchantable volume at tree- and stand-level is becoming important for plantation owners. Although there are many studies on the total tree volume estimation from allometric models, these allometric models cannot predict tree- and stand-level merchantable volume at any merchantable height, and the stand-level merchantable volume model was not seen yet in Chinese fir plantations. This study aimed to develop (1) a compatible taper function for tree-level merchantable volume estimation, and (2) a stand-level merchantable volume model for Chinese fir plantations. This “taper function system” consisted in a taper function, a merchantable volume equation and a total tree volume equation. 46 Chinese fir trees were felled to develop the taper function in Shitai County, Anhui province, China. A second-order continuous autoregressive error structure corrected the inherent serial autocorrelation of different observations in one tree. The taper function and volume equations were fitted simultaneously after autocorrelation correction. The compatible taper function fitted well to our data and had very good performances in diameter and total tree volume prediction. The stand-level merchantable volume equation based on the ratio approach was developed using basal area, dominant height, quadratic mean diameter and top diameter (ranging from 0 to 30 cm) as independent variables. At last, a total stand-level volume table using stand basal area and dominant height as variables was proposed for local forest managers to simplify the stand volume estimation.     

黑龙江不同区域人工红松心边材及树皮削度可加性模型系统构建

基于黑龙江省孟家岗林场、东京城林业局和林口林业局不同林分条件下103株人工红松解析木的2977个圆盘数据,结合林业研究中常见的Kozak(1988)、Muhairwe(1999)、Lee(2003)、Kozak(2004)可变指数削度方程,构建带皮直径、心材直径、边材宽度、树皮厚度的削度方程,并对比选出最优基础模型;采用SAS软件PROC MODEL模块中似乎不相关回归(SUR),建立带皮直径、心材直径、边材宽度和树皮厚度削度方程的可加性模型系统,同时将区域作为哑变量引入模型中,通过调整确定系数(R_adj²)、均方根误差(RMSE)、赤池信息准则(AIC)、贝叶斯信息准则(BIC)等模型评价指标,对模型进行综合评价。结果表明: 带皮直径、心材直径、边材宽度和树皮厚度最优基础模型均为Kozak(2004);可加性模型系统在满足各分量与总量可加性的基础上,也得到较好的预测效果,预估精度均达到98%以上,引入哑变量后,可加性模型系统预测能力均有不同程度的提升,尤其心材直径和边材宽度预测能力提升更显著;不同区域带皮直径和树皮厚度削度差异较小,而心材直径、边材宽度的削度存在明显差异。本研究构建的包含哑变量可加性模型系统,不但模型预测精度较高,还满足带皮直径、心材直径、边材宽度和树皮厚度之间的可加性逻辑,为人工红松心边材及树皮材积的准确估测提供了基础。     

人工红松树干内部节子体积预测模型

基于黑龙江省林口林业局林场和东京城林业局林场29块标准地中49株人工红松1207个节子数据,使用图片处理软件Digimizer对节子纵剖面图片数据进行提取,将节子形状用一个二维散点图表示。根据节子二维形状散点图,把人工红松节子分为3种类型: 活节(整个节子为健全节)、未包藏死节(节子由健全节和疏松节组成)和包藏死节(节子的健全节和疏松节部分被树干包藏)。3个类型节子的健全节体积通过对健全节形状参数方程求积得到;疏松节体积利用圆柱体的体积计算得到;节子总体积等于健全节体积与疏松节体积之和。最后,基于节子变量(节子直径、节子相对高、节子总长度)和树木变量(胸径),采用样地和树木水平的线性混合模型建立了红松人工林健全节体积、疏松节体积和节子总体积的预测模型。与基础模型相比,考虑样地和树木水平的混合效应所建立的健全节体积、疏松节体积和节子总体积预测模型,其参数估计更精准,残差分布更均匀,拟合精度明显提高。检验结果表明,基础模型预估精度均在90%以上,引入样地和树木效应的混合模型的预估精度均在93%以上,说明所建模型可以很好地预测红松人工林节子体积大小。     

A new look at computation of the complexity index in mangroves: do disturbed forests have clues to analyze canopy height patchiness?

