Last modified: 2014-03-28

#### Abstract

A new model for phenotypic plasticity is introduced. The model takes up the approach of Lynch and Gabriel (1987) and Gabriel et al. (2005) who generalized the common idea of tolerance curves that show performance of a species in dependence of an environmental state. They assumed that different phenotypes of the same genotype can have different tolerance curves with Gaussian shape and created models for phenotypic plasticity where adaptations to changed environmental conditions occur either irreversibly at birth or instantaneously after a delay during life. This models however showed certain unrealistic properties and since many traits seem to adapt in a continuous fashion, a model for continuous adaptation has been created.

For the new model it is assumed that the parameters of the tolerance curve (that determine mode and breadth of adaptation and the area included under the curve) can shift linearly and a technique to calculate the fitness of plastic individuals and genotypes is introduced and applied for a periodically changing environment. For that calculations it is assumed that the life-time fitness of an organism results from multiplying performance during different phases of life (consider survival probabilities) and that the genotype fitness is given by the geometric mean of the fitness of a large number of generations (consider their offspring numbers). Constraints of adaptation are made concerning the speed of adaptation and the area under the tolerance curve what implies a trade-off between being well adapted at the mode of the tolerance curve and having a wider breadth of adaptation. Assuming that evolution selects for the highest fitness, parameters belonging to genotypes are optimized numerically and the results are used for comparisons with the preceding models.

The new model is characterized by continuity of adaptation, the presence of intermediate phenotypes, duration of transformations that depend on their extend, a higher robustness with respect to assumptions about environmental fluctuations, applicability to continuously changing environments and no restrictions on the minimum duration of an environmental state.

Finally we discuss how the framework can be generalized. That includes different constraints of plasticity (for example speed and maximal extent of adaptation or direct costs of plasticity) and different ways of how tolerance curves can adapt (including non-linear adaptations and change between generations for evolutionary questions). By considering continuously changing tolerance curves to describe environment dependent performance not only of organisms but also of populations, single organs, cells or organelles the framework could inspire models from different areas of biology that include evolutionary biology, population dynamics and physiology.