Measuring Probabilistic Reaction Norms for Age and Size at Maturation

Abstract

We present a new probabilistic concept of reaction norms for age and size at maturation that is applicable when observations are carried out at discrete time intervals. This approach can also be used to estimate reaction norms for age and size at metamorphosis or at other ontogenetic transitions. Such estimations are critical for understanding phenotypic plasticity and life-history changes in variable environments, assessing genetic changes in the presence of phenotypic plasticity, and calibrating size-and age-structured population models. We show that previous approaches to this problem, based on regressing size against age at maturation, give results that are systematically biased when compared to the probabilistic reaction norms. The bias can be substantial and is likely to lead to qualitatively incorrect conclusions; it is caused by failing to account for the probabilistic nature of the maturation process. We explain why, instead, robust estimations of maturation reaction norms should be based on logistic regression or on other statistical models that treat the probability of maturing as a dependent variable. We demonstrate the utility of our approach with two examples. First, the analysis of data generated for a known reaction norm highlights some crucial limitations of previous approaches. Second, application to the northeast arctic cod (Gadus morhua) illustrates how our approach can be used to shed new light on existing real-world data.

Fisheries-induced evolutionary changes in maturation reaction norms in North Sea sole Solea solea

“…The PMRN is defined as the probability that an immature individual becomes mature at a particular age and size (Heino et al 2002) and thereby overcomes the confounding effects of growth and mortality. For sole, first time and repeat spawners cannot be distinguished, therefore the probability of becoming mature (p) was estimated by a refinement of the PMRN method (Barot et al 2004) and is estimated on a yearly basis as: (2) where m is the probability of being mature, a is age, s is size, x is any other additional factor possibly affecting maturation (e.g.…”

“…5), but noise in our data was too high to detect a trend in the slopes. Based on the assumption that environmental variation affecting maturation is reflected in variations in the growth rate (Heino et al 2002), a genetic change in maturation can thus be disentangled from the phenotypic plasticity in maturation. The length-based reaction norm is more reliably estimated than the weight-based reaction norm because of the ambiguity in data selection and corresponding interpretation for the weights (see below) resulting in a smaller data set and higher uncertainty in the reaction norm estimates.…”

“…By definition, a reaction norm describes which phenotypes will be expressed by a genotype under a certain range of environmental conditions (Stearns & Koella 1986). The probabilistic maturation reaction norm (PMRN) has been developed recently to estimate the reaction norm for the onset of maturity conditional on age and size (Heino et al 2002, Barot et al 2004). Environmental variability is assumed to be reflected in differences in somatic growth, so that any environmental factor that has its effect through growth can be disentangled.…”

Phenotypic convergence along a gradient of predation risk

“…The first group of analyses tends to focus on single traits in a few environments (for a review see [4]) or at most two traits (e.g. [5,6]). These studies demonstrate the potential role of plasticity in generating or reducing variation in a trait among different environments.…”

Change in individual growth rate and its link to gill-net fishing in two sympatric whitefish species

“…Fishing is typically size-selective (Myers and Hoenig 1997;Fukuwaka and Morita 2008), and since significant heritabilities have been reported for traits that could be size-related in many fish [up to h 2 = 0.5, see (Theriault et al 2007;Carlson and Seamons 2008)], size-selective fishing is expected to induce rapid evolutionary changes (Palumbi 2001;Smith and Bernatchez 2008;Darimont et al 2009). Traits such as age or size at maturation (Heino et al 2002;Grift et al 2003;Sharpe and Hendry 2009), average reproductive effort (Yoneda and Wright 2004;Thomas et al 2009), or individual growth rates (Handford et al 1977;Ricker 1981;Swain et al 2007; Thomas and Eckmann 2007;Nusslé et al 2009) are likely to evolve in response to size-selective fishing. Such fishing has therefore been termed a 'large-scale experiment in life-history evolution' (Rijnsdorp 1993;Law 2000;Stokes and Law 2000), and studies on fishery-induced evolution have increased in numbers during the last decade (see Jørgensen et al (2007) for a review of phenotypic change attributed to fishery-induced selection).…”