Function-valued adaptive dynamics and optimal control theory

Evolutionary-developmental (evo-devo) dynamics of hominin brain size

Brain size tripled in the human lineage over four million years, but why this occurred remains uncertain. Here, to study what caused this brain expansion, I mathematically model the evolutionary and developmental (evo-devo) dynamics of hominin brain size. The model recovers (1) the evolution of brain and body sizes of seven hominin species starting from brain and body sizes of the australopithecine scale, (2) the evolution of the hominin brain-body allometry and (3) major patterns of human development and evolution. I show that the brain expansion recovered is not caused by direct selection for brain size but by its genetic correlation with developmentally late preovulatory ovarian follicles. This correlation is generated over development if individuals experience a challenging ecology and seemingly cumulative culture, among other conditions. These findings show that the evolution of exceptionally adaptive traits may not be primarily caused by selection for them but by developmental constraints that divert selection.

A mathematical framework for evo-devo dynamics

Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Classic evolutionary theory indicates that development affects evolution by modulating the genetic covariation upon which selection acts, thus affecting genetic constraints. However, whether genetic constraints are relative, thus diverting adaptation from the direction of steepest fitness ascent, or absolute, thus blocking adaptation in certain directions, remains uncertain. This limits understanding of long-term evolution of developmentally constructed phenotypes. Here we formulate a general, tractable mathematical framework that integrates age progression, explicit development (i.e., the construction of the phenotype across life subject to developmental constraints), and evolutionary dynamics, thus describing the evolutionary and developmental (evo-devo) dynamics. The framework yields simple equations that can be arranged in a layered structure that we call the evo-devo process, whereby five core elementary components generate all equations including those mechanistically describing genetic covariation and the evo-devo dynamics. The framework recovers evolutionary dynamic equations in gradient form and describes the evolution of genetic covariation from the evolution of genotype, phenotype, environment, and mutational covariation. This shows that genotypic and phenotypic evolution must be followed simultaneously to yield a dynamically sufficient description of long-term phenotypic evolution in gradient form, such that evolution described as the climbing of a fitness landscape occurs in "geno-phenotype" space. Genetic constraints in geno-phenotype space are necessarily absolute because the phenotype is related to the genotype by development. Thus, the long-term evolutionary dynamics of developed phenotypes is strongly non-standard: (1) evolutionary equilibria are either absent or infinite in number and depend on genetic covariation and hence on development; (2) developmental constraints determine the admissible evolutionary path and hence which evolutionary equilibria are admissible; and (3) evolutionary outcomes occur at admissible evolutionary equilibria, which do not generally occur at fitness landscape peaks in geno-phenotype space, but at peaks in the admissible evolutionary path where "total genotypic selection" vanishes if exogenous plastic response vanishes and mutational variation exists in all directions of genotype space. Hence, selection and development jointly define the evolutionary outcomes if absolute mutational constraints and exogenous plastic response are absent, rather than the outcomes being defined only by selection. Moreover, our framework provides formulas for the sensitivities of a recurrence and an alternative method to dynamic optimization (i.e., dynamic programming or optimal control) to identify evolutionary outcomes in models with developmentally dynamic traits. These results show that development has major evolutionary effects.

Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

How development affects evolution

Natural selection acts on developmentally constructed phenotypes, but how does development affect evolution? This question prompts a simultaneous consideration of development and evolution. However, there has been a lack of general mathematical frameworks mechanistically integrating the two, which may have inhibited progress on the question. Here, we use a new mathematical framework that mechanistically integrates development into evolution to analyse how development affects evolution. We show that, while selection pushes genotypic and phenotypic evolution up the fitness landscape, development determines the admissible evolutionary pathway, such that evolutionary outcomes occur at path peaks rather than landscape peaks. Changes in development can generate path peaks, triggering genotypic or phenotypic diversification, even on constant, single-peak landscapes. Phenotypic plasticity, niche construction, extra-genetic inheritance, and developmental bias alter the evolutionary path and hence the outcome. Thus, extra-genetic inheritance can have permanent evolutionary effects by changing the developmental constraints, even if extra-genetically acquired elements are not transmitted to future generations. Selective development, whereby phenotype construction points in the adaptive direction, may induce adaptive or maladaptive evolution depending on the developmental constraints. Moreover, developmental propagation of phenotypic effects over age enables the evolution of negative senescence. Overall, we find that development plays a major evolutionary role.

