Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant's fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.

4Evolutionary graph theory has grown to be an area of intense study. Despite the 5 amount of interest in the field, it seems to have grown separate from other subfields 6 of population genetics and evolution. In the current work I introduce the concept of

ALMA observations have revealed nuclear dusty molecular disks/tori with characteristic sizes 15-40 pc in the few Seyferts and low luminosity AGN studied so far. These structures are generally decoupled both morphologically and kinematically from the host galaxy disk. We present ALMA observations of the CO(2-1) and CO(3-2) molecular gas transitions and associated (sub)-millimeter continua of the nearby Seyfert 1.5 galaxy NGC 3227 with angular resolutions 0.085 − 0.21 (7-15 pc). On large scales the cold molecular gas shows circular motions as well as streaming motions on scales of a few hundred parsecs associated with a large scale bar. We fitted the nuclear ALMA 1.3 mm emission with an unresolved component and an extended component. The 850 µm emission shows at least two extended components, one along the major axis of the nuclear disk and the other along the axis of the ionization cone. The molecular gas in the central region (1 ∼ 73 pc) shows several CO clumps with complex kinematics which appears to be dominated by non-circular motions. While we cannot demonstrate conclusively the presence of a warped nuclear disk, we also detected non-circular motions along the kinematic minor axis. They reach line-of-sight velocities of v − v sys = 150 − 200 km s −1 . Assuming that the radial motions are in the plane of the galaxy, then we interpret them as a nuclear molecular outflow due to molecular gas in the host galaxy being entrained by the AGN wind. We derive molecular outflow rates of 5 M yr −1 and 0.6 M yr −1 at projected distances of up to 30 pc to the northeast and southwest of the AGN, respectively. At the AGN location we estimate a mass in molecular gas of 5 × 10 5 M and an equivalent average column density N(H 2 ) = 2 − 3 × 10 23 cm −2 in the inner 15 pc. The nuclear CO(2-1) and CO(3-2) molecular gas and sub-mm continuum emission of NGC 3227 do not resemble the classical compact torus. Rather, these emissions extend for several tens of parsecs and appear connected with the circumnuclear ring in the host galaxy disk, as found in other local AGN.

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We study the evolution of a pair of competing behavioural alleles in a structured population when there are non-additive or 'synergistic' fitness effects. Under a form of weak selection and with a simple symmetry condition between a pair of competing alleles, Tarnita et al. provide a surprisingly simple condition for one allele to dominate the other. Their condition can be obtained from an analysis of a corresponding simpler model in which fitness effects are additive. Their result uses an average measure of selective advantage where the average is taken over the long-term-that is, over all possible allele frequencies-and this precludes consideration of any frequency dependence the allelic fitness might exhibit. However, in a considerable body of work with non-additive fitness effects-for example, hawk-dove and prisoner's dilemma games-frequency dependence plays an essential role in the establishment of conditions for a stable allele-frequency equilibrium.Here, we present a frequency-dependent generalization of their result that provides an expression for allelic fitness at any given allele frequency p. We use an inclusive fitness approach and provide two examples for an infinite structured population. We illustrate our results with an analysis of the hawk-dove game.Keywords: evolutionary game theory; non-additive games; relatedness; allele frequency; Price equation; frequency dependence INTRODUCTIONAn enormous body of significant work constructs analytical models for the genetical evolution of social behaviour. The key relationship here is the dependence of focal fitness on the behaviour (phenotypic value) of a number of interactants. These behaviours are typically correlated with individual genotypic values, and the resulting connection between fitness and genotype allows us to get hold of the manner in which selection changes the frequency of alleles coding for alternative behaviours. The central tool in this analysis is the covariance formula of Price [1]. It requires us to calculate the covariance between focal fitness and focal genotype, and the dependence of the former on the genotypic values of neighbouring individuals reduces the problem to one of calculating covariances between neighbouring genotypes (or between expressions involving neighbouring genotypes) and the focal genotype. In building models of genetic change, we make explicit assumptions about how focal fitness depends on local genotypic values. The simplest models we work with are linear; that is, fitness effects among interactants are assumed to be additive-if Y and Z both interact with X, the effect on the fitness of X is the sum of the individual effects of Yand Z. If X, Y and Z have genotypic values x, y and z, then the Price equation requires calculation of the covariances cov(x, y) and cov(x, z). Much of the work on cooperation and altruism makes this assumption. Other models use nonlinear functions and considerable attention has been paid to quadratic expressions [2]. These arise naturally in haploid models in which genotypic values ar...