Nonlinear social evolution and the emergence of collective action.

Organisms from microbes to humans engage in a variety of social behaviors, which affect fitness in complex, often nonlinear ways. The question of how these behaviors evolve has consequences ranging from antibiotic resistance to human origins. However, evolution with nonlinear social interactions is challenging to model mathematically, especially in combination with spatial, group, and/or kin assortment. We derive a mathematical condition for natural selection with synergistic interactions among any number of individuals. This result applies to populations with arbitrary (but fixed) spatial or network structure, group subdivision, and/or mating patterns. In this condition, nonlinear fitness effects are ascribed to collectives, and weighted by a new measure of collective relatedness. For weak selection, this condition can be systematically evaluated by computing branch lengths of ancestral trees. We apply this condition to pairwise games between diploid relatives, and to dilemmas of collective help or harm among siblings and on spatial networks. Our work provides a rigorous basis for extending the notion of "actor", in the study of social evolution, from individuals to collectives.

Fixation probabilities in graph-structured populations under weak selection.

A population's spatial structure affects the rate of genetic change and the outcome of natural selection. These effects can be modeled mathematically using the Birth-death process on graphs. Individuals occupy the vertices of a weighted graph, and reproduce into neighboring vertices based on fitness. A key quantity is the probability that a mutant type will sweep to fixation, as a function of the mutant's fitness. Graphs that increase the fixation probability of beneficial mutations, and decrease that of deleterious mutations, are said to amplify selection. However, fixation probabilities are difficult to compute for an arbitrary graph. Here we derive an expression for the fixation probability, of a weakly-selected mutation, in terms of the time for two lineages to coalesce. This expression enables weak-selection fixation probabilities to be computed, for an arbitrary weighted graph, in polynomial time. Applying this method, we explore the range of possible effects of graph structure on natural selection, genetic drift, and the balance between the two. Using exhaustive analysis of small graphs and a genetic search algorithm, we identify families of graphs with striking effects on fixation probability, and we analyze these families mathematically. Our work reveals the nuanced effects of graph structure on natural selection and neutral drift. In particular, we show how these notions depend critically on the process by which mutations arise.

Quantifying the Contribution of Habitats and Pathways to a Spatially Structured Population Facing Environmental Change.

The consequences of environmental disturbance and management are difficult to quantify for spatially structured populations because changes in one location carry through to other areas as a result of species movement. We develop a metric, G, for measuring the contribution of a habitat or pathway to network-wide population growth rate in the face of environmental change. This metric is different from other contribution metrics, as it quantifies effects of modifying vital rates for habitats and pathways in perturbation experiments. Perturbation treatments may range from small degradation or enhancement to complete habitat or pathway removal. We demonstrate the metric using a simple metapopulation example and a case study of eastern monarch butterflies. For the monarch case study, the magnitude of environmental change influences the ordering of node contribution. We find that habitats within which all individuals reside during one season are the most important to short-term network growth under complete removal scenarios, whereas the central breeding region contributes most to population growth over all but the strongest disturbances. The metric G provides for more efficient management interventions that proactively mitigate impacts of expected disturbances to spatially structured populations.

Transient amplifiers of selection and reducers of fixation for death-Birth updating on graphs.

The spatial structure of an evolving population affects the balance of natural selection versus genetic drift. Some structures amplify selection, increasing the role that fitness differences play in determining which mutations become fixed. Other structures suppress selection, reducing the effect of fitness differences and increasing the role of random chance. This phenomenon can be modeled by representing spatial structure as a graph, with individuals occupying vertices. Births and deaths occur stochastically, according to a specified update rule. We study death-Birth updating: An individual is chosen to die and then its neighbors compete to reproduce into the vacant spot. Previous numerical experiments suggested that amplifiers of selection for this process are either rare or nonexistent. We introduce a perturbative method for this problem for weak selection regime, meaning that mutations have small fitness effects. We show that fixation probability under weak selection can be calculated in terms of the coalescence times of random walks. This result leads naturally to a new definition of effective population size. Using this and other methods, we uncover the first known examples of transient amplifiers of selection (graphs that amplify selection for a particular range of fitness values) for the death-Birth process. We also exhibit new families of "reducers of fixation", which decrease the fixation probability of all mutations, whether beneficial or deleterious.

Ecosystem service flows from a migratory species: Spatial subsidies of the northern pintail.

