The Gompertz model is well known and widely used in many aspects of biology. It has been frequently used to describe the growth of animals and plants, as well as the number or volume of bacteria and cancer cells. Numerous parametrisations and re-parametrisations of varying usefulness are found in the literature, whereof the Gompertz-Laird is one of the more commonly used. Here, we review, present, and discuss the many re-parametrisations and some parameterisations of the Gompertz model, which we divide into Ti (type I)- and W0 (type II)-forms. In the W0-form a starting-point parameter, meaning birth or hatching value (W0), replaces the inflection-time parameter (Ti). We also propose new “unified” versions (U-versions) of both the traditional Ti -form and a simplified W0-form. In these, the growth-rate constant represents the relative growth rate instead of merely an unspecified growth coefficient. We also present U-versions where the growth-rate parameters return absolute growth rate (instead of relative). The new U-Gompertz models are special cases of the Unified-Richards (U-Richards) model and thus belong to the Richards family of U-models. As U-models, they have a set of parameters, which are comparable across models in the family, without conversion equations. The improvements are simple, and may seem trivial, but are of great importance to those who study organismal growth, as the two new U-Gompertz forms give easy and fast access to all shape parameters needed for describing most types of growth following the shape of the Gompertz model.

Aim This paper reviews possible candidate models that may be used in theoretical modelling and empirical studies of species-area relationships (SARs). The SAR is an important and well-proven tool in ecology. The power and the exponential functions are by far the models that are best known and most frequently applied to species-area data, but they might not be the most appropriate. Recent work indicates that the shape of species-area curves in arithmetic space is often not convex but sigmoid and also has an upper asymptote.Methods Characteristics of six convex and eight sigmoid models are discussed and interpretations of different parameters summarized. The convex models include the power, exponential, Monod, negative exponential, asymptotic regression and rational functions, and the sigmoid models include the logistic, Gompertz, extreme value, Morgan-Mercer-Flodin, Hill, Michaelis-Menten, Lomolino and Chapman-Richards functions plus the cumulative Weibull and beta-P distributions.Conclusions There are two main types of species-area curves: sample curves that are inherently convex and isolate curves, which are sigmoid. Both types may have an upper asymptote. A few have attempted to fit convex asymptotic and/or sigmoid models to species-area data instead of the power or exponential models. Some of these or other models reviewed in this paper should be useful, especially if species-area models are to be based more on biological processes and patterns in nature than mere curve fitting. The negative exponential function is an example of a convex model and the cumulative Weibull distribution an example of a sigmoid model that should prove useful. A location parameter may be added to these two and some of the other models to simulate absolute minimum area requirements.

Species‐area relationships (SARs) are among the most studied phenomena in ecology, and are important both to our basic understanding of biodiversity and to improving our ability to conserve it. But despite many advances to date, our knowledge of how various factors contribute to SARs is limited, searches for single causal factors are often inconclusive, and true predictive power remains elusive. We believe that progress in these areas has been impeded by 1) an emphasis on single‐factor approaches and thinking of factors underlying SARs as mutually exclusive hypotheses rather than potentially interacting processes, and 2) failure to place SAR‐generating factors in a scale‐dependent framework. We here review mathematical, ecological, and evolutionary factors contributing to species‐area relationships, synthesizing major hypotheses from the literature in a scale‐dependent context. We then highlight new research directions and unanswered questions raised by this scale‐dependent synthesis.

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This paper has extended and updated my earlier list and analysis of candidate models used in theoretical modelling and empirical examination of species–area relationships (SARs). I have also reviewed trivariate models that can be applied to include a second independent variable (in addition to area) and discussed extensively the justifications for fitting curves to SARs and the choice of model. There is also a summary of the characteristics of several new candidate models, especially extended power models, logarithmic models and parameterizations of the negative‐exponential family and the logistic family. I have, moreover, examined the characteristics and shapes of trivariate linear, logarithmic and power models, including combination variables and interaction terms. The choice of models according to best fit may conflict with problems of non‐normality or heteroscedasticity. The need to compare parameter estimates between data sets should also affect model choice. With few data points and large scatter, models with few parameters are often preferable. With narrow‐scale windows, even inflexible models such as the power model and the logarithmic model may produce good fits, whereas with wider‐scale windows where inflexible models do not fit well, more flexible models such as the second persistence (P2) model and the cumulative Weibull distribution may be preferable. When extrapolations and expected shapes are important, one should consider models with expected shapes, e.g. the power model for sample area curves and the P2 model for isolate curves. The choice of trivariate models poses special challenges, which one can more effectively evaluate by inspecting graphical plots.