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 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 functions from R+ to the finite subsets of some "trait" space ⊂ Rk. 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) tells which subsets of 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 an invasion. Fitnesses supposedly satisfy certain constraints derived from their population/community dynamical origin, such as the fact that all mutants which are equal to some "resident", i.e., element of the present adaptive condition, have zero fitness. Apart from that we suppose that s is as smooth as can possibly be 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 origins of s, and the reasons 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 framework in its simplest guise, that is, when we assume that is 1dimensional, 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 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. ________________________________ 1) Institute of Evolutionary and Ecological Sciences EEW, section Theoretical Biology Kaiserstraat 63, 2311 GP Leiden, the Netherlands 2) ADN, IIASA, A-2361 Laxenburg, Austria 3) Population Biology Group, Department of Atomic Physics Eötvös University, 1088 Budapest Múzeum krt. 4/a, Hungary 4) present address: Population Biology Group, Department of Genetics Eötvös University, 1088 Budapest Múzeum krt. 4/a, Hungary 2 1. The larger context 1.1. Evolutionary basics The most conspicuous, if not the defining, properties of life are that living objects (1) reproduce almost faithfully, and (2) die. It is a mathematical necessity that the independent reproduction of particles leads to exponential population growth (or to rapid extinction, but such populations habitually escape our attention) (Jagers, 1975, 1991, 1995). Therefore in any finite world organisms will (3) interact, both directly through jostling or fighting, and indirectly through the consumption of resources and the sharing of predators. The consequence of (1) to (3) is that life evolves: Those types that do a better job in contributing to future generations will inherit the earth. Until a copying error during the reproductive act creates a still "better adapted" type. Evolution will grind to a halt only when it has reached a combination of types which cannot be bettered under the current condition of the environment. Simple though it may seem, this scenario becomes interestingly complicated due to the fact that those same types are (co-)instrumental in creating the current environmental condition. Remark: That there is no sign yet that evolution on this earth is going to freeze has two causes. The easy one is that the physical configuration of the world keeps changing. But it usually does so relatively slowly. Much to the biologist's luck, since it allows him/her (sometimes) to predict organismal properties from evolutionary considerations. The second cause is more involved: (a) There is no need that ecology drives evolution to a point attractor, even in models which only consider simple external (phenotypic) representations of organisms. But if we assume that too extreme phenotypes are weak survivors, as is generally the case in the real world, we may expect at least convergence to some nice attractor. However, there is a snag. (b) Since the internal (genotypic) representation of organisms is almost infinitely complicated, the map from genotype to any simple phenotypic representation is very many to one. Dolphins, Ichthyosaurs, tuna, and sharks may look similar, but underneath they are very different creatures. Consequently the mutational supply (due to copying errors of the genetic material) of new phenotypic variation shows considerable history dependence. (a) and (b) together make that when the evolutionary process is looked at in somewhat greater detail, it appears that non-point attractors with some recurrence property just don't exist. Evolution either halts, or progresses indefinitely, though not necessarily progressively. Luckily, here again, proper modes of abstraction as well as time scale differences come to the rescue of those who nevertheless want to make predictions. 1. 2. History: the changes in attention paid to ecological and genetic complexity The mechanistic theory of evolution started public life with the publication of Charles Darwin's "On the Origin of Species" in 1859. The one flaw in the reasoning of the early Darwinists was their, lukewarm, adherence to the concept of blending inheritance (the blending of the properties of the parents in their offspring), since by mathematical necessity evolution can only occur among particles which reproduce sufficiently faithfully. But they clearly saw evolution as driven by the interaction between individuals, as is proved by Darwin's statement that he owed his idea of the "struggle for existence" to the writings of Thomas Malthus. At the turn of the century the inheritance problem was solved by the rediscovery of a piece of contract research by a Moravian monk with physicist leanings, Gregor Mendel. It aren't the organisms which reproduce almost faithfully, but their genes. This considerably complexifies the logic, since the genes inhabiting one organism affect each other's reproductive potential. In the twenties a reconciliation of the Mendelian and Darwinian paradigms was effected by the three great mathematical population geneticists, Sir Ronald Fisher, J.B.S Haldane, and Sewall Wright. The hand-waving linking up in the forties and fifties of the resulting circle of ideas with those of the paleontologists and taxonomists of the day is now referred to as the Modern Synthesis. The strength of that link is still among the biologists' articles of faith. Ironically the mathematical framework underlying the Modern Synthesis dealt almost exclusively with the genetics of populations of non-interacting individuals. For this was one of the main simplifications made by the early theoretical population geneticists in order to cope with the complexities of realistic inheritance laws. It is even more ironical that this assumption of noninteraction makes it particularly hard on model populations to split into lines going their separate ways. The origin of species was, and is, still one of the less well understood problems of population genetics. 