Adaptive Dynamics: A Geometrical Study of the Consequences of Nearly Faithful Reproduction

We set out to explore a class of stochastic processes called, 'adaptive dynamics', which supposedly capture some of the essentials of the long-term biological evolution. These processes have a strong deterministic component. This allows a classification of their qualitative features which in many aspects is similar to classifications from the theory of deterministic dynamical systems. But, they also display a good number of clear-cut novel dynamical phenomena. The sample functions of an adaptive dynamics are piece-wise constant function from R+ to the finite subsets of some 'trait' space X in R k. Those subsets we call 'adaptive conditions'. Both the range and the jumps of a sample function are governed by a function s, called 'fitness', mapping the present adaptive condition and the trait value of a potential 'mutant' to R. Sign(s) tell us which subsets of X qualify as adaptive conditions, which mutants can potentially 'invade', leading to a jump in the sample function, and which adaptive condition(s) can result from such invasion. Fitness supposedly satisfy certain constraints derived from their population/ community dynamical origin, such as the fact that all mutants which are equal to some 'residents', i.e., element of the present adaptive condition, have zero fitness. Apart from that, we suppose that s is as smooth as can be possibly condoned by its community dynamical origin. Moreover, we assume that a mutant can differ but little from its resident 'progenitor'. In Sections 1 and 2, we describe the biological background of our mathematical framework. In Section 1, we deal with the position of our framework relative to present and past evolutionary research. In Section 2, we discuss the community dynamical origin of s, and the reason for making a number of specific simplifications relative to the full complexity seen in nature. In Sections 3 and 4, we consider some general, mathematical, as well as biological conclusions that can be drawn from our frame work in its simplest guise, that is, when we assume that X is 1-dimensional, and that the cardinality of the adaptive conditions stays low. The main result is a classification of the adaptively singular points. These points comprise both the adaptive point attractors, as well as the points where the adaptive trajectory can branch; thus, attaining its characteristic tree-like shape. In Section 5, we discuss how adaptive dynamics relate through a limiting argument to stochastic models in which individual organisms are represented as separate entities. It is only through such a limiting procedure that any class of population or evolutionary models can eventually be justified. Our basic assumptions are : (i) clonal reproduction, i.e., the resident individuals reproduce faithfully without any of the complications of sex or Mendelian genetics, except for the occasional occurrence of a mutant, (ii) a large system size and an even rarer occurrence of mutations per birth event, (iii) uniqueness and global attractiveness of any interior attractor of the community dynamics in the limit of the infinite system size. In Section 6, we try to delineate, by a tentative listing of 'axioms', the largest possible class of processes that can result from the kind of limiting considerations spelled out in Section 5. And in Section 7, we heuristically derive some very general predictions about macro-evolutionary patterns, based on those weak assumptions only. In the final Section 8, we discuss (i) how the results from the preceding sections may fit into a more encompassing view of biological evolution, and (ii) some directions for further research.