Testing a method for analyzing the effect of parameter change in climate driven ecological systems

In this study, we consider a herbivore-predator metapopulation model, consisting of two patches. Only the herbivores are allowed to migrate between the patches. Furthermore, the intrinsic growth rates of the herbivores depend on climatic fluctuations. Our objective is the conservation of the herbivores. We are interested in finding the parameter perturbations that, when applied, will reduce the extinction risk of one of the subpopulations the most. The risk of extinction is measured in terms of the fifth percentiles of the subpopulations. This is the value below which the subpopulation is found 5 out of 100 times in a series of values taken at fixed time intervals. We make use of the short-term behavior of the model by formulating the tangent linear equations. In the neighborhood of a short interval of a reference orbit the linear error growth can then be calculated. The parameter perturbation that causes the largest error growth, the so-called first singular vector, can be computed with the use of adjoint equations. The adjoint system acts as a backward integration. It turns out that at certain moments in time such a singular vector has a direction for which a parameter perturbation is likely effective in changing the dynamics in a long model simulation. The selection is based on a specific local behavior of the error growth. This adjoint method is compared with a method in which parameter perturbations are randomly chosen. Here we carry out a test for a model with only few parameters. However, it can be applied to models with a very large number of parameters, making the adjoint method an interesting alternative for the random method as then the required number of runs cannot be realized within a realistic computing time. A same objection holds for the use of a systematic method of finding an optimum, e.g. by computing the gradient in the fifth percentile with respect to the parameter values.