This paper proposes some guidelines to compute complexity index in those mangroves where either seasonal or strong disturbances have occurred. We surveyed 31 mangrove localities in Buenaventura Bay, Central Pacific Coast of Colombia, where structural parameters were measured within a 0.1 ha plot. Also, most likely disturbances were noted for each plot. Complexity index was calculated in its classical form using the arithmetic mean of the three tallest trees(maximal mean) and alternatively using: a) the total mean: the height average of all trees recorded in each plot; and b) the mode: the most frequent tree height class (10 cm intervals) within each plot. Afterwards, we compared the three computations and discussed there liability of each one according to the current state by plot. In addition, all structural parameters were sorted in two diameter at breast height (dbh) cohorts (2.5–10 cm and ≥ 10 cm) to figure out which contributes more to the forest structure. We conclude the following: (a) The mean of the three tallest trees is not a good estimator of forest development when seasonal or strong disturbances occur since complexity index based in it always overestimates forest structure. (b) Seasonal disturbances and recruitment produce mosaic forests. The best estimator of this condition is the mean height which encompasses both central tendency and variability. (c) The modal height is also helpful to establish the dominant cohort when forests show two or more storied-canopies, or intermediate cohorts are missing. It also applies when a new stock of recruits is entering in a mature forest (the modal or maximal heights can be used interchangeably in this case). (d) Maximal height is the best estimator for uniformly developed forests with closed canopies and/or a single dominant tall-cohort. (e) If one is not confident about which height type to include to compute the complexity index, we recommend to sort out structure data by dbh-cohorts and calculate indices for both of them. This will show which cohort is contributing most to forest complexity. Finally, we suggest to exclude non-mangrove species from the complexity index computation since they mostly do not contribute significantly to forest basal area. This revised version was published online in July 2006 with corrections to the Cover Date.     

Physiological effects of tapering in highly trained athletes.

This study examined some of the physiological and performance effects of three different tapers in highly trained athletes. After 8 wk of training, nine male middle-distance runners were randomly assigned to one of three different 7-day tapers: a high-intensity low-volume taper (HIT), a low-intensity moderate-volume taper (LIT), or a rest-only taper (ROT). After the first taper, subjects resumed training for 4 wk and performed a second taper and then resumed training for 4 wk and completed the remaining taper, so that each subject underwent all three tapers. Performance was measured before and after each taper by a treadmill run to fatigue at a velocity equivalent each subject's best 1,500-m time. Voluntary isometric strength and evoked contractile properties of the quadriceps were measured before and after each taper, as were muscle glycogen concentration and citrate synthase activity (from needle biopsies) and total blood and red cell volume by 125I and 51Cr tagging. Maximal O2 consumption was unaffected by all three tapers, but running time to fatigue increased significantly after HIT (+22%). It was unaffected by LIT (+6%) and ROT (-3%) procedure. Citrate synthase activity increased significantly with HIT and decreased significantly with ROT. Muscle glycogen concentration increased significantly after ROT and HIT, and strength increased after all three tapers. Total blood volume increased significantly after HIT and decreased after ROT.(ABSTRACT TRUNCATED AT 250 WORDS)     

Optimizing athletic performance by influence curves.

Recent application of modeling techniques to physical training has opened the possibility for prediction from training. Solution of the inverse problem, determining a training program to produce a desired performance at a specific time, is also possible and may yield strategies for achieving better training and tapering (complete or relative rest for a period before competition) regimens for competitive athletes. A mathematical technique derived from model theory is described in this paper that allows the design of an optimal strategy of physical preparation for an individual to do well in a single future competitive event or cluster of events. Simulation results, using default parameters of a training model, suggest that presently accepted forms of taper for competition may remain too rigorous and short in duration to achieve the best result possible from the training undertaken.     

