A White Noise Approach to Evolutionary Ecology

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity, it is reasonable to consider the evolutionary response to demographic stochasticity and its relation to random genetic drift. Here we close this gap in the context of quantitative genetics by deriving the dynamics of the distribution of a quantitative character and the abundance of a biological population from a stochastic partial differential equation driven by space-time white noise. In the process we develop a useful set of heuristics to operationalize the powerful, but abstract theory of white noise and measure-valued stochastic processes. This approach allows us to compute the full implications of demographic stochasticity on phenotypic distributions and abundances of populations. We demonstrate the utility of our approach by deriving a quantitative genetic model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by rates of interactions, competition coefficients, or selection gradients. Analyzing the relationship between selection gradients and competition coefficients reveals independence between linear selection gradients and competition coefficients. In contrast, absolute values of linear selection gradients and quadratic selection gradients tend to be positively correlated with competition coefficients. That is, competing species that strongly affect each other’s abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models that underlie rigorous investigations of the relationship between feedbacks of biological processes and the patterns of diversity they produce.

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity, it is reasonable to consider the evolutionary response to demographic stochasticity and its relation to random genetic drift. Here we close this gap in the context of quantitative genetics by deriving the dynamics of the distribution of a quantitative character and the abundance of a biological population from a stochastic partial differential equation driven by space-time white noise. In the process we develop a useful set of heuristics to operationalize the powerful, but abstract theory of white noise and measure-valued stochastic processes. This approach allows us to compute the full implications of demographic stochasticity on phenotypic distributions and abundances of populations. We demonstrate the utility of our approach by deriving a quantitative genetic model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by rates of interactions, competition coefficients, or selection gradients. Analyzing the relationship between selection gradients and competition coefficients reveals independence between linear selection gradients and competition coefficients. In contrast, absolute values of linear selection gradients and quadratic selection gradients tend to be positively correlated with competition coefficients. That is, competing species that strongly affect each other's abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models that underlie rigorous investigations of the relationship between feedbacks of biological processes and the patterns of diversity they produce.

Introduction
Current mathematical approaches to synthesize the dynamics of abundance and evolution in populations have capitalized on the fact that biological 12 fitness plays a key role in determining both sets of dynamics. In particular, while covariance of fitness and genotype is the basis of evolution by natural 14 selection, the mean fitness across all individuals in a population determines the growth, stasis or decline of abundance. Although this connection has been 16 established in the contexts of population genetics Kimura, 1970, Roughgarden, 1979), evolutionary game theory (Hofbauer and Sigmund, 1998, 18 Lion, 2018, Nowak, 2006, quantitative genetics (Doebeli, 1996, Lande, 1982, Lion, 2018 and a unifying framework for these three distinct approaches to 20 evolutionary theory (Champagnat et al., 2006), there remains a gap in incorporating the intrinsically random nature of abundance into the evolution of 22 continuous traits. Specifically, in theoretical quantitative genetics the derivation of a population's response to random genetic drift is derived in discrete 24 time under the assumption of constant effective population size using arguments based on properties of random samples (Lande, 1976). Though this 26 approach conveniently mimics the formalism provided by the Wright-Fisher model of population genetics, real population sizes fluctuate over time. Fur-28 thermore, since these fluctuations are themselves stochastic, it seems natural to derive expressions for the evolutionary response to demographic stochas-30 ticity and consider how the results relate to characterizations of random genetic drift. This can be done in continuous time for population genetic models 32 without too much technical overhead, assuming a finite number of alleles (Gomulkiewicz et al., 2017, Lande et al., 2009, Parsons et al., 2010. However, for 34 populations with a continuum of types, such as a quantitative trait, finding a formal approach to derive the evolutionary response to demographic stochas-36 ticity has remained a vexing mathematical challenge. In this paper we close this gap by combining the calculus of white noise with results on rescaled Since MVBP are abstract mathematical objects and their rigorous study requires elaborate mathematical machinery, the use of MVBP in mainstream theoretical evolutionary ecology has been limited. However, they provide 54 natural models of biological populations by capturing various mechanistic details. In particular, MVBP generalize classical birth-death processes, such 56 as the Galton-Watson process (Kimmel andAxelrod, 2015, Dawson, 1993), to model populations of discrete individuals that carry some value in a given 58 type-space. Selection can then be modelled by associating these values with average reproductive output and mutation can be incorporated using a model 60 that determines the distribution of offspring values given their parental value. For population genetic models the type-space is the discrete set of possible 62 alleles individuals can carry. In quantitative genetic models tracking the evolution of d-dimensional phenotypes, this type-space is typically set to the Eu-64 clidean space R d . By starting with branching processes we can implement mechanistic models of biological fitness that account for the phenotype of the 66 focal individual along with the phenotypes and number of all other individuals in a population or community. By taking a rescaled limit, we can then 68 use these detailed individual-based models to derive population-level models tracking the dynamics of population abundance and phenotypic distribution 70 driven by selection, mutation and demographic stochasticity. Hence, rescaled limits of MVBP provide a means to derive mathematically tractable, yet bio-72 logically mechanistic models of eco-evolutionary dynamics.

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(1998) and Champagnat et al. (2006) have shown that rescaled limits for a large class of MVBP converge to solutions of SPDE. Although cases in which d ≥ 1 76 can be treated using the so-called martingale problem formulation (Dawson, 1993), the SPDE formulation provides a more intuitive description of the bi-78 ological processes involved. We therefore focus on the case d = 1 here. This allows us to introduce a concrete set of heuristics for deriving SDE track-80 ing the dynamics of abundance, phenotypic mean and phenotypic variance to a wide audience of mathematical evolutionary ecologists. Following our 82 approach to simplify notation and develop heuristics for calculations, future work can possibly use the martingale formulation to extend the results pre-84 sented here for d > 1 and even for infinite-dimensional traits (Dawson, 1993, Stinchcombe et al., 2012. Rigorous introductions to SPDE and rescaled limits 86 of MVBP have been respectively provided by Da Prato and Zabczyk (2014) and Etheridge (2000).

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In this paper we begin in §2 by introducing the basic framework of our approach. We first outline the essential ideas behind deriving evolutionary 90 dynamics from abundance dynamics using a deterministic partial differential equation (PDE). In SM §3.1 we review rescaled limits of MVBP, their asso-92 ciated SPDE and introduce an approach to derive SDE tracking the dynamics of abundance, phenotypic mean and phenotypic variance. This approach 94 requires performing calculations with respect to space-time white noise processes and we provide heuristics for doing so in SM §2.1. In §2.2 we discuss 96 consequences of the derived SDE for general phenotypic distributions and simplify their expressions by assuming normally distributed phenotypes. For added biological relevance, we incorporate models of inheritance and development following classical quantitative genetics. To demonstrate how our ecology. In this section we briefly outline derivations of the replicator-mutator equation and trait dynamics from abundance dynamics in the deterministic 122 case. We then extend these formula along with related results to the case of random reproductive output (i.e., demographic stochasticity). 124 2.1. Deterministic Dynamics Finite Number of Types. We start by considering the dynamics of an asexually 126 reproducing population in a homogeneous environment. For simplicity, we first assume individuals are haploid and carry one of K alleles each with a dif-128 ferent fitness expressed as growth rate before introducing a model involving a quantitative trait. Under these assumptions, the derivation of the evolution 130 of allele frequencies due to natural selection can be derived from expressions of exponential growth. This, and a few related approaches, have been pro-132 vided by Crow and Kimura (1970). Mutation can be included using a matrix of transition rates. Specifically, denoting ν i the abundance of individuals with 134 allele i, m i the growth rate of allele i (called the Malthusian parameter in Crow and Kimura, 1970), µ ij the mutation rate from allele i to allele j and assuming 136 selection and mutation are decoupled (Bürger, 2000), we have Starting from this model, we get the total abundance of the population as 138 N = ∑ i ν i , the frequency of allele i as p i = ν i /N and the mean Malthusian fitness of the population asm = ∑ i p i m i . Note we have used the abbreviation 140 ∑ i = ∑ K i=1 to simplify inline notation. Observing ∑ ij µ ji ν j = ∑ ij µ ij ν i , we use linearity of differentiation to derive the dynamics of abundance dN/dt as To derive the dynamics of the allele frequencies p 1 , . . . , p K , we use the quotient rule of elementary calculus to find Two important observations of these equations include: (i) Mean Malthusian fitnessm is equivalent to the population growth rate and thus determines 146 the abundance dynamics of the entire population. (ii) Selection for allele i occurs when m i >m and selection against allele i occurs when m i <m. Hence,148 as mentioned in the introduction, fitness plays a key role in determining both abundance dynamics and evolution.

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Equation (3) is known in the field of evolutionary game theory as a replicatormutator equation (Nowak, 2006). Instead of being explicitly focused on alleles, 152 the replicator-mutator equation describes the fluctuations of relative abundances of various types in a population in terms of replication and annihila-154 tion rates of each type and hence can be used to model dynamical systems outside of evolutionary biology (Nowak, 2006).

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Continuum of Types. Inspired by equations (1)-(3), we derive an analog of the replicator-mutator equation for a continuum of types (that is, for a quanti-158 tative trait). In particular, we model a continuously reproducing population with trait values x ∈ R and an abundance density ν(x, t) that represents the 160 amount of individuals in the population with trait value x at time t. Hence, the abundance density satisfies N(t) = ν(x, t)dx and p(x, t) = ν(x, t)/N(t) 162 is the relative density of trait x which we also refer to as the phenotypic distribution. Note we have used the abbreviation = +∞ −∞ to simplify inline 164 notation.
In analogy with the growth rates m i for equation (1) we write m(ν, x) as 166 the growth rate associated with trait value x which depends on the abundance density ν. We assume mutation is captured by diffusion with coefficient µ 2 .
168 Hence, we model the demographic dynamics of a population and the dynamics of a quantitative character simultaneously by the PDE Equation (4) qualifies both as a semilinear evolution equation and also a scalar reaction-diffusion equation. Although the general theory of such equations is quite rich, it is also quite difficult (Evans, 2010, Zheng, 2004. Hence, to stay within the realms of analytical tractability and biological plausibility, 174 we require a set of technical assumptions which we list in SM §1.1. These assumptions guarantee solutions to equation (4) exist for all finite time t > 0 176 and, hence, let us investigate the ecological and evolutionary dynamics of biological populations.
178 Equation (4) can be seen as an analog of equation (1) for a continuum of types. By assuming mutation acts via diffusion, the effect of mutation causes 180 the abundance density ν(x, t) to flatten out across phenotypic space. In fact, if the growth rate is constant across x, then this model of mutation will cause 182 ν(x, t) to converge to a flat line in x as t → ∞. Interpreting the trait value x as location in geographic space, equation (4) becomes a well-studied model of 184 spatially distributed population dynamics (Cantrell and Cosner, 2004).
Although clearly an idealized representation of biological reality, this model 186 is sufficiently general to capture a large class of dynamics including density dependent growth and frequency dependent selection. As an example, lo-188 gistic growth combined with stabilizing selection can be captured using the growth rate where a > 0 the is strength of abiotic stabilizing selection around the phenotypic optimum θ, c > 0 is the strength of intraspecific competition and we 192 refer to R as the innate growth rate (see §3.3 below). In the language of population ecology, r = R − a 2 (θ − x) 2 is the intrinsic growth rate of the population 194 (Chesson, 2000). This model assumes competitive interactions cause the same reduction in fitness regardless of trait value.

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This exemplary fitness function has a few convenient properties. First, the effect of competition induces a local carrying capacity on the population, 198 leading to a finite equilibrium abundance over bounded subsets of phenotypic (or geographic) space. Second, abiotic selection prevents the abundance 200 density from diffusing too far from the abiotic optimum. In particular, when R > 1 2 √ aµ > 0,x(0) is finite, σ 2 (0) is non-negative and finite and N(0) is 202 positive and finite, this leads to a unique stable equilibrium given bŷ We demonstrate this result in SM §1.2. The equilibrial phenotypic variance 204 predicted by this model coincides with a classic quantitative genetic result predicted by modelling the combined effects of Gaussian stabilizing selection 206 and the Gaussian allelic model of mutation (Bürger, 2000, Johnson and Barton, 2005, Lande, 1975, Walsh and Lynch, 2018.

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To derive a replicator-mutator equation from equation (4), we employ integration-by-parts and the chain rule from calculus. Writing for the mean fitness, we find Equation 8b result closely resembles Kimura's continuum-of-alleles model (Kimura, 1965). The primary difference being that our model utilizes diffusion 212 instead of convolution with an arbitrary mutation kernel. However, our model of mutation can be derived as an approximation to Kimura's model, which 214 has been referred to as the Gaussian allelic approximation in reference to the distribution of mutational effects on trait values at each locus in a genome 216 (Lande, 1975, Bürger, 1986, Bürger, 2000, Johnson and Barton, 2005, the infinitesimal genetics approximation in reference to modelling continuous traits 218 as being encoded by an infinite number of loci each having infinitesimal effect (Fisher, 1919, Barton et al., 2017 and the Gaussian descendants approxima-220 tion in reference to offspring trait values being normally distributed around their parental values (Bulmer, 1971, Turelli, 2017.

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To distinguish this model from previous models of phenotypic evolution we refer to PDE (4) from which (8b) was derived as the Deterministic Asex-224 ual Gaussian allelic model with Abundance dynamics (abbreviated DAGA). Later, we will extend this model to include the effects of demographic stochas-226 ticity, which we refer to as the Stochastic Asexual Gaussian allelic model with Abundance dynamics (abbreviated SAGA).

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Evolutionary Dynamics. We now apply DAGA to derive the dynamics of mean traitx and phenotypic variance σ 2 . Both of these dynamics are expressible in 230 terms of covariances with fitness. For an abundance distribution ν(x) and associated phenotypic distribution p(x), the covariance of fitness and phenotype 232 across the population is defined as Following this, we again apply integration-by-parts and the chain rule 234 from calculus to find the dynamics of the mean traitx as Equation (10) is a continuous time analog of the well known Robertson-Price equation without transmission bias (Frank, 2012, Lion, 2018, Price, 1970, Queller, 2017, Robertson, 1966. Whether or not the covariance of fitness and 238 phenotype creates change inx to maximize mean fitnessm depends on the degree to which selection is frequency dependent (Lande, 1976). Since this 240 change is driven by a covariance with respect to phenotypic diversity, the response in mean trait to selection is mediated by the phenotypic variance. In 242 particular, when σ 2 = 0,x will not respond to selection. Following the approach taken to calculate the evolution ofx, we find the 244 response of phenotypic variation to this model of mutation and selection is In the absence of mutation equation (11)  we see that the response in σ 2 to selection can be expressed as a covariance of fitness and square error, which is defined in analogy to Cov(m(ν, x), x). Just 250 as for the evolution ofx, this covariance also creates change in σ 2 that can either increase or decrease mean fitnessm, depending on whether or not se-252 lection is frequency dependent. The effect of selection on phenotypic variance can be positive or negative depending on whether selection is stabilizing or 254 disruptive.

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In SM §4, we extend these results to include the effects of demographic stochasticity. The idea is to add an appropriate noise term to DAGA. Hence, 258 we wish to study stochastic partial differential equations (SPDE) that provide natural generalizations of DAGA. Fortunately, rigorous first principle deriva-260 tions of such SPDE have been provided by Li (1998) and Champagnat et al. (2006). The noise terms driving these SPDE are space-time white noise pro-262 cesses, denotedẆ(x, t), which are random processes uncorrelated in both space and time. In SM §2.1, we provide a set of heuristics for performing 264 calculations with respect to space-time white noise including methods to derive SDE from SPDE in analogy to our derivations of ordinary differential 266 equations (ODE) from PDE above. Since our aim is to present this material to a wide audience of mathematical evolutionary ecologists, our treatment 268 of space-time white noise and stochastic integration deviates from standard definitions to remove the need for a detailed technical treatment. However, 270 in SM §2.2, we show our heuristics are consistent with the rigorous infinitedimensional stochastic calculus presented in Da Prato and Zabczyk (2014).
Using our simplified approach, the reader will only need some elementary probability and an intuitive understanding of SDE, including Brownian mo-274 tion, in addition to the notions of Riemann integration and partial differentiation already employed.

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To understand how SPDE can be derived from biological first principles, we provide in SM §3.1 an informal discussion of measure-valued branch-278 ing processes (MVBP) (which provide individual-based models) and their diffusion-limits (which provide population-level models). Diffusion-limits of 280 MVBP return so-called superprocesses which track the evolution of abundance and phenotypic distribution (Etheridge, 2000). For univariate traits and 282 under biologically natural conditions, these superprocesses admit abundance densities satisfying SPDE. Under the simplifying assumptions inherited from 284 our treatment of deterministic dynamics and the additional assumption that the variance of individual reproductive output, denoted by V ≥ 0, is indepen-286 dent of trait values, we obtain as a special case the relatively simple expression for an SPDE that generalizes DAGA We refer to this special case as the Stochastic Asexual Gaussian allelic model with Abundance dynamics (SAGA). The simplicity of SAGA allows 290 us to use properties of space-time white noise processes to derive a set of SDE that generalize equations (8a), (10) and (11) to include the effects of demo-292 graphic stochasticity (see SM §3.2 and SM §4). In particular, we find where W N , Wx and W σ 2 are standard Brownian motions and barred expres-294 sions such as (x −x) 4 are averaged quantities with respect to the phenotypic distribution p(x, t). Intuitively, one can interpret equations (13) as if they are 296 ordinary differential equations, but this is not technically rigorous since Brownian motion is nowhere differentiable with respect to time. In SM §4 we show 298 that in general W N is independent of both Wx and W σ 2 , but Wx and W σ 2 may covary depending on the shape of p(x, t).

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Many known results follow directly from expressions (13). Firstly, assuming no variance in reproductive output so that V = 0 recovers the determin-302 istic dynamics derived in §2.1. Alternatively, one can take N → ∞ to recover the deterministic dynamics forx and σ 2 . Characteristically, we note the effect 304 of demographic stochasticity on abundance grows with √ N. Hence, dividing by N, we find the effects of demographic stochasticity on the per-capita 306 growth rate diminish with increased abundance. Relating the response to demographic stochasticity derived here to the effect of random genetic drift 308 derived in classic quantitative genetic theory, if we set σ 2 and N constant with respect to time, then integrating the stochastic term in equation (13b) over a single unit of time returns a normally distributed random variable with mean zero and variance equal to Vσ 2 /N. In particular, assuming perfect inheritance, 312 when reproductive variance is unity (V = 1) this random variable coincides with the effect of random genetic drift on the change in mean trait over a 314 single generation derived using sampling arguments (Lande, 1976). There is also an interesting connection with classical population genetics. A funda-316 mental result from early population genetic theory is the expected reduction in diversity due to the chance loss of alleles in finite populations (Fisher, 1923, 318 Wright, 1931. This expected reduction in diversity due to random genetic drift is captured by the third term in the deterministic component of expres-

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These expressions can be used to investigate the dynamics of the mean and variance for a very general set of phenotypic distributions. However, in 326 the next subsection we simplify these expressions by assuming normally distributed trait values, known as the Gaussian population assumption (Turelli 328 2017). In SM §4 we show that under the Gaussian case W N , Wx and W σ 2 are independent. Hence, although the Gaussian population assumption is very re-330 strictive as a model of phenotypic diversity and, except for very special cases of growth rates, is not formally justified, its exceedingly convenient properties 332 make it an important initial approximation.

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By assuming normally distributed trait values, expressions (13) transform into efficient tools for deriving the dynamics of populations given a fitness 336 function m(ν, x). Gaussian phenotypic distributions can be formally obtained through Gaussian, exponential or weak selection approximations together 338 with a simplified model of mutation, genotype-phenotype mapping and asexual reproduction or random mating (Bürger, 2000, Lande, 1980, Turelli, 2017, 340 1986, 1984. Hence, given appropriate assumptions on selection, mutation and reproduction, the abundance density ν(x, t) can be approximated as a 342 Gaussian curve in x when the ratio V/N is small (i.e., when the variance in reproductive output is much smaller than the population size). As with any 344 diffusion approximation, this requires a sufficiently large abundance to accurately reflect the dynamics of populations. Therefore, models developed in 346 this framework are not suitable for studies involving very small population sizes. Allowing for these restrictions, we assume Under this assumption, covariances with fitness can be written in terms of fitness gradients. In particular, we find and (x −x) 4 = 3σ 4 . These results imply trait dynamics can be rewritten as These equations allow us to derive the response in trait mean and variance 352 by taking derivatives of fitness, a much more straightforward operation than calculating a covariance for general phenotypic distributions. Note that in the 354 above expressions, the partial derivatives ofm represent frequency independent selection and the averaged partial derivatives of m represent frequency 356 dependent selection. This relationship has already been pointed out by Lande (1976) for the evolution of the mean trait in discrete time, but here we see an 358 analogous relationship holds in continuous time and also for the evolution of trait variance.

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In SM §5 we generalize this result to the case when traits are imperfectly inherited. In this case, the phenotypic variance σ 2 is replaced by a genetic 362 variance G. This genetic variance represents the component of σ 2 explained by additive effects among genetic loci encoding for the focal phenotype (Bulmer, 364 1971, Roughgarden, 1979, Walsh and Lynch, 2018. It is therefore fitting that G is referred to as the additive genetic variance. Following classical quantitative 366 genetic assumptions we find From expressions (17) we see that, under our simple treatment of inheri-368 tance, focusing on additive genetic variance G instead of the variance in expressed traits σ 2 makes no structural changes to the basic equations describing 370 the dynamics of populations. Instead we see the role played by the variance of expressed traits is now being played by the additive genetic variance. In the next section, we make use of these expressions to develop a model of diffuse coevolution in a guild of S species competing along a resource continuum.

A Model of Diffuse Coevolution
In this section we demonstrate the use of our framework by developing 376 a model of diffuse coevolution across a guild of S species whose interactions are mediated by resource competition along a single niche axis. Because our 378 approach treats abundance dynamics and evolutionary dynamics simultaneously, this model allows us to investigate the relationship between selection 380 gradients and competition coefficients, which we carry out in §3.3.

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The dynamics of phenotypic distributions and abundances have been derived above and so the only task remaining is the formulation of a fitness 384 function. Our approach mirrors closely the theory developed by MacArthur and Levins (1967), Levins (1968) and MacArthur (1972MacArthur ( , 1970MacArthur ( , 1969. The most 386 significant difference, aside from allowing evolution to occur, is our treatment of resource availability. In particular, we assume resources are replenished 388 rapidly enough to ignore the dynamics of their availability. A derivation from the MVBP framework is provided in SM §6.

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Abiotic Selection and Competition. For species i we inherit the above notation for trait value, distribution, average, variance, abundance, etc., except with an We further assume the niche center x i is normally distributed among indi-400 viduals in species i, but the niche breadth w i and total niche utilization U i are constant across individuals in species i and therefore cannot evolve. We as-402 sume resources are distributed along the niche gradient and that each species experiences heterogeneous fitness benefits at different niche locations. Taking 404 into account both resource availability and fitness benefits, we suppose individuals of species i maximize their benefits by sampling resources at niche 406 location θ i ∈ R. We assume the benefits for individuals of species i derived from resources with value ζ ∈ R decreases as (ζ − θ i ) 2 increases at a rate 408 A i ≥ 0. In the absence of competition, we further suppose individuals leave on average Q i offspring when their utilization curve is concentrated at θ i (that 410 is, when x i = θ i and w i = 0). Combining these assumptions, we denote by e i (ζ) the fitness benefits for individuals sampling at niche location ζ so that The effect of abiotic stabilizing selection on the fitness for an individual of species i with niche location x i is then given by To determine the potential for competition between individuals with niche locations x i and x j , belonging to species i and j respectively, we compute the (21) To map the degree of niche overlap to fitness, we assume competition 418 between individuals with niche locations x i and x j decreases the expected reproductive output for the individual in species i at the rate c i O ij (x i − x j ) for 420 some c i > 0. We refer to c i as the strength of competition for species i.
The Fitness Function. Assuming the effects due to competitive interactions and 422 abiotic stabilizing selection on the expected reproductive output of individuals accumulates multiplicatively, we derive in SM §6 an expression for the 424 expected reproductive output of individuals in each. Applying a series of diffusion-limits, we then find the following expressions for the growth rate 426 associated with trait value x for species i along with the population growth rate of species i: where a i is the strength of abiotic stabilizing selection on species i. The variablesb ij , b ij determine the sensitivity of competitive effects on species i to 430 differences in niche locations between species i and j. We refer to R i as the innate growth rate of species i to distinguish it from the intrinsic growth rate 432 commonly referred to in the field of population ecology. These are composite parameters given by the following expressions:

The Model
In SM §6 we combine equations (13a), (17) and (22) to find Together, equations (24) provide a synthetic model capturing the dynamics of abundance and evolution from common biological mechanisms.

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Model Behavior. Despite the convoluted appearance of system (24), there are some nice features that reflect biological reasoning. For example, the dynam-440 ics of abundance generalize Lotka-Volterra dynamics. In particular, the effect of competition with species j on the fitness of species i grows linearly with N j .

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However, as biotic selection pushesx i away fromx j , the effect of competition with species j on the fitness of species i rapidly diminishes due to the Gaussian weights capturing a reduction in niche overlap. These Gaussian weights have been usefully employed to capture interaction preference in recent investigations of coevolution in mutualistic networks (de Andreazzi et al., 2019, Medeiros et al., 2018, Guimarães et al., 2017. The divergence ofx i andx j due 448 to competition is referred to in the community ecology literature as character displacement (Brown and Wilson, 1956). We also see that the fitness of species i drops quadratically with the difference betweenx i and the abiotic optimum θ i . Hence, abiotic selection acts to pullx i towards θ i .

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The response in mean traitx i to natural selection is proportional to the amount of heritable variation in the population, represented by the additive 454 genetic variance G i . However, we have that G i is itself a dynamic quantity. Under our model, abiotic stabilizing selection erodes away heritable variation 456 at a rate that is independent of both N i andx i . The effect of competition on G i is a bit more complicated. When b ij (x i −x j ) 2 < 1, competition with 458 species j acts as diversifying selection which tends to increase the amount of heritable variation. However, when b ij (x i −x j ) 2 > 1, competition with species 460 j acts as directional selection and reduces G i . In the following subsections we demonstrate the behavior of system (24) by plotting numerical solutions and 462 investigate implications for the relationship between the strength of ecological interactions and selection.

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Community Dynamics. For the sake of illustration we numerically integrated system (24) for a richness of S = 100 species. We assumed homogeneous 466 model parameters across species in the community as summarized by Table  1. We repeated numerical integration under the two scenarios of weak and 468 strong competition. For the first scenario of weak competition we set c = 1.0 × 10 −7 and for the second scenario of strong competition we set c = 5.0 × 10 −6 .

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With these two sets of model parameters, we simulated our model for 1000.0 units of time. For both scenarios, we initialized the trait means tox i = 0.0, 472 additive genetic variances to G i = 10.0 and abundances to N i = 1000.0 for each i = 1, . . . , S.

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Temporal dynamics for each scenario are provided in Figure 1. This figure suggests weaker competition leads to smoother dynamics and a higher degree 476 of organization within the community. Considering expression (24a) we note that, all else equal, relaxed competition allows for larger growth rates which 478 promote greater abundances. From (24a) we also note that the per-capita effects on demographic stochasticity diminish with abundance. To see this,  variance of reproductive output 5.0 effects can be ignored for large populations, we find that minute asymmetries generated by demographic stochasticity remain significant drivers of community structure. In particular, although we initialized each species with identical state variables and model parameters, we found an enormous amount 492 of asymmetry in both the evolutionary and abundance dynamics and even some peculiar synchronized shifts. Although future work may show these 494 bizarre features always dissipate after the system has been given sufficient time to evolve, we see demographic stochasticity has pronounced effects on 496 communities experiencing non-equilibrium dynamics.
Although Figure 1 suggests interesting patterns in the dynamics of abun-498 dance and trait evolution, a more formal investigation is needed to better understand the relationship between them. In the following subsection we take 500 a step in this direction by approximating correlations between competition coefficients and components of selection gradients induced by interspecific 502 interactions.

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Here we investigate the relationship between competition coefficients, which measure the effect of ecological interactions on abundance dynamics, with se-506 lection gradients, which measure the magnitude and direction of selection on mean trait and trait variance. We start by considering the expressions of abso-508 lute competition coefficients implied by equations (24). However, it turns out absolute competition coefficients display some unfortunate behaviour with 510 respect to our model. We therefore introduce a slightly modified form of absolute competition coefficients. We then provide formula for the components 512 of linear and quadratic selection coefficients corresponding to the effects of interspecific interactions. Lastly, we use a high-richness (large S) approxima-514 tion to determine correlations between competition coefficients and selection gradients across the community. Associated calculations are provided in SM 516 §7.3.
Competition coefficients. Relating our treatment of resource competition to the-518 oretical community ecology, the absolute competition coefficientα ij , which measures the effect of species j on the growth rate of species i (sensu Chesson, 520 2000), becomes a dynamical quantity that can be written as is the intrinsic growth rate of species i. Then, dN i (t) can be expressed as Following our model, the classically defined absolute competition coeffi-524 cient for species i is parameterized with the intrinsic growth rate of species i appearing in the denominator. In turn, these intrinsic growth rates depend 526 on the balance between the innate growth rate R i and the effect of abiotic stabilizing selection. However, this balance further depends on mean trait and 528 additive genetic variance, which evolve freely. This leads to the potential for the signage of r i to switch between positive and negative which implies the 530 potential for infinite absolute competition coefficients. Furthermore, we see these competition coefficients are influenced by abiotic stabilizing selection in-532 stead of solely capturing the effects of inter/intraspecific interactions. Hence, we find it necessary to introduce a modification of the absolute competition 534 coefficientα ij that avoids these caveats. In particular, we define We call α ij the specific competition coefficient mediating the effects of 536 species j on the growth rate of species i. Under this parameterization, the abundance dynamics of species i is now expressed as Selection Gradients. Linear and quadratic selection gradients have been defined by Lande and Arnold (1983). While the linear selection gradient β 540 measures the effect of selection on mean trait evolution, the stabilizing selection gradient γ measures the effect of selection on additive genetic or phe-542 notypic variance. Since these quantities are classically defined with respect to discrete-time models of trait evolution, we provide the analogous definitions 544 for continuous-time models in SM §7.1. Following our model of diffuse coevolution, we then show these selection gradients can be additively partitioned 546 into components due to interactions with each species and abiotic stabilizing selection. In particular, we find the components of linear and quadratic 548 selection gradients for species i induced by species j are given respectively by With these expressions, the dynamics of mean trait and additive genetic 550 variance simplify to High-Richness Approximation. We now make use of the expressions derived 552 for competition coefficients and selection gradients to investigate their relationship. As a first pass, we assume the niche-breadths w i and intraspecific 554 variances σ 2 i are equivalent across species so that the sensitivity parameters b ij = 1/(w i + w j + σ 2 i + σ 2 j ) = b are constant across interacting pairs of species.

556
We also assume abundances N i , niche-use parameters U i , strengths of competition c i and mean traitsx i are distributed independently of each other with 558 respective means and variances denoted byN, V N ,Ū, V U ,c, V c ,x, Vx. We further assume that richness S is large and the distribution of mean trait values 560 is approximately normal. Under these assumptions we obtained analytical approximations for the 562 correlations between specific competition coefficients α ij and selection gradients β ij , γ ij . These calculations are provided in SM §7.3. In particular, we found linear selection gradients are not associated with competition coefficients (Corr(α, β) ≈ 0). However, we did find a non-trivial relationship 566 between the magnitudes of linear selection gradients and competition coefficients (Corr(α, |β|) = 0) and also between quadratic selection gradients and 568 competition coefficients (Corr(α, γ) = 0). Their expressions can be found in SM §7.3.

570
To understand if associations between competition coefficients and selection gradients tend to be positive or negative, we visualized these relation-572 ships in Figure 2. We fixed w,c, V c ,Ū, V U ,N and V N and allowed the amounts of intraspecific trait variance σ 2 and interspecific trait variance Vx to vary. We 574 found, for biologically realistic areas of parameter space, absolute values of linear selection gradients and quadratic selection gradients tend to be posi-576 tively associated with competition coefficients. Hence, if we know of competing species that strongly effect each others abundances then we can guess they 578 also impose directional and diversifying selection on one another. However, based on this information alone, we cannot guess at the direction of selection.

Conclusion
We have introduced a novel approach to derive eco-evolutionary mod-582 els using the calculus of white noise and diffusion-limits of measure-valued branching processes (MVBP) and coined SAGA, a SPDE model of phenotypic 584 evolution that accounts for demographic stochasticity. From SAGA we derived SDE that track the dynamics of abundance, mean trait and additive 586 genetic variance. Observing the expressions of these SDE, we find the effects of demographic stochasticity on the evolution of mean trait and additive 588 genetic variance characterize the effects of random genetic drift. Although Lande (1976) has previously characterized the effects of random genetic drift 590 on mean trait evolution in quantitative genetic models, the approach taken assumed constant effective population size and discrete non-overlapping gen-592 erations. In contrast, our approach shows random genetic drift is a result of demographic stochasticity for continuously reproducing populations with 594 fluctuating abundances.
To illustrate the relevance of our approach to studies of evolutionary ecol-596 ogy, we combined our SDE with classical competition theory to derive a model of diffuse coevolution. We then used this model to investigate the relationship 598 between standardized selection gradients and competition coefficients. We found absolute values of linear selection gradients and raw values of quadratic 600 selection gradients are positively related with competition coefficients. In the process, we derived expressions for competition coefficients and components 602 of selection gradients due to pairwise interactions as functions of niche-use parameters (niche breadth, total use and mean and variance of niche loca-604 tion), strength of competitive interactions and abundance. Although the framework outlined here holds great potential for develop-606 ing a synthetic theory of coevolving ecological communities, there are two technical gaps in the mathematical foundations of our approach. Firstly, we were unable to derive formal conditions under which trait means and variances remain finite for finite time. However, a result due to Evans and Perkins 610 (1994) shows that the diffusion-limit for a pair of interacting MVBP following our simple niche-based treatment of competition exist when growth rates, as 612 functions of trait values and abundances, are bounded above. This result can be easily extended to finite sets of competing species and therefore formally 614 establishes the existence of abundances as diffusion processes. Further work is needed to determine the conditions under which trait means and variances 616 exist as diffusion processes. The models studied here provide likely sufficient conditions. In particular, since diffusive mutation does not lead to "heavy-618 tailed" phenotypic distributions, we expect the mean trait and trait variance to remain finite so long as total abundance is positive, given finite initial values 620 for trait mean and variance. That is, since we have not included any processes that would cause blow-up either in mean trait or trait variance, we expect so-622 lutions of the SDE (13) to exist for all finite time t such that N(t) > 0 when |x(0)|, σ 2 (0) < +∞. This assumption appears especially well-founded under 624 quadratic stabilizing selection. Since fitness indefinitely decreases as individual trait value becomes indefinitely large (see equation (22)), the diversifying 626 effects of mutation and competition will eventually be overwhelmed by stabilizing selection. Hence quadratic stabilizing selection prevents the abundance 628 densities of populations from venturing indefinitely far from their phenotypic optima.

630
Secondly, although SDE derived under the assumption of normally distributed phenotypes provide particularly useful formula by replacing covari-632 ances between phenotype and fitness with fitness gradients, this assumption is mathematically rigorous only under deterministic dynamics and when the 634 growth rate is a linear or concave-down quadratic function of trait value. However, following our derivation based on classical competition theory, we 636 found the associated growth rate is highly non-linear. While this extreme nonlinearity is mathematically inconvenient, it also captures important biological 638 details and thus allows for a more realistic model of community dynamics. In spite of this inconsistency in our model formulation, we found resulting 640 dynamics under the assumption of normally distributed trait values retained well-founded biological intuition. Furthermore, previous work in the field of 642 theoretical quantitative genetics has demonstrated the assumption of normally distributed trait values is robust to fitness functions that select for non-normal 644 trait distributions when inheritance is given a more realistic treatment and when populations reproduce sexually (Turelli and Barton, 1994, Barton et al., 646 2017). Hence, future work is needed to extend our approach to account for sexual reproduction, more realistic models of inheritance and to investigate 648 the community-level consequences of non-normally distributed trait values.
Overall, this work demonstrates that connecting contemporary theoretical 650 approaches of evolutionary ecology with some fundamental results in the theory of measure-valued branching processes and their diffusion-limits allows 652 for the development of a rigorous, yet flexible approach to synthesizing the dynamics of abundance and distribution of quantitative characters. In particu-lar, equations (13a) and (17) provide a fundamental set of equations for deriving stochastic eco-evolutionary models involving quantitative traits. However, 656 these equations require an expression for growth rates associated with each trait value. Conveniently, equation (SM.34) in SM §3.1 provides a means to 658 derive such growth rates from individual based models. Taken together, these results provide a means to derive analytically tractable dynamics from mecha-