A nonlinear L^2 stability analysis for a spatial ecology model based on Luckinbill's experiment

In the last 30 years the use of appropriate reaction terms in (spatial) ecology models has attracted the attention of several researchers. In particular, for prey-predator type systems, the choice of reaction terms has been based on the results of several experimental studies made to test the effect of predator’s density changes upon the feeding rate per unit predator. Many experimental results indicate that the feeding rate per unit predator react strongly to changes in density of predators. Beddington (1975) and De Angelis et al (1975), contemporarely but independently, suggested to include negative effects of increased predator biomass on the feeding rate through a very simple term that can simulate many cases of trophic interaction characterized by the predator interface. In the present communication the L^2 stability of the coexistence equilibrium state have been performed through a new methodology introduced by Prof. S. Rionero which consists in using some peculiar functionals which link stability - instability results directly to the eigenvalues of the linear operator of the problem at hand.