Calculating second derivatives of population growth rates for ecology and evolution
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KeywordEigenvalues; Hessian matrix; Invasion exponent; Matrix population models; Net reproductive rate; Sensitivity analysis
Second derivatives of the population growth rate measure the curvature of its response to demographic, physiological or environmental parameters. 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. 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 R0, which measures per-generation growth. 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. We also illustrate several ecological and evolutionary applications for these second derivative calculations with a case study for the tropical herb Calathea ovandensis.
© The Author(s), 2014. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Methods in Ecology and Evolution 5 (2014): 473–482, doi:10.1111/2041-210X.12179.
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