不同林分起源的相容性生物量模型构建

目前为止已有不同方法构建生物量相容性模型,但不同林分起源的生物量相容性模型很少报道。针对此问题,以150株南方马尾松(Pinus masson iana)地上生物量数据为例,利用比例平差法和非线性联立方程组法建立不同起源地上生物量以及干材、干皮、树枝和树叶各分项生物量相容的通用性模型。根据分配层次不同,两种方法又各自考虑总量直接控制和分级联合控制两种方案。从直径、树高、地径、枝下高和冠幅5个林分变量中选取不同的变量构建一元、二元和三元生物量模型,并利用加权最小二乘回归法消除生物量模型中存在的异方差性。结果为:比例平差法和非线性联立方程组法都能有效保证各分项生物量总和等于总生物量,模型预测精度满足要求。总体而言,非线性联立方程组方法比比例平差方法精度高,同时两种方法中总量直接控制法比分级联合控制法预测效果好;各分项生物量模型本身作为权函数能有效消除异方差;各分项对应的三元生物量模型预测精度最高,其次是二元生物量模型,最低是一元生物量模型,但这些差异不是很大。总之,为权衡考虑模型预测精度和调查成本,建议把直径和树高作为协变量利用总量直接控制非线性联立方程组法对不同起源生物量建模。     

Resource patch size and flexible foraging in white-faced capuchins (Cebus capucinus)

White-faced capuchins (Cebus capucinus)on Barro Colorado Island, Panama, have a flexible foraging strategy. Typically, foraging party size is small and individuals feed dispersed from one another. When seasonal fruiting of large volume trees occurs, the majority of the group forages simultaneously. As C. capucinusdo not display a rigorous dominance structure and there are few indications that individuals or coalitions monopolize food patches,individuals are expected to display scramble strategies instead of high frequencies of contest competition. I recorded foraging party size (simultaneous foragers), the total number of animals to feed successively, and the diameter at breast height (DBH) of fruit trees used in two habituated troops. Individuals in each group spent a substantial amount of time — 65 and 48% of foraging time for each group — foraging in party sizes of one. Monkeys predominantly foraged alone in small trees (0- to 20- cm DBH), successively in medium trees (21- to 60- cm DBH), and simultaneously in large trees (>61- cm DBH). They used small trees more frequently than all other tree classes. In medium-sized trees, although fruit was plentiful, space was limited. In these trees Cebusforaged successively. In large-volume trees, space and fruit were abundant and several individuals fed together. As the DBH of fruiting trees increased, the average foraging party size increased exponentially. Cebus capucinusat Barro Colorado Island modify their foraging party size to adapt to the seasonal patterns of fruit production.     

小兴安岭红松针阔混交林立木掘根特征

测定了小兴安岭红松针阔混交林76个样方(20 m×20 m)内127株掘根倒木的形态指标、根系指标和立地条件,分析掘根差异的影响因子以及掘根木外形参数与根部土壤的关系.结果表明：不同树种的掘根数量差异显著,以冷杉受掘根危害最重,红松次之,榆树最轻.阔叶树种的抗掘根性能优于针叶树种.树木对掘根的抗性随胸径和树高的增加先快速降低后逐渐升高,分别在20 cm径级和14 m高度级处达到最低;树木尖削度和树冠投影面积越小,抗掘根性能越好;掘根率与林分密度呈显著负相关.处于湿地、缓坡、中低海拔处或迎风坡的树木掘根风险更大.掘根时受扰动土壤的深度、面积及体积与倒木胸径、树高及材积之间呈显著相关.     

Ecological adaptation of the shea butter tree (Vitellaria paradoxa C.F. Gaertn.) along climatic gradient in Bénin,West Africa

The ecological adaptation of shea butter trees was assessed based on their dendrometric and production traits in four shea butter tree parks occurring in different climatic zones of Bénin. A total of 99 rectangular plots of 50 × 30 m were established within the four parks according to a random sampling scheme. In each plot, all trees with a diameter at breast height (dbh) >10 cm were inventoried and measured for stem and crown diameters, and total height. The production of 120 productive shea butter trees was quantified. Collected data were used to compute structural parameters for each park. Moreover, stem diameter and height structures of the trees were established. Principal component analysis was performed on the dendrometric variables, and the first three components were correlated with the climatic parameters. Results revealed significant differences between parks in most of the dendrometric and production parameters of shea butter trees. For all the four parks, stem diameter and height structures present a Gaussian shape with left dissymmetry. In the Guinean zone, shea butter trees develop large crowns but produce little quantities of fruits, whereas in the Sudanian regions, the opposite trend was observed.     