Migratory species provide important benefits to society, but their cross-border conservation poses serious challenges. By quantifying the economic value of ecosystem services (ESs) provided across a species' range and ecological data on a species' habitat dependence, we estimate spatial subsidies-how different regions support ESs provided by a species across its range. We illustrate this method for migratory northern pintail ducks in North America. Pintails support over $101 million USD annually in recreational hunting and viewing and subsistence hunting in the U.S. and Canada. Pintail breeding regions provide nearly $30 million in subsidies to wintering regions, with the "Prairie Pothole" region supplying over $24 million in annual benefits to other regions. This information can be used to inform conservation funding allocation among migratory regions and nations on which the pintail depends. We thus illustrate a transferrable method to quantify migratory species-derived ESs and provide information to aid in their transboundary conservation.

A general modeling framework for describing spatially structured population dynamics.

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network-based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life-history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network-based population is modeled with discrete time steps. Using both theoretical and real-world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network-based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles.

The limits of weak selection and large population size in evolutionary game theory.

Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the [Formula: see text] limit, in which weak selection is applied before the large population limit, and the [Formula: see text] limit, in which the order is reversed. Formal mathematical definitions of the [Formula: see text] and [Formula: see text] limits are provided. Applying these definitions to the Moran process of evolutionary game theory, we obtain asymptotic expressions for fixation probability and conditions for success in these limits. We find that the asymptotic expressions for fixation probability, and the conditions for a strategy to be favored over a neutral mutation, are different in the [Formula: see text] and [Formula: see text] limits. However, the ordering of limits does not affect the conditions for one strategy to be favored over another.

The molecular clock of neutral evolution can be accelerated or slowed by asymmetric spatial structure.

Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, K, at which these substitutions accumulate in the population coincides with the mutation rate, u, at which they arise in individuals: K = u. This identity enables genetic sequence data to be used as a "molecular clock" to estimate the timing of evolutionary events. While the molecular clock is known to be perturbed by selection, it is thought that K = u holds very generally for neutral evolution. Here we show that asymmetric spatial population structure can alter the molecular clock rate for neutral mutations, leading to either Ku. Our results apply to a general class of haploid, asexually reproducing, spatially structured populations. Deviations from K = u occur because mutations arise unequally at different sites and have different probabilities of fixation depending on where they arise. If birth rates are uniform across sites, then K ≤ u. In general, K can take any value between 0 and Nu. Our model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze over 900 Twitter networks to study the effect of network topology on the fixation of neutral innovations in social evolution.

Formation of morphogen gradients: local accumulation time.

Spatial regulation of cell differentiation in embryos can be provided by morphogen gradients, which are defined as the concentration fields of molecules that control gene expression. For example, a cell can use its surface receptors to measure the local concentration of an extracellular ligand and convert this information into a corresponding change in its transcriptional state. We characterize the time needed to establish a steady-state gradient in problems with diffusion and degradation of locally produced chemical signals. A relaxation function is introduced to describe how the morphogen concentration profile approaches its steady state. This function is used to obtain a local accumulation time that provides a time scale that characterizes relaxation to steady state at an arbitrary position within the patterned field. To illustrate the approach we derive local accumulation times for a number of commonly used models of morphogen gradient formation.

Local kinetics of morphogen gradients.

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production. We present analytical results that characterize the dynamics of this process and are in quantitative agreement with numerical solutions of the underlying nonlinear equations. The derived results provide an explicit connection between the parameters of the problem and the time needed to reach a steady state value at a given position. Our approach can be used for the quantitative analysis of tissue patterning by morphogen gradients, a subject of active research in biophysics and developmental biology.

How long does it take to establish a morphogen gradient?

A morphogen gradient is defined as a concentration field of a molecule that acts as a dose-dependent regulator of cell differentiation. One of the key questions in studies of morphogen gradients is whether they reach steady states on timescales relevant for developmental patterning. We propose a systematic approach for addressing this question and illustrate it by analyzing several models that account for diffusion and degradation of locally produced chemical signals.

Multiscale modeling of diffusion in the early Drosophila embryo.

We developed a multiscale approach for the computationally efficient modeling of morphogen gradients in the syncytial Drosophila embryo, a single cell with multiple dividing nuclei. By using a homogenization technique, we derived a coarse-grained model with parameters that are explicitly related to the geometry of the syncytium and kinetics of nucleocytoplasmic shuttling. One of our main results is an accurate analytical approximation for the effective diffusivity of a morphogen molecule as a function of the nuclear density. We used this expression to explore the dynamics of the Bicoid morphogen gradient, a signal that patterns the anterior-posterior axis of the embryo. A similar approach can be used to analyze the dynamics of all three maternal morphogen gradients in Drosophila.