![]() ![]() When fitness is a linear function of a trait, its second derivatives are zero, and there is selection to shift the trait's mean value. The first derivatives of fitness are selection gradients (Lande 1982). The second derivatives of fitness with respect to trait values have consequences for selection. Characterizing nonlinear selection processes 1 of Carslake, Townley & Hodgson ( 2008), for example, adding the second-order terms may actually reduce the accuracy of the approximation. (eqn 1)We caution that although this may, in some cases, provide a more accurate calculation, this is not guaranteed. The sensitivity of the elasticity of growth rate to parameters similarly depends on second derivatives (Caswell, 1996, 2001). These second-order effects are quantified by the sensitivity, with respect to a parameter θ j, of the sensitivity of λ to another parameter θ i, that is, by the second derivatives. However, such perturbations also affect the sensitivity itself, that is, sensitivity is 'situational' (Stearns 1992). The sensitivity of growth rate provides insight into the population response to parameter perturbations. ![]() Second-order sensitivity analysis and growth rate estimation Several of these applications are summarized in Table 1 and described in the following sections. characterizing nonlinear selection gradients and evolutionary equilibria). assessing and improving recommendations from sensitivity analysis, approximating the sensitivities of stochastic growth rates) and evolution (e.g. The second derivatives of growth rates have applications in both ecology (e.g. Applications of second derivatives of growth rates with λ as a fitness measure, the sensitivity of λ with respect to a parameter is the selection gradient on that parameter) (Caswell 2001). These first derivatives are used to project the effects of vital rate changes due to environmental or management perturbations, uncertainty in parameter estimates and phenotypic evolution (i.e. Sensitivities (first partial derivatives) of λ with respect to relevant parameters quantify how population growth responds to vital rate perturbations. The discrete-time population growth rate λ is given by the dominant eigenvalue of the population projection matrix. Measures of population growth rate, including the discrete-time growth rate λ, the continuous-time growth rate r = log λ and the net reproductive rate R 0, are of particular interest. Using matrix population models, ecological indices can be calculated as functions of vital rates such as survival or fertility. We also illustrate several ecological and evolutionary applications for these second derivative calculations with a case study for the tropical herb Calathea ovandensis.We present a suite of formulae for the second derivatives of each growth rate and show how to compute these derivatives with respect to projection matrix entries and to lower-level parameters affecting those matrix entries.Using matrix calculus, we derive the second derivatives of three population growth rate measures: the discrete-time growth rate λ, the continuous-time growth rate r = log λ and the net reproductive rate R 0, which measures per-generation growth.The second derivatives quantify the response of sensitivity results to perturbations, provide a classification of types of selection and provide one way to calculate sensitivities of the stochastic growth rate. Second derivatives of the population growth rate measure the curvature of its response to demographic, physiological or environmental parameters.With the example of American ginseng it was shown that evalution of r by definition and model approaches could produce opposite results. The calculation of r by definition requires the data on the dynamics of population numbers, whereas calculation on the basis of the model requires the demographic tables of birth and death rate, but not the population numbers. However this model requires simultaneous realization of several assumptions improbable for natural populations: exponential change in population size, stable age structure and maintaining constant age-dependent birth and death rates. The fundamentally different approach is based on the calculation of r within the framework of demographic model, realized as Euler - Lotka equation or population projection matrices. The common opinion considering the equation as suitable only for exponentially growing population was found to be incorrect. It was shown that well known equation r = ln/(t2 - t1) is the definition of the average value of intrinsic growth rate of population r within any given interval of time t2-t1 and changing arbitrarity its numbers N(t). ![]()
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