This handbook is the fourth volume in a series of volumes devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience. It covers a variety of problems in ordinary differential equations. It provides pure mathematical and real world applications. It is written for mathematicians and scientists of many related fields.

Ordinary differential equations

Body condition is assumed to influence an animal’s health and fitness. Various non-destructive methods based on body mass and a measure of body length have been used as condition indices (CIs), but the dominant method amongst ecologists is currently the calculation of residuals from an ordinary least squares (OLS) regression of body mass against length. Recent studies of energy reserves in small mammals and starlings claimed to validate this method, although we argue that they did not include the most appropriate tests since they compared the CI with the absolute size of energy reserves. We present a novel CI (the ‘scaled mass index’) based on the central principle of scaling, with important methodological, biological and conceptual advantages. Through a reanalysis of data from small mammals, starlings and snakes, we show that the scaled mass index is a better indicator of the relative size of energy reserves and other body components than OLS residuals, performing better in all seven species and in 19 out of 20 analyses. We also present an empirical and theoretical comparison of the scaled mass index and OLS residuals as CIs. We argue that the scaled mass index is a useful new tool for ecologists.

New perspectives for estimating body condition from mass/length data: the scaled mass index as an alternative method

In this review I show that the '3/4-power scaling law' of metabolic rate is not universal, either within or among animal species. Significant variation in the scaling of metabolic rate with body mass is described mainly for animals, but also for unicells and plants. Much of this variation, which can be related to taxonomic, physiological, and/or environmental differences, is not adequately explained by existing theoretical models, which are also reviewed. As a result, synthetic explanatory schemes based on multiple boundary constraints and on the scaling of multiple energy-using processes are advocated. It is also stressed that a complete understanding of metabolic scaling will require the identification of both proximate (functional) and ultimate (evolutionary) causes. Four major types of intraspecific metabolic scaling with body mass are recognized [based on the power function R=aMb, where R is respiration (metabolic) rate, a is a constant, M is body mass, and b is the scaling exponent]: Type I: linear, negatively allometric (b 1) to one or two later phases of negative allometry (b<1). Ontogenetic changes in the metabolic intensity of four component processes (i.e. growth, reproduction, locomotion, and heat production) appear to be important in these different patterns of metabolic scaling. These changes may, in turn, be shaped by age (size)-specific patterns of mortality. In addition, major differences in interspecific metabolic scaling are described, especially with respect to mode of temperature regulation, body-size range, and activity level. A 'metabolic-level boundaries hypothesis' focusing on two major constraints (surface-area limits on resource/waste exchange processes and mass/volume limits on power production) can explain much, but not all of this variation. My analysis indicates that further empirical and theoretical work is needed to understand fully the physiological and ecological bases for the considerable variation in metabolic scaling that is observed both within and among species. Recommended approaches for doing this are discussed. I conclude that the scaling of metabolism is not the simple result of a physical law, but rather appears to be the more complex result of diverse adaptations evolved in the context of both physico-chemical and ecological constraints.

Beyond the '3/4-power law': variation in the intra- and interspecific scaling of metabolic rate in animals.

Many investigators use the reduced major axis (RMA) instead of ordinary least squares (OLS) to define a line of best fit for a bivariate relationship when the variable represented on the X-axis is measured with error. OLS frequently is described as requiring the assumption that X is measured without error while RMA incorporates an assumption that there is error in X. Although an RMA fit actually involves a very specific pattern of error variance, investigators have prioritized the presence versus the absence of error rather than the pattern of error in selecting between the two methods. Another difference between RMA and OLS is that RMA is symmetric, meaning that a single line defines the bivariate relationship, regardless of which variable is X and which is Y, while OLS is asymmetric, so that the slope and resulting interpretation of the data are changed when the variables assigned to X and Y are reversed. The concept of error is reviewed and expanded from previous discussions, and it is argued that the symmetry-asymmetry issue should be the criterion by which investigators choose between RMA and OLS. This is a biological question about the relationship between variables. It is determined by the investigator, not dictated by the pattern of error in the data. If X is measured with error but OLS should be used because the biological question is asymmetric, there are several methods available for adjusting the OLS slope to reflect the bias due to error. RMA is being used in many analyses for which OLS would be more appropriate.