USE OF NATIVE PLANTS FOR REMEDIATION OF TRICHLOROETHYLENE: I. DECIDUOUS TREES

Phytoremediation of trichloroethylene (TCE) can be accomplished using fast-growing, deep-rooting trees. The most commonly used tree for phytoremediation of TCE has been the hybrid poplar. This study looks at native southeastern trees of the United States as alternatives to the use of hybrid poplar. The use of native trees for phytoremediation allows for simultaneous restoration of contaminated sites. A 2-mo, greenhouse-based study was conducted to determine if sycamore (Plantanus L.), eastern cottonwood (Populus deltoides), sweetgum (Liquidambar styraciflua L.), and willow (Salix sachalinensis) trees possess the ability to degrade TCE by assessing TCE metabolite formation in the plant tissue. In addition to the metabolic capabilities of each tree species, growth parameters were measured including change in height, water usage, total fresh weight of each tissue type, and calculated total leaf surface area. Willow trees had the greatest increase in height among all trees tested; however, at higher concentrations TCE inhibits growth. Sycamore trees had the highest overall leaf surface area and total biomass, which correlated with sycamore trees also having the highest average water usage over the course of the experiment. Carbon tubes used to sample transpiration gases from sycamore, sweetgum, and cottonwood trees did not contain detectable levels of TCE. Tenex sample collection tubes used to sample willow trees during TCE exposure showed average TCE concentrations of up to 0.354 ng TCE cm^?2 leaf tissue. All exposed trees contained TCE in the root, stem, and leaf tissues. The concentration of TCE remaining in tissues at the conclusion of the experiment varied, with the highest levels found in the roots and the lowest levels found in the leaves. Metabolites were also observed in different tissue types of all trees tested. The highest concentrations of trichloroacetic acid were observed in the leaves of the sycamore trees and cottonwood trees. Based on the growth parameters tested and the ability to metabolize TCE, sycamore and native cottonwood species are the best candidates for phytoremediation from this study.     

Size-dependent Allometry of Tree Height, Diameter and Trunk-taper

The allometry of tree height with respect to trunk diameterand the allometry of trunk diameter with respect to distancefrom the top of the tree (i.e. trunk taper) were determinedfor 27 Robinia pseudoacacia trees differing in age and sizegrowing in an open field. The allometric (scaling) exponentfor height was > 1 for small and young trees and decreasedto 2/3 and then 1/2 as tree size and age increased. Similarly,the exponent for taper was > 1 near the tips of young andold trunks and converged onto values of 2/3 and 1/2 toward thebase of mature tree trunks. These observations indicate thata single 'optimal mechanical design principle' (i.e. elastic,stress or geometric self-similarity) neither holds true throughoutthe lifetime of R. pseudoacacia trees, nor does a single designprinciple govern the taper of a trunk throughout its entirelength. Rather, over the course of growth and development, theallometry of R. pseudoacacia tree height and trunk taper progressivelychanges, complying with geometric self-similarity for youngplants (and young portions of old plants) and subsequently givingthe appearance of elastic or stress self-similarity as plants(or portions of plants) get older and therefore larger. Analysesof published (and new) data suggest that the conclusions drawnfor R. pseudoacacia trees are likely to hold true for othertree species because stem growth in diameter is 'indeterminate'whereas growth in overall tree height is asymptotic and thereforeessentially 'determinate'.Copyright 1995, 1999 Academic Press Scaling, woody plants, Robinia pseudoacacia     

Allometric Models for Tree Volume and Total Aboveground Biomass in a Tropical Humid Forest in Costa Rica1

Allometric equations for the estimation of tree volume and aboveground biomass in a tropical humid forest were developed based on direct measurements of 19 individuals of seven tree species in Northern Costa Rica. The volume and the biomass of the stems represented about two‐thirds of the total volume and total aboveground biomass, respectively. The average stem volume varied between 4 and 11 Mg/tree and the average total aboveground biomass ranged from 4 to 10 mg/tree. The mean specific gravity of the sampled trees was 0.62 ± 0.06 (g/cm³). The average biomass expansion factor was 1.6 ± 0.2. The best‐fit equations for stem and total volume were of logarithmic form, with diameter at breast height (R²= 0.66 ? 0.81) as an independent variable. The best‐fit equations for total aboveground biomass that were based on combinations of diameter at breast height, and total and commercial height as independent variables had R² values between 0.77 and 0.87. Models recommended for estimating total aboveground biomass are based on diameter at breast height, because the simplicity of these models is advantageous. This variable is easy to measure accurately in the field and is the most common variable recorded in forest inventories. Two widely used models in literature tend to underestimate aboveground biomass in large trees. In contrast, the models developed in this study accurately estimate the total aboveground biomass in these trees.     