3 The second point on which the population genetics of the time fell short as a cornerstone for the theory of adaptive evolution is that it almost exclusively concentrated on the changes in the relative frequencies of types from a fixed genetic repertoire. For this is the scale where contact could be made between theory and genetic observations on real populations. Yet, the overall features of long term adaptive evolution crucially depend on the existence of a continual trickle of new mutants. The stream of novel adaptive variation is that small and fickle, that it is essentially beyond direct observation. But its effects can be seen in overwhelming profusion. We are but one instance Around 1970 both conceptual omissions were rectified by W.D. Hamilton (1967), G.R. Price and John Maynard Smith (Maynard Smith & Price, 1973; Maynard Smith, 1982), who put to the fore the concept of Evolutionarily Unbeatable Strategy. An EUS is a strategy which when played by everybody prevents all comparable strategies from increasing in numbers. Such strategies are the natural longer term evolutionary traps. (By now EUSes are more often called Evolutionarily Stable Strategies. Unfortunately this is a misnomer as EUSes need not be stable in the dynamic sense.) Of course there was a price. Only the statics of adaptive evolution was considered. Moreover, it became common usage to assume clonal reproduction (i.e., the almost faithful reproduction of individuals), in order to concentrate on behavioural interactions. Luckily later research has shown that a good number of the general results kept their ground for more realistic types of inheritance. But exceptions that are neither trivial nor contrived have been found as well. 1.3. About this paper In this paper we set out to construct in a general manner the simplest possible dynamical counterpart to the EUS concept. Since we primarily want to cope with general types of ecological complexities we stick to the by now time-honoured assumption of clonal reproduction. Moreover we assume that the ecological and evolutionary time scales are clearly separated. Finally we shall assume that the types can be characterised by a finite number of numerical traits, that the ecology satisfies some continuity conditions (to be expounded below) and that mutation only produces small steps in trait space. 1.4. Relation to present day views of the evolutionary process No doubt red-blooded biologists will find our assumptions artificial. To them we have the following three remarks to make in our defence. (i) It is always better to start hunting for patterns in some well chosen caricature of reality, and to leave it for a second stage to see to how those patterns modify when additional realism is added, than not to see any wood for the trees. (However, till we reach that second stage our conclusions about long term evolution should be taken with a pinch of salt.) (ii) The least we do is develop an internally consistent picture of a class of evolutionary processes, well worth of study in their own right. It is only by studying various classes of evolutionary processes that one may ever hope to bring out their essence. (iii) Our picture is the simplest one allowing the eventual development of a bifurcation theory of EUSes. Anyone who knows what bifurcation theory has done for differential equations will appreciate the usefulness of such a development. For mathematicians we may add that there is a wholly new, and rather unusual, class of dynamical systems waiting to be explored. As a final point we should make clear that we are by no means the first to venture on the present path. Some notable forerunners are Ilan Eshel (1983, 1991,1995; & Feldman, 1982, 1984), Jonathan Roughgarden (1976, 1979, 1983), Freddy Bugge Christiansen (1984, 1988, 1991; & Loeschcke, 1980, 1987, Loeschcke & -, 1984a,b), Peter Taylor (1989), Karl Sigmund (Hofbauer & -, 1990, Nowak & -, 1990), Si Levin (Cohen & -, 1987; Ludwig & -, 1992), Peter Hammerstein (1995, & Selten, 1994), and Carlo Matessi (& Di Pascuale, 1995). The main difference of our effort from theirs is that we strive to construct a clear mathematical framework that should abstractly encompass a greater deal of ecological complexity (but at the cost of highly oversimplifying the genetical end). Tom Vincent and co-workers (1990; & Brown, 1984, 1987, 1988, 1989; Brown & -, 1987a,b, 1992; & Fisher, 1988; et al., 1993) followed a line of thought that superficially is rather similar to ours. Our approach differs from theirs both in its greater formal abstraction and in that we try to stick to formalisms that consistently allow an interpretation in individual-based terms concordant with the basic philosophy with which we started this discourse (see also Metz & De Roos, 1992). 4 2. Reconciling the population dynamical and taxonomical viewpoints