Robust optimal observation of a metapopulation

A metapopulation consists of interacting populations, each occupying distinct spatially separated patches of habitat. Modelling these populations has become increasingly important because anthropogenic impacts on spatially homogeneous populations have led to increased habitat fragmentation and accidental introduction of invasive species. We employ a two-parameter continuous-time Markovian model for patch occupancy, which has been used previously to study the spread of disease in closed populations. This model takes into account proximity of patches, rather than any detailed spatial structure. Our model will be particularly suited to studying species occupying marine environments that are subject to complex ocean currents, where the connectivity between habitat patches is dynamic, and most applicable to invertebrate and plant species that are dispersed passively by these currents. We address the problem of designing an optimal sampling scheme that specifies when to observe the population in order to obtain the most accurate and precise estimates of the parameters. Our approach is based on approximating the true likelihood of observing the numbers of occupied patches at a particular sequence of times by a much simpler Gaussian likelihood, obtained from a diffusion approximation of the underlying discrete-state Markov process. This approximation is known to be highly accurate when the number of patches is sufficiently large. Furthermore, when coupled with an appropriate optimization technique such as the Cross-Entropy Method (used here), the approximation gives rise to a robust procedure for determining the optimal observation schedule, one which is considerably simpler than would otherwise be possible. We investigate the performance of two design criteria: ED-optimality and a particular Maximin-optimality criterion. ED optimal design stems from the well known D-optimal design, but where the design is chosen to maximize the expected value of the D-optimality criterion (the determinant of the Fisher Information matrix), given priors on the parameters. The Maximin-optimal design maximizes the minimum value of the D-optimality criteria given the priors. These criteria can yield robust designs, which allow one to incorporate prior belief about the parameter values before any data is collected. We investigate ‘informative’ and ‘less informative’ priors for each parameter in combination with both design criteria. By assuming a true parameter set for a given metapopulation, we are able to compare kernel density estimates of the maximum likelihood estimator under the various combinations of optimality criteria and prior distributions. We illustrate our methods with reference to a model for the spread of crown of thorns starfish (Acanthaster planci) amongst the 55 islands comprising the Ryukyu group in Japan. Our results suggest that ED-optimality criteria can sometimes lead to bimodality in distribution of the maximum likelihood estimator. However, estimation appears to be greatly improved if the ED criteria is used in conjunction with less informative priors for the parameter that governs the rate of drift towards equilibrium. Interestingly, our Maximin-optimality criteria does not give rise to such bimodality, but does result in a considerable loss in precision.