Increasing empirical evidence suggests that many genetic variants influence multiple distinct phenotypes. When cross-phenotype effects exist, multivariate association methods that consider pleiotropy are often more powerful than univariate methods that model each phenotype separately. Although several statistical approaches exist for testing cross-phenotype effects for common variants, there is a lack of similar tests for gene-based analysis of rare variants. In order to fill this important gap, we introduce a statistical method for cross-phenotype analysis of rare variants using a nonparametric distance-covariance approach that compares similarity in multivariate phenotypes to similarity in rare-variant genotypes across a gene. The approach can accommodate both binary and continuous phenotypes and further can adjust for covariates. Our approach yields a closed-form test whose significance can be evaluated analytically, thereby improving computational efficiency and permitting application on a genome-wide scale. We use simulated data to demonstrate that our method, which we refer to as the Gene Association with Multiple Traits (GAMuT) test, provides increased power over competing approaches. We also illustrate our approach using exome-chip data from the Genetic Epidemiology Network of Arteriopathy.

Structured population models, particularly size- or age-structured, have a long history of informing conservation and natural resource management. While size is often easier to measure than age and is the focus of many management strategies, age-structure can have important effects on population dynamics that are not captured in size-only models. However, relatively few studies have included the simultaneous effects of both age- and size-structure. To better understand how population structure, particularly that of age and size, impacts restoration and management decisions, we developed and compared a size-structured integral projection model (IPM) and an age- and size-structured IPM, using a population of Crassostrea gigas oysters in the northeastern Pacific Ocean. We analyzed sensitivity of model results across values of local retention that give populations decreasing in size to populations increasing in size. We found that age- and size-structured models yielded the best fit to the demographic data and provided more reliable results about long-term demography. Elasticity analysis showed that population growth rate was most sensitive to changes in the survival of both large (>175 mm shell length) and small (<75 mm shell length) oysters, indicating that a maximum size limit, in addition to a minimum size limit, could be an effective strategy for maintaining a sustainable population. In contrast, the purely size-structured model did not detect the importance of large individuals. Finally, patterns in stable age and stable size distributions differed between populations decreasing in size due to limited local retention and populations increasing in size due to high local retention. These patterns can be used to determine population status and restoration success. The methodology described here provides general insight into the necessity of including both age- and size-structure into modeling frameworks when using population models to inform restoration and management decisions.

Positive density dependence (i.e., Allee effects) can create a threshold of population states below which extinction of the population occurs. The existence of this threshold, which can often be a complex, multi-dimensional surface, rather than a single point, is of particular importance in degraded populations for which there is a desire for successful restoration. Here, we incorporated positive density dependence into a closed, size- and age-structured integral projection model parameterized with empirical data from an eastern oyster, Crassostrea virginica, population in Pamlico Sound, North Carolina, USA. To understand the properties of the threshold surface, and implications for restoration, we introduced a general method based on a linearization of the threshold surface at its unique, unstable equilibrium. We estimated the number of oysters of a particular age (i.e., stock enhancement), or the surface area of dead shell substrate required (i.e., habitat enhancement) to move a population from an extinction trajectory to a persistence trajectory. The location of the threshold surface was strongly affected by changes in the amount of local larval retention. Traditional stock enhancement with oysters <1 yr old (i.e., spat) required three times as many oysters relative to stock enhancement with oysters between ages 3 and 7 yr old, while the success of habitat enhancement depended upon the initial size distribution of the population. The methodology described here demonstrates the importance of considering positive density dependence in oyster populations, and also provides insights into effective management and restoration strategies when dealing with a high dimensional threshold separating extinction and persistence.

At the spatial scale relevant to many field studies and management policies, populations may experience more external recruitment than internal recruitment. These sources of recruitment, as well as local demography, are often subject to stochastic fluctuations in environmental conditions. Here, we introduce a class of stochastic models accounting for these complexities, provide analytic methods for understanding their long‐term behaviour and illustrate the application of methods to two marine populations. The population state n(x) of these stochastic models is a function or vector keeping track of densities of individuals with continuous (e.g. size) or discrete (e.g. age) traits x taking values in a compact metric space. This state variable is updated by a stochastic affine equation nt+1 = At+1nt + bt+1 where is At+1 is a time varying operator (e.g. an integral operator or a matrix) that updates the local demography and bt+1 is a time varying function or vector representing external recruitment. When the realized per‐capita growth rate of the local demography is negative, we show that all initial conditions converge to the same time‐varying trajectory. Furthermore, when A1,A2,… and b1,b2,… are stationary sequences, this limiting behaviour is determined by a unique stationary distribution. When the stationary sequences are periodic, uncorrelated or a mixture of these two types of stationarity, we derive explicit formulas for the mean, within‐year covariance and autocovariance of the stationary distribution. Sensitivity formulas for these statistical features are also given. The analytic methods are illustrated with applications to discrete size‐structured models of space‐limited coral populations and continuously size‐structured models of giant clam populations.

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