The stagnation paradox: the ever-improving but (more or less) stationary population fitness

Fisher's fundamental theorem states that natural selection improves mean fitness. Fitness, in turn, is often equated with population growth. This leads to an absurd prediction that life evolves to ever-faster growth rates, yet no one seriously claims generally slower population growth rates in the Triassic compared with the present day. I review here, using non-technical language, how fitness can improve yet stay constant (stagnation paradox), and why an unambiguous measure of population fitness does not exist. Subfields use different terminology for aspects of the paradox, referring to stasis, cryptic evolution or the difficulty of choosing an appropriate fitness measure; known resolutions likewise use diverse terms from environmental feedback to density dependence and ‘evolutionary environmental deterioration’. The paradox vanishes when these concepts are understood, and adaptation can lead to declining reproductive output of a population when individuals can improve their fitness by exploiting conspecifics. This is particularly readily observable when males participate in a zero-sum game over paternity and population output depends more strongly on female than male fitness. Even so, the jury is still out regarding the effect of sexual conflict on population fitness. Finally, life-history theory and genetic studies of microevolutionary change could pay more attention to each other.

The Fundamental Theorem of Natural Selection

Suppose we have n different types of self-replicating entity, with the population Pi of the ith type changing at a rate equal to Pi times the fitness fi of that type. Suppose the fitness fi is any continuous function of all the populations P1,⋯,Pn. Let pi be the fraction of replicators that are of the ith type. Then p=(p1,⋯,pn) is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.

Acceleration of Evolutionary Processes by Learning and Extended Fisher's Fundamental Theorem

Natural selection is general and powerful concept not only to explain evolutionary processes of biological organisms but also to design engineering systems such as genetic algorithms and particle filters.There is a surge of interest, both from biology and engineering, in considering natural selection of intellectual agents that can learn individually. Learning by individual agents of better behaviors for survival may accelerate the evolutionary processes by natural selection. We have accumulating pieces of evidence that organisms can transmit its information to the next generation via epigenetic states or memes. Also, such idea is important for engineering applications to improve the genetic algorithms and the particle filter. To accelerate the evolutionary process, an agent should change their strategy so that the population fitness increases the most. Equivalently, an agent should update the strategy towards a gradient (derivative) of the population fitness with respect to the strategy. However, it has not yet been clarified whether and how an agent can estimate the gradient and accelerate the evolutionary process. We also lack methodology to quantify the acceleration to understand and predict the impact of learning. In this paper, we address these problems. We show that an learning agent can accelerate the evolutionary process by proposing ancestral learning, which uses the information transmitted from the ancestor (ancestral information) via epigenetic states or memes. Numerical experiments show that ancestral learning actually accelerates the evolutionary process. We next show that the ancestral information is sufficient to estimate the gradient. In particular, learning can accelerate the evolutionary process without communications between agents. Finally, to quantify the acceleration, we extend the Fisher's fundamental theorem (FF-thm) for natural selection to ancestral learning. The conventional FF-thm relates the speed of evolution by natural selection to the variety of the individual fitness in the population. Our extended FF-thm relates the acceleration of the evolutionary process to the variety of individual fitness of the agent. By the theorem, we can quantitatively understand when and why learning is beneficial.

Fisher’s Fundamental Theorem of Natural Selection Isn’t Fundamental After All

Fisher’s Fundamental Theorem of Natural Selection (FTNS) was called “biology’s central theorem” (Fisher, 1930, pgs. 36–37; Brockman, 2011; Royal Society, 2020). FTNS might possibly have been accorded this status for decades because Fisher himself declared his own theorem to be fundamental to biology (Fisher, 1930, pgs. 36–37). However, the idea that Fisher’s theorem is biology’s central theorem is by-and-large a myth promoted by popu- lar science writers like Richard Dawkins (Brockman, 2011). Joseph Felsenstein, when delivering the 2018 Fisher Memorial Lecture declared that FTNS was “alas, not so fundamental” (Felsenstein, 2018; Felsenstein, 2017, pg. 94. One may be hard-pressed to find a biology textbook or biology student who can explain how FTNS helps them understand biology. Even the meaning and proof of the FTNS have re- mained contentious even to this day (Price, 1972; Basener and Sanford, 2018). Not only does FTNS do little to nothing to explain biological evolution, but like most population genetic and evolutionary literature, FTNS relies on a definition of fit- ness in terms of population growth rates rather than the biophysical notions of fitness which are more in line with the common-sense intuitions of the medical and engineering communities. From the perspective of the biophysical (rather than the population growth) notion of fitness, natural selection might be more accurately described as an agent against the increase of complexity rather than an agent for it. Thus, metaphorically speaking, some sort of anti-Weasel model of natural selection might better describe how selection actu- ally works in nature rather than Dawkins’ Weasel or other man-made genetic algorithms. However, the main focus of this communication is to pro- vide some pedagogical insights through simple numerical illustrations of Fisher’s Theorem. The hope is that this will show the general irrelevance of FTNS to the question of the evolution of complexity by means of natural selection, and thus show that Fisher’s Theorem is not so fundamental after all.