Wind-induced stresses in cherry trees: evidence against the hypothesis of constant stress levels

We calculated the wind-induced bending moments and stresses generated in the stems of five Prunus serotina conspecifics differing in height and canopy shape and size (based on detailed measurements of stem projected area and location with respect to ground level) to test the hypothesis that wind-loads generate uniform and constant stress levels along the lengths of tree twigs, branches, and trunks. These calculations were performed using five different wind speed profiles to evaluate the relative importance of the shape of wind speed profiles versus the ’geometry’ of tree shape on stem stress distributions and magnitudes. Additionally, we evaluated the effect of absolute tree size and stem taper on wind- induced stresses by scaling the size of smaller conspecifics to the absolute height of the largest of the five trees yet retaining the original stem proportions (i.e., diameter relative to stem length) for each plant. Finally, we also determined how the factor of safety for wind-loading (i.e., the quotient of stem yield stress and wind-load stress) changed as a function of tree size (and, presumably, age). Our results indicate that wind-load stress levels (1) vary along stem length even for the same wind speed profile and the same maximum wind speed; (2) would increase to dangerous levels with increasing tree height if it were not for ontogenetic changes in stem taper and canopy shape that reduce stress intensities to manageable levels; (3) tend to be more dependent on stem taper and canopy shape and size than on the shape of the wind speed profile; and (4) the factor of safety against wind-induced mechanical failure decreases as trees get larger, but varies along the length of large trees such that preferential stem failure is likely and functionally adaptive. We thus (1) reject the hypothesis of constant wind-induced stress levels; (2) support the view that size-dependent changes in stem taper are required to maintain wind-load mechanical reliability; and (3) suggest that certain portions of mature trees are ’designed’ to fail under high winds speeds, thereby reducing drag and the bending moments and stresses experienced by trunks. Received: 24 May 1999 / Accepted: 8 October 1999     

Improving accuracy of breeding values by incorporating genomic information in spatial-competition mixed models

Climate change and the increasing demand for sustainable energy resources require urgent strategies to increase the accuracy of selection in tree breeding (associated with higher gain). We investigated the combined pedigree and genomic-based relationship approach and its impact on the accuracy of predicted breeding values using data from 5-year-old Eucalyptus grandis progeny trial. The number of trees that can be genotyped in a tree breeding population is limited; therefore, the combined approach can be a feasible and efficient strategy to increase the genetic gain and provide more accurate predicted breeding values. We calculated the accuracy of predicted breeding values for two growth traits, diameter at breast height and total height, using two evaluation approaches: the combined approach and the classical pedigree-based approach. We also investigated the influence of two different trait heritabilities as well as the inclusion of competition genetic effects or environmental heterogeneity in an individual-tree mixed model on the estimated variance components and accuracy of breeding values. The genomic information of genotyped trees is automatically propagated to all trees with the combined approach, including the non-genotyped mothers. This increased the accuracy of overall breeding values, except for the non-genotyped trees from the competition model. The increase in the accuracy was higher for the total height, the trait with low heritability. The combined approach is a simple, fast, and accurate genomic selection method for genetic evaluation of growth traits in E. grandis and tree species in general. It is simple to implement in a traditional individual-tree mixed model and provides an easy extension to individual-tree mixed models with competition effects and/or environmental heterogeneity.     

