Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data.

For theoretical studies, reaction norm evolution in a changing environment can be modeled by means of the multivariate breeder's equation, with the reaction norm parameters treated as traits in their own right. This is, however, not a feasible approach for the use of field data, where the intercept and slope values are not available. An alternative approach is to use infinite-dimensional characters and smooth covariance function estimates found by, e.g., random regression. This is difficult because of the need to find, for example, polynomial basis functions that fit the data reasonably well over time, and because reaction norms in multivariate cases are correlated, such that they cannot be modeled independently. Here, I present an alternative approach based on a multivariate linear mixed model of any order, with dynamical incidence and residual covariance matrices that reflect the changing environment. From such a mixed model follows a dynamical BLUP model for the estimation of the individual reaction norm parameter values at any given parent generation, and for updating of the mean reaction norm parameter values from generation to generation by means of Robertson's secondary theorem of natural selection. This will, for example, make it possible to disentangle the microevolutionary and plasticity components in climate change responses. The BLUP model incorporates the additive genetic relationship matrix in the usual way, and overlapping generations can easily be accommodated. Additive genetic and environmental model parameters are assumed to be known and constant, but it is discussed how they can be estimated by means of a prediction error method. The identifiability by the use of field or laboratory data containing environmental, phenotypic, fitness, and additive genetic relationship data is an important feature of the proposed model.

The important choice of reference environment in microevolutionary climate response predictions.

It is well documented that individuals of wild populations can adjust to climate change by means of phenotypic plasticity, but few reports on adaptation by means of genetically based microevolution caused by selection. Disentanglement of these separate effects requires that the reference environment (the environmental zero point) is defined, and this should not be done arbitrarily. The problem is that an error in the reference environment may lead to large errors in predicted microevolution. Together with parameter values and initial mean trait values, the reference environment can be estimated from environmental, phenotypic and fitness data. A prediction error method for this purpose is described, with the feasibility shown by simulations. As shown in a toy example, an estimated reference environment may have large errors, especially for small populations. This may still be a better choice than use of an initial environmental value in a recorded time series, or the mean value, which is often used. Another alternative may be to use the mean value of a past and stationary stochastic environment, which the population is judged to have been fully adapted to, in the sense that the expected geometric mean fitness was at a global maximum. Exceptions are cases with constant phenotypic plasticity, where the microevolutionary changes per generation follow directly from phenotypic and environmental data, independent of the chosen reference environment.

The environmental zero-point problem in evolutionary reaction norm modeling.

There is a potential problem in present quantitative genetics evolutionary modeling based on reaction norms. Such models are state-space models, where the multivariate breeder's equation in some form is used as the state equation that propagates the population state forward in time. These models use the implicit assumption of a constant reference environment, in many cases set to zero. This zero-point is often the environment a population is adapted to, that is, where the expected geometric mean fitness is maximized. Such environmental reference values follow from the state of the population system, and they are thus population properties. The environment the population is adapted to, is, in other words, an internal population property, independent of the external environment. It is only when the external environment coincides with the internal reference environment, or vice versa, that the population is adapted to the current environment. This is formally a result of state-space modeling theory, which is an important theoretical basis for evolutionary modeling. The potential zero-point problem is present in all types of reaction norm models, parametrized as well as function-valued, and the problem does not disappear when the reference environment is set to zero. As the environmental reference values are population characteristics, they ought to be modeled as such. Whether such characteristics are evolvable is an open question, but considering the complexity of evolutionary processes, such evolvability cannot be excluded without good arguments. As a straightforward solution, I propose to model the reference values as evolvable mean traits in their own right, in addition to other reaction norm traits. However, solutions based on an evolvable G matrix are also possible.