Mathematical Structure of Evolutionary Theory

Here we postulate three laws which form a mathematical framework for the evolutionary dynamics in biology. The second law is most quantitative and is explicitly expressed in a unique form of stochastic differential equation. Salient features of Darwinian evolutionary dynamics are captured by this law: the probabilistic nature of evolution, ascendancy, and the adaptive landscape. Four dynamical elements are involved in the present formulation: the ascendant matrix, the transverse matrix, the Wright fitness function, and the stochastic drive. The first law may be regarded as a special case of the second law. It gives the reference point to discuss the evolutionary dynamics. The third law describes the relationship between the focused level of description to its lower and higher ones, and also defines the dichotomy of deterministic and stochastic drives. A precise description of Wright's adaptive landscape is given and a new interpretation of Fisher's fundamental theorem of natural selection is provided. The generality of the proposed laws is supported by the demonstration of their equivalence to a general non-equilibrium dynamics formulation, based on a recently proved theorem. Though the proposed laws are not the rigorous description, they should be regarded as mathematically what the laws of evolution might be. In addition, they provide a coherence framework to discuss several current evolutionary problems, such as speciation and stability, readily showing that statistical physics tools can be applied to Darwinian dynamics.