d'Alembert's direct and inertial forces acting on populations: the Price equation and the fundamental theorem of natural selection

摘要

I develop a framework for interpreting the forces that act on any populationdescribed by frequencies. The conservation of total frequency, or totalprobability, shapes the characteristics of force. I begin with Fisher'sfundamental theorem of natural selection. That theorem partitions the totalevolutionary change of a population into two components. The first component isthe partial change caused by the direct force of natural selection, holdingconstant all aspects of the environment. The second component is the partialchange caused by the changing environment. I demonstrate that Fisher'spartition of total change into the direct force of selection and the forcesfrom the changing environmental frame of reference is identical to d'Alembert'sprinciple of mechanics, which separates the work done by the direct forces fromthe work done by the inertial forces associated with the changing frame ofreference. In d'Alembert's principle, there exist inertial forces from a changein the frame of reference that exactly balance the direct forces. I show thatthe conservation of total probability strongly shapes the form of the balancebetween the direct and inertial forces. I then use the strong results forconserved probability to obtain general results for the change in any systemquantity, such as biological fitness or energy. Those general results derivefrom simple coordinate changes between frequencies and system quantities.Ultimately, d'Alembert's separation of direct and inertial forces provides deepconceptual insight into the interpretation of forces and the unification ofdisparate fields of study.