Exchange of plant growth energy and hydrodynamical energy in a Riverine ecosystem
A second order ordinary differential equations model (SODE) of a vegetated riverine ecosystem, amalgamating hydrological equations with ecological equations, is presented for the first time. After formulating plant production equations based on minimization of Medawar Growth Energy, hydrodynamical equations are introduced describing ideal water flow under a potential gradient and quadratic disapative terms, the latter due to vegetated drag from the plant´s deep rooted flexible stems. The drag coefficients are seen to depend on the plant biomass, and therefore, couple the 4 equations into an integrated whole indicated by the non-vanishing Berwald-Gauss curvature. Sediment impact on growth occurs via transport of clay in suspension, carrying minerals beneficial to the plants and is incorporated into the equations (SODE) via anholonomic frames and semi-symmetric connections. The set of constant total energy trajectories forms a Finslerian manifold with the Synge action-metric and their stability is studied. Several consequences are derived. In particular, it is proved that the hydrdynamical subsystem is unstable unless the elevation above sea level is an explicit function of both length along the riverbed and its orthoganal extent, the so-called wetted perimeter.