Finite element approximation of a surface-subsurface coupled problem arising in forest dynamics

We propose a finite element approximation for an evolution model describing the spatial population distribution of two salt tolerant plant species, such as mangroves, which are affected by inter- and intra-specific competition (Lotka-Volterra), population pressure (cross-diffusion) and environmental heterogeneity (environmental potential). The environmental potential and the Lotka-Volterra terms are assumed to depend on the salt concentration in the roots region, which may change as a result of mangroves ability for uptaking fresh water and leave the salt of the solution behind, in the saturated porous medium. Consequently, partial differential equations modeling the population dynamics on the surface are coupled with Darcy-transport equations modeling the salt and pressure-velocity distribution in the subsurface. We provide a numerical discretization based on a stabilized mixed finite element method for the transport-Darcy flow problem coupled to a finite element method for a regularized version of the cross-diffusion population model, which we use to numerically demonstrate the behavior of the system.