We present a new method to find parameter sets that allow all populations to co-exist in multi-trophic level food web models in which the outcome of competition between populations at each trophic level is determined by R* theory. The method involves sequentially destabilising an eigenvalue at the boundary equilibrium point of the winning population at each trophic level. We illustrate the procedure on a six population, three trophic level ecosystem model of a pelagic Antarctic ecosystem.
We used the method to find an initial parameter set for which all populations coexisted. Only three model evaluations were required to find a parameter set that allowed coexistence. In contrast, a random search of parameter space required an average of 250 model evaluations to find each coexistence parameter set. The method is useful for identifying regions of parameter space that have high densities of coexistence solutions.