Use and misuse of the reduced major axis for line-fitting

The theory of asymptotic speeds of spread and monotone traveling waves is established for a class of monotone discrete and continuous-time semiflows and is applied to a functional differential equation with diffusion, a time-delayed lattice population model and a reaction-diffusion equation in an infinite cylinder. c � 2006 Wiley Periodicals, Inc.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.20221

Asymptotic speeds of spread and traveling waves for monotone semiflows with applications

Body condition can have important fitness consequences, but measuring body condition of live animals from wild populations has been the subject of much recent debate. Using the residuals from a regression of body mass on a linear measure of body size is one of the most common methods of measuring condition and has been used in many vertebrate taxa. Recently, the use of this method has been criticized because assumptions are likely violated. We tested several assumptions regarding the use of this method with body composition and morphometric data from five species of small mammals and with statistical simulations. We tested the assumptions that the relationship between body mass and body size is linear, and that the proportion of mass associated with energy reserves is independent of body size. In addition, we tested whether the residuals from reduced major axis (RMA) regression or major axis (MA) regression performed better than the residuals from ordinary least squares (OLS) regression as indices of body condition. We found no evidence of nonlinear relationships between body mass and body size. Relative energy reserves (fat and lean dry mass) were generally independent or weakly dependent on body size. Residuals from MA and RMA regression consistently explained less variation in body composition than OLS regression. Using statistical simulations, we compared the effects of violations of the assumption that true condition and residual indices are independent of body size on the OLS, MA, and RMA procedures and found that OLS performed better than the RMA and MA procedures. Despite recent criticisms of residuals from mass-size OLS regressions, these indices of body condition appear to satisfy critical assumptions. Although some caution is warranted when using residuals, especially when both inter- individual variation in body size and measurement error are high, we found no reason to reject OLS residuals as legitimate indices of body condition.

Restitution of mass–size residuals: validating body condition indices

http://cermics.enpc.fr/~al-hajm/ref-tr-wave/b.z_1992.pdf

Existence of traveling wavefront solutions for the discrete Nagumo equation

Traveling wavefronts for the discrete Fisher's equation

http://research.amnh.org/users/rfr/koons05a.pdf

Transient population dynamics:relations to life history and initial population state

Matrix population models have become standard tools for the demographic analysis of age- or stage-structured populations. Although age-classified (Leslie) matrix models make maximum use of age-specific demographic data, age at first reproduction, which has been suggested to be an important life-history variable, does not appear as an explicit parameter in these models. Consequently, the sensitivity of population growth rate to changes in age at first reproduction cannot be calculated using standard techniques. Age-specific demographic data to parameterize age-structured models are difficult to collect, and models that can be parameterized with partial demographic data (“partial life-cycle models”) have been developed. Partial life-cycle models are based on life-history stages, and these models can also be used to calculate sensitivity of population growth rate to changes in various life-history variables, including ages at first and last reproduction. Here, we present a partial life-cycle model appropriate for situations where demographic data are collected immediately after the birth pulse (post-breeding census). We present methods of parameterizing the partial life-cycle model, and derive formulas for calculating the sensitivity of the population growth rate to changes in model parameters, including ages at first and last reproduction. We analyzed life-table data for several species of mammals using the partial life-cycle model and found that results of our partial life-cycle model compare favorably with those obtained from the corresponding age-classified models.

https://www.jstor.org/stable/pdfplus/2679913.pdf

Bertram Zinner

Papers

Restitution of mass–size residuals: validating body condition indices

Existence of traveling wavefront solutions for the discrete Nagumo equation

Traveling wavefronts for the discrete Fisher's equation

Transient population dynamics:relations to life history and initial population state

Partial life-cycle analysis: a model for birth-pulse populations