Testing a method for reconstructing structural development of even-aged Abies sachalinensis stands

Accuracy of a stand reconstruction technique was examined by comparing the estimated values of the aboveground biomass, total stem volume, stem volume growth, and stand density of Abies sachalinensis stands to those observed between 1980 and 1998 in Hokkaido, northern Japan. Census data from two stands established in 1973, one fertilized and the other unfertilized, were used for the examination. The stand statistics in the past were estimated from the DBH and height of individual trees measured in 1998, data on the aboveground biomass and stem volume with bark for nine living trees of various sizes harvested in each plot in 1998 or in 1999, and data from the stem analysis of the same harvested trees. We showed that the reconstructed patterns of the frequency distribution in aboveground biomass and in stem volume were generally the same as those observed in both plots and in any year in the past (except for 1982 and/or 1980), and that the reproduced patterns of stand development over time were similar to those observed directly in the past. Accuracy in predicting stand statistics was generally in the order of ±10% relative error. We consider that the present method of stand reconstruction could be used to estimate aboveground biomass, total stem volume, and stem volume growth of a stand in the past. Interpretation of results for the early years (1982 and 1980) and for the stand density requires caution.     

Estimating Stem Volume Using QuickBird Imagery and Allometric Relationships for Open Populus xiaohei Plantations

There has been a great deal of interest in studying the crown of trees using remote sensing data. In this study, crown width was extracted from high-resolution QuickBird images for open Populus xiaohei plantations. Regression models for predicting the individual stem volumes of Populus xiaohei were established using extracted crown width, as well as estimated tree parameters （i.e. diameter at breast height [DBH] and tree height） as predictors. Our results indicated that crown width could be accurately extracted from QuickBird images using a multi-scale segmentation approach with a mean relative error of 5.74%, especially for wide-spacing stands. Using either extracted crown width alone or with estimated DBH and tree height can successfully estimate individual stem volume of Populus xiaohei with the R2 value ranging from 0.87 to 0.92 depending on different model forms. In particular, the two second-order polynomial models （model 2 and model 6）, based on QuickBird image-derived crown widths and estimated DBH and tree heights, respectively, were the best at describing the relationship between stem volume and tree characteristics.     

Estimating Stem Volume Using QuickBird Imagery and Allometric Relationships for Open Populus xiaohei Plantations

There has been a great deal of interest in studying the crown of trees using remote sensing data.In this study,crownwidth was extracted from high-resolution QuickBird images for open Populus xiaohei plantations.Regression modelsfor predicting the individual stem volumes of Populus xiaohei were established using extracted crown width,as well asestimated tree parameters(i.e.diameter at breast height[DBH]and tree height)as predictors.Our results indicated thatcrown width could be accurately extracted from QuickBird images using a multi-scale segmentation approach with a meanrelative error of 5.74%,especially for wide-spacing stands.Using either extracted crown width alone or with estimatedDBH and tree height can successfully estimate individual stem volume of Populus xiaohei with the R~2 value ranging from0.87 to 0.92 depending on different model forms.In particular,the two second-order polynomial models(model2 andmodel 6),based on QuickBird image-derived crown widths and estimated DBH and tree heights,respectively,were the bestat describing the relationship between stem volume and tree characteristics.     

The evolution of alternative mating strategies in variable environments

Summary We assessed the influence of phenotypic plasticity in age at maturity on the maintenance of alternative mating strategies in male Atlantic salmon,Salmo salar. We calculated the fitness,r, associated with the parr and the anadromous strategies, using age-specific survival data from the field and strategy-specific fertilization data from the laboratory. The fitness of each strategy depended largely on mate competition (numbers of parr per female, i.e. parr frequency) and on age at maturity. Fitness declined with increasing numbers of parr per female with equilibrium frequencies (at which the fitnesses of each strategy are equal) being within the range observed in the wild. Equilibrium parr frequencies declined with decreasing growth rate and increasing age at maturity. Within populations, the existence of multiple age-specific sets of fitness functions suggests that the fitnesses of alternative strategies are best represented as multidimensional surfaces. The points of intersection of these surfaces, whose boundaries encompass natural variation in age at maturity and mate competition, define an evolutionarily stable continuum (ESC) of strategy frequencies along which the fitnesses associated with each strategy are equal. We propose a simple model that incorporates polygenic thresholds of a largely environmentally-controlled trait (age at maturity) to provide a mechanism by which an ESC can be maintained within a population. An indirect test provides support for the prediction that growth-rate thresholds for parr maturation exist and are maintained by stabilizing selection. Evolutionarily stable continua, maintained by negative frequency-dependent selection on threshold traits, provide a theoretical basis for understanding how alternative life histories can evolve in variable environments.