Computing the selection gradient and evolutionary response of an infinite-dimensional trait

Beder, Jay H.; Gomulkiewicz, Richard

doi:10.1007/s002850050102

Cited by 26 publications

(20 citation statements)

References 15 publications

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“…The theory of optimal control has been used in the context of function-valued traits in game theory (Hamelin and Lewis, 2010), in quantitative genetics (Gomulkiewicz and Kirkpatrick, 1992;Gomulkiewicz and Beder, 1996;Beder and Gomulkiewicz, 1998;Jaffrézic and Pletcher, 2000;Kingsolver et al, 2001), and in life-history theory (Perrin and Sibly, 1993;Gilchrist et al, 2006). The novel feature considered in this article is the extension of the methods of optimal control theory to problems with environmental feedback, which is essential for tackling biologically realistic models.…”

Section: Introductionmentioning

confidence: 99%

Function-valued adaptive dynamics and optimal control theory

2012

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In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler's equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin's maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.

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Section: Introductionmentioning

confidence: 99%

Function-valued adaptive dynamics and optimal control theory

2012

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“…(b) Response to selection Given a population's additive genetic covariance function and either the selection differential or the selection gradient, we can calculate the population's mean function in the next generation (Kirkpatrick & Heckman 1989;Beder & Gomulkiewicz 1998) M 0 ðtÞ Z MðtÞ C Ð Gðt; xÞjðxÞ dx:…”

Section: Selection On Curvesmentioning

confidence: 99%

Up hill, down dale: quantitative genetics of curvaceous traits

Meyer

Kirkpatrick

2005

Phil. Trans. R. Soc. B

100

107

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'Repeated' measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as 'functionvalued' (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances.Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multidimensional analyses are discussed.

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“…Equation (1 b) represents the selection response in terms of the selection gradient, β(x). This is a measure of the strength of directional selection (Lande & Arnold, 1983 ;Kirkpatrick & Heckman, 1989 ;Kirkpatrick, 1993 ;Gomulkiewicz & Beder, 1996 ;Beder & Gomulkiewicz, 1998). Multiplying a selection gradient by the corresponding phenotypic standard deviation gives the selection intensity, a non-dimensional quantity that is widely used by breeders (Falconer & Mackay, 1996, p. 189).…”

Section: The Infinite-dimensional Frameworkmentioning

confidence: 99%

Artificial selection on phenotypically plastic traits

Kirkpatrick

Bataillon²

1999

Genet. Res.

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Many phenotypes respond physiologically or developmentally to continuously distributed environmental variables such as temperature and nutritional quality. Information about phenotypic plasticity can be used to improve the efficiency of artificial selection. Here we show that the quantitative genetic theory for 'infinite-dimensional' traits such as reaction norms provides a natural framework to accomplish this goal. It is expected to improve selection responses by making more efficient use of information about environmental effects than do conventional methods. The approach is illustrated by deriving an index for mass selection of a phenotypically plastic trait. We suggest that the same approach could be extended directly to more general and efficient breeding schemes, such as those based on general best linear unbiased prediction. Methods for estimating genetic covariance functions are reviewed.

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Computing the selection gradient and evolutionary response of an infinite-dimensional trait

Cited by 26 publications

References 15 publications

Function-valued adaptive dynamics and optimal control theory

Function-valued adaptive dynamics and optimal control theory

Up hill, down dale: quantitative genetics of curvaceous traits

Artificial selection on phenotypically plastic traits

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