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We develop a novel advanced Particle Markov chain Monte Carlo algorithm thatis capable of sampling from the posterior distribution of non-linear statespace models for both the unobserved latent states and the unknown modelparameters. We apply this novel methodology to five population growth models,including models with strong and weak Allee effects, and test if it canefficiently sample from the complex likelihood surface that is often associatedwith these models. Utilising real and also synthetically generated data sets weexamine the extent to which observation noise and process error may frustrateefforts to choose between these models. Our novel algorithm involves anAdaptive Metropolis proposal combined with an SIR Particle MCMC algorithm(AdPMCMC). We show that the AdPMCMC algorithm samples complex, high-dimensionalspaces efficiently, and is therefore superior to standard Gibbs or MetropolisHastings algorithms that are known to converge very slowly when applied to thenon-linear state space ecological models considered in this paper.Additionally, we show how the AdPMCMC algorithm can be used to recursivelyestimate the Bayesian Cram\'er-Rao Lower Bound of Tichavsk\'y (1998). We deriveexpressions for these Cram\'er-Rao Bounds and estimate them for the modelsconsidered. Our results demonstrate a number of important features of commonpopulation growth models, most notably their multi-modal posterior surfaces anddependence between the static and dynamic parameters. We conclude by samplingfrom the posterior distribution of each of the models, and use Bayes factors tohighlight how observation noise significantly diminishes our ability to selectamong some of the models, particularly those that are designed to reproduce anAllee effect.

Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC)