Optimising the carrying capacity in logistic diffusive models: Some qualitative results
Résumé
We revisit the problem of optimising (either maximising or minimising) the total biomass or wider classes of criteria in logistic-diffusive models. Since [24], a lot of effort has been devoted to the qualitative understanding of the following question: how should one spread resources in order to maximise or minimise the total biomass? This question was studied in detail, mostly in situations where the carrying capacity of the environment and the growth rate of the population are equal. Following recent contributions [7], [11], [40], we propose a mathematical treatment of this question in contexts where the growth rate and the carrying capacity are de-correlated. We investigate the optimisation of generic criteria with respect to the carrying capacity. We settle the problem of existence by providing the proper relaxation of the optimisation problems, using homogenisation-type arguments. At a qualitative level, we obtain theoretical results backing up the observations and other results of [7], [11], [40] regarding the optimality of homogeneous environments. We similarly investigate in details the question of minimising the total biomass. We also provide several stationarity results (in the context of large diffusivities) for the minimisation of the biomass and related criteria.