The Origin of Additive Genetic Variance Driven by Positive Selection

Fisher’s fundamental theorem of natural selection predicts no additive variance of fitness in a natural population. Consistently, studies in a variety of wild populations show virtually no narrow-sense heritability (h2) for traits important to fitness. However, counterexamples are occasionally reported, calling for a deeper understanding on the evolution of additive variance. In this study, we propose adaptive divergence followed by population admixture as a source of the additive genetic variance of evolutionarily important traits. We experimentally tested the hypothesis by examining a panel of ∼1,000 yeast segregants produced by a hybrid of two yeast strains that experienced adaptive divergence. We measured >400 yeast cell morphological traits and found a strong positive correlation between h2 and evolutionary importance. Because adaptive divergence followed by population admixture could happen constantly, particularly in species with wide geographic distribution and strong migratory capacity (e.g., humans), the finding reconciles the observation of abundant additive variances in evolutionarily important traits with Fisher’s fundamental theorem of natural selection. Importantly, the revealed role of positive selection in promoting rather than depleting additive variance suggests a simple explanation for why additive genetic variance can be dominant in a population despite the ubiquitous between-gene epistasis observed in functional assays.

The Price equation and reproductive value

The Price equation is widely recognized as capturing conceptually important properties of natural selection, and is often used to derive versions of Fisher’s fundamental theorem of natural selection, the secondary theorems of natural selection and other significant results. However, class structure is not usually incorporated into these arguments. From the starting point of Fisher’s original connection between fitness and reproductive value, a principled way of incorporating reproductive value and structured populations into the Price equation is explained, with its implications for precise meanings of (two distinct kinds of) reproductive value and of fitness. Once the Price equation applies to structured populations, then the other equations follow. The fundamental theorem itself has a special place among these equations, not only because it always incorporated class structure (and its method is followed for general class structures), but also because that is the result that justifies the important idea that these equations identify the effect of natural selection. The precise definitions of reproductive value and fitness have striking and unexpected features. However, a theoretical challenge emerges from the articulation of Fisher’s structure: is it possible to retain the ecological properties of fitness as well as its evolutionary out-of-equilibrium properties? This article is part of the theme issue ‘Fifty years of the Price equation’.

The source of additive genetic variance of evolutionarily important traits

Fisher’s fundamental theorem of natural selection predicts no additive variance of fitness in a natural population. Consistently, observations in a variety of wild populations show virtually no narrow-sense heritability (h2) for traits important to fitness. However, counterexamples are occasionally reported, calling for a deeper understanding on the evolution of additive variance. In this study we propose adaptive divergence followed by population admixture as a source of the additive genetic variance of evolutionarily important traits. We experimentally tested the hypothesis by examining a panel of ~1,000 yeast segregants produced by a hybrid of two yeast strains that experienced adaptive divergence. We measured over 400 yeast cell morphological traits and found a strong positive correlation between h2 and evolutionary importance. Because adaptive divergence followed by population admixture could happen constantly, particularly in some species such as humans, the finding reconciles the observation of abundant additive variances in evolutionarily important traits with Fisher’s fundamental theorem of natural selection. It also suggests natural selection may effectively promote rather than suppress additive genetic variance in species with wide geographic distribution and strong migratory capacity.

Additive genetic variance for lifetime fitness and the capacity for adaptation in an annual plant

The immediate capacity for adaptation under current environmental conditions is directly proportional to the additive genetic variance for fitness, VA(W). Mean absolute fitness, , is predicted to change at the rate , according to Fisher’s Fundamental Theorem of Natural Selection. Despite ample research evaluating degree of local adaptation, direct assessment of VA(W) and the capacity for ongoing adaptation is exceedingly rare. We estimated VA(W) and in three pedigreed populations of annual Chamaecrista fasciculata, over three years in the wild. Contrasting with common expectations, we found significant VA(W) in all populations and years, predicting increased mean fitness in subsequent generations (0.83 to 6.12 seeds per individual). Further, we detected two cases predicting “evolutionary rescue”, where selection on standing VA(W) was expected to increase fitness of declining populations ( < 1.0) to levels consistent with population sustainability and growth. Within populations, interannual differences in genetic expression of fitness were striking. Significant genotype-by-year interactions reflected modest correlations between breeding values across years (all r < 0.490), indicating temporally variable selection at the genotypic level; that could contribute to maintaining VA(W). By directly estimating VA(W) and total lifetime , our study presents an experimental approach for studies of adaptive capacity in the wild.