Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi--Dirac statistic functions

If the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter-Gum\-mel scheme \cite{Sch_Gu}, understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator \nabla \cdot {\bal \ff(v)} \nabla , $v$ chemical potential, while the hole density is defined by $p={\cal F}(v)\le e^v.$ A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of {\bal \ff(v)} and ${\cal F}(v)$

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented