Gaussian process framework for temporal dependence and discrepancy functions in Ricker-type population growth models

Abstract

Density dependent population growth functions are of central importance to population dynamics modelling because they describe the theoretical rate of recruitment of new individuals to a natural population. Traditionally, these functions are described with a fixed functional form with temporally constant parameters and without species interactions. The Ricker stock-recruitment model is one such function that is commonly used in fisheries stock assessment. In recent years, there has been increasing interest in semiparametric and temporally varying population growth models. The former are related to the general statistical approach of using semiparametric discrepancy functions, such as Gaussian processes (GP), to model deviations around the expected parametric function. In the latter, the reproductive rate, which is a key parameter describing the population growth rate, is assumed to vary in time. In this work, we introduce how these existing Ricker population growth models can be formulated under the same statistical approach of hierarchical GP models. We also show how the time invariant semiparametric approach can be extended and combined with the time varying reproductive rate using a GP model. Then we extend these models to the multispecies setting by incorporating cross-covariances among species with a continuous time covariance structure using the linear model of coregionalization. As a case study, we examine the productivity of three Pacific salmon populations. We compare the alternative Ricker population growth functions using model posterior probabilities and leave-one-out cross validation predictive densities. Our results show substantial temporal variation in maximum reproductive rates and reveal temporal dependence among the species, which have direct management implications. However, our results do not support inclusion of semiparametric discrepancy function and they suggest that the semiparametric discrepancy functions may lead to challenges in parameter identifiability more generally.