Hamilton's rule, gradual evolution, and the optimal (feedback) control of phenotypically plastic traits

Most traits expressed by organisms, such as gene expression profiles, developmental trajectories, behavioural sequences and reaction norms are function-valued traits (colloquially "phenotypically plastic traits"), since they vary across an individual's age and in response to various internal and/or external factors (state variables). Furthermore, most organisms live in populations subject to limited genetic mixing and are thus likely to interact with their relatives. We here formalise selection on genetically determined function-valued traits of individuals interacting in a group-structured population, by deriving the marginal version of Hamilton's rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton's rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for their uninvadability (evolutionary stability). We develop a novel analysis using optimal control theory and differential game theory, to simultaneously characterise and compare the first-order conditions of (i) open-loop traits - functions of time (or age) only, and (ii) closed-loop (state-feedback) traits - functions of both time and state variables. We show that closed-loop traits can be represented as the simpler open-loop traits when individuals do not interact or when they interact with clonal relatives. Our analysis delineates the role of state-dependence and interdependence between individuals for trait evolution, which has implications to both life-history theory and social evolution.

The Evolution of Immigration Strategies Facilitates Niche Expansion by Divergent Adaptation in a Structured Metapopulation Model

Local adaptation and habitat choice are two key factors that control the distribution and diversification of species. Here we model habitat choice mechanistically as the outcome of dispersal with nonrandom immigration. We consider a structured metapopulation with a continuous distribution of patch types and determine the evolutionarily stable immigration strategy as the function linking patch type to the probability of settling in the patch on encounter. We uncover a novel mechanism whereby coexisting strains that only slightly differ in their local adaptation trait can evolve substantially different immigration strategies. In turn, different habitat use selects for divergent adaptations in the two strains. We propose that the joint evolution of immigration and local adaptation can facilitate diversification and discuss our results in the light of niche conservatism versus niche expansion.

Host-virus-predator coexistence in a grey-box model with dynamic optimization of host fitness

Lytic viruses are believed to affect both flow patterns and host diversity in microbial food webs. Models resolving host and virus communities into subgroups can represent both aspects. However, when flow pattern is the prime interest, such models may seem unnecessary complex. This has led to proposals of black-box models using only total community sizes as state variables. This simplification creates a coexistence problem, however, since predator and virus communities then compete for the same, shared, prey = host community. Mathematically, this problem can be solved by introducing feedbacks allowing community-level properties to adapt. The different mathematical alternatives for such feedback represent different ecological assumptions and thus different hypotheses for how the balance between predators and viruses is controlled in nature. We here explore a model where the feedback works through an increase in host community resistance in response to high virus abundances, thereby reducing virus production. We use a dynamic "strategy" index S to describe the balance between defensive and competitive abilities in the host community, and assume the rate of change in S to be proportional to the local slope of the per capita fitness gradient for the host. We explore how such a "grey-box" model can allow stable coexistence of viruses and predators, and how equilibrium food web structure, virus-to-host ratio, and partitioning of host production varies; both as functions of host community traits, and as functions of external bottom-up and top-down drivers.

Variability in life-history switch points across and within populations explained by Adaptive Dynamics

Understanding the factors that shape the timing of life-history switch points (SPs; e.g. hatching, metamorphosis and maturation) is a fundamental question in evolutionary ecology. Previous studies examining this question from a fitness optimization perspective have advanced our understanding of why the timing of life-history transitions may vary across populations and environments. However, in nature we also often observe variability among individuals within populations. Optimization theory, which typically predicts a single optimal SP under physiological and environmental constraints for a given environment, cannot explain this variability. Here, we re-examine the evolution of a single life-history SP between juvenile and adult stages from an Adaptive Dynamics (AD) perspective, which explicitly considers the feedback between the dynamics of population and the evolution of life-history strategy. The AD model, although simple in structure, exhibits a diverse range of evolutionary scenarios depending upon demographic and environmental conditions, including the loss of the juvenile stage, a single optimal SP, alternative optimal SPs depending on the initial phenotype, and sympatric coexistence of two SP phenotypes under disruptive selection. Such predictions are consistent with previous optimization approaches in predicting life-history SP variability across environments and between populations, and in addition they also explain within-population variability by sympatric disruptive selection. Thus, our model can be used as a theoretical tool for understanding life-history variability across environments and, especially, within species in the same environment.

