Parameter Estimation in Complex Plankton Models using the Boundary Eigenvalue Nudging - Genetic Algorithm (BENGA) Method

The analysis of trophically complex mathematical ecosystem models is typically carried out using numerical techniques because it is considered that the number and nonlinear nature of the equations involved makes the use of analytic techniques virtually impossible. In particular, building such models is a notoriously difficult task; most competing populations in many ecosystem models collapse (i.e. have multiple spurious extinctions). This is a realisation in silico of the Principle of Competitive Exclusion (Gause 1932, 1934) that stipulates that only as many competing populations can coexist as there are resources to support them. The incongruity of Gause's laboratory findings with the natural world led Hutchinson (1961) to propose the Paradox of the Plankton. This paradox articulated the observation that many populations of plankton coexisted in the ocean in apparent contradiction of Gause's Principle and the predictions of the simple theoretical models of the day. Whilst many solutions to the Paradox of the Plankton of varying robustness have been proposed (for example, Huisman and Weissing 1999, Schippers et al. 2001, Cropp and Norbury 2012b) the parameterization of complex ecosystem models remains a significant practical challenge (Cropp and Norbury 2013) that applied modelers need to overcome (for example, Gabric et al. 2008). The conservative normal (CN) framework articulates a number of ecological axioms that govern ecosystems by exploiting the properties of systems that are written in Kolmogorov form. Previous work has shown that trophically simple models developed within the CN framework are mathematically tractable, simplifying analysis. By exploiting the properties of Kolmogorov ecological systems it is possible to ensure models have particular properties, such as all populations remaining extant, into an ecological model. Here we demonstrate the usefulness of analytical results known for models of Kolmogorov form to construct a trophically complex ecosystem model. We also show that the properties of Kolmogorov ecological systems can be exploited to create ecosystem models with structural coexistence, that is, models for which no population goes extinct for all realistic (i.e. positive) parameter sets. We utilize this property in conjunction the key attribute of Kolmogorov systems, that closed form analytical expressions for the eigenvalues associated with populations that are zero at an equilibrium point are trivially obtained, to provide a computationally efficient method for the refinement of model parameters. Genetic Algorithms (GAs) are commonly used to estimate parameter sets that allow ecological models to reproduce observed data or theoretical objectives (Kristensen et al. 2003). It is a well-known property of GAs and other optimization approaches that convergence slows as the dimension of the solution space increases (Fournier et al. 2011), that is, as the number of parameters required to be estimated increases. We combine the properties of Kolmogorov systems with a GA to increase the rate of convergence by preconditioning the problem to a region of the solution space where desirable solutions reside. The method can be used to precondition parameter values used in standard optimization techniques, such as genetic algorithms, to significantly improve convergence towards a target equilibrium state. We use an equilibrium state community composition as this is an observed property of systems with structural coexistence, and provides a useful and easily derived goal function in the absence of measured data. However, the method is equally applicable to any chosen point in the solution space of a model.