Mathematical tools for controlling invasive species in protected areas

A challenging task in the management of Protected Areas is to control the spread of invasive species, either floristic or faunistic, and the preservation of indigenous endangered species, typically competing for the use of resources in a fragmented habitat. In this paper, we present some mathematical tools that have been recently applied to contain the worrying diffusion of wolf-wild boars in a Southern Italy Protected Area belonging to the Natura 2000 network. They aim to solve the problem according to three different and in some sense complementary approaches: (i) the qualitative one, based on the use of dynamical systems and bifurcation theory; (ii) the Z-control, an error-based neural dynamic approach; (iii) the optimal control theory. In the case of the wild-boars, the obtained results are illustrated and discussed. To refine the optimal control strategies, a further development is to take into account the spatio-temporal features of the invasive species over large and irregular environments. This approach can be successfully applied, with an optimal allocation of resources, to control an invasive alien species infesting the Alta Murgia National Park: Ailanthus altissima. This species is one of the most invasive species in Europe and its eradication and control is the object of research projects and biodiversity conservation actions in both protected and urban areas [11]. We lastly present, as a further example, the effects of the introduction of the brook trout, an alien salmonid from North America, in naturally fishless lakes of the Gran Paradiso National Park, study site of an on-going H2020 project (ECOPOTENTIAL).