Twelve fundamental life histories evolving through allocation-dependent fecundity and survival

An organism's life history is closely interlinked with its allocation of energy between growth and reproduction at different life stages. Theoretical models have established that diminishing returns from reproductive investment promote strategies with simultaneous investment into growth and reproduction (indeterminate growth) over strategies with distinct phases of growth and reproduction (determinate growth). We extend this traditional, binary classification by showing that allocation‐dependent fecundity and mortality rates allow for a large diversity of optimal allocation schedules. By analyzing a model of organisms that allocate energy between growth and reproduction, we find twelve types of optimal allocation schedules, differing qualitatively in how reproductive allocation increases with body mass. These twelve optimal allocation schedules include types with different combinations of continuous and discontinuous increase in reproduction allocation, in which phases of continuous increase can be decelerating or accelerating. We furthermore investigate how this variation influences growth curves and the expected maximum life span and body size. Our study thus reveals new links between eco‐physiological constraints and life‐history evolution and underscores how allocation‐dependent fitness components may underlie biological diversity.

Adaptive significance of critical weight for metamorphosis in holometabolous insects

Holometabolous insect larvae become committed to metamorphosis when they reach a critical weight. Although the physiological mechanisms involved in this process have been well-studied, the adaptive significance of the critical weight remains unclear. Here, we developed a life history model for holometabolous insects and evaluated it from the viewpoint of optimal energy allocation. We found that, without a priori assumptions about critical weight, the optimal growth schedule is always biphasic: larval tissues grow predominately until they reach a certain threshold, after which the imaginal tissues begin rapid growth, suggesting that the emergence of a critical weight as a phase-transition point is a natural consequence of optimal growth scheduling. Our model predicts the optimal timing of critical-weight attainment, in agreement with observations in phylogenetically-distinct species. Furthermore, it also predicts the scaling of growth scheduling against environmental change, i.e., the relative value and timing of the critical weight should be constant, thus providing a general interpretation of observed phenotypic plasticity. This scaling relationship allows the classification of adaptive responses in critical weight into five possible types that reflect the ecological features of focal insects. In this manner, our theory and its consistency with experimental observations demonstrate the adaptive significance of critical weight.

Why Darwin would have loved evolutionary game theory

Humans have marvelled at the fit of form and function, the way organisms' traits seem remarkably suited to their lifestyles and ecologies. While natural selection provides the scientific basis for the fit of form and function, Darwin found certain adaptations vexing or particularly intriguing: sex ratios, sexual selection and altruism. The logic behind these adaptations resides in frequency-dependent selection where the value of a given heritable phenotype (i.e. strategy) to an individual depends upon the strategies of others. Game theory is a branch of mathematics that is uniquely suited to solving such puzzles. While game theoretic thinking enters into Darwin's arguments and those of evolutionists through much of the twentieth century, the tools of evolutionary game theory were not available to Darwin or most evolutionists until the 1970s, and its full scope has only unfolded in the last three decades. As a consequence, game theory is applied and appreciated rather spottily. Game theory not only applies to matrix games and social games, it also applies to speciation, macroevolution and perhaps even to cancer. I assert that life and natural selection are a game, and that game theory is the appropriate logic for framing and understanding adaptations. Its scope can include behaviours within species, state-dependent strategies (such as male, female and so much more), speciation and coevolution, and expands beyond microevolution to macroevolution. Game theory clarifies aspects of ecological and evolutionary stability in ways useful to understanding eco-evolutionary dynamics, niche construction and ecosystem engineering. In short, I would like to think that Darwin would have found game theory uniquely useful for his theory of natural selection. Let us see why this is so.

The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin's maximum principle

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370–389, 2006) and Parvinen et al. (J Math Biol 67: 509–533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin’s maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.