arXiv subjects
Publications and source records attributed to Leoncio Rodriguez Quinones.
We describe a shape derivative approach to provide a candidate for an optimal domain among non-simply connected planar domains with two boundary components. This approach is an adaptation of the work on the extremal eigenvalue problem for the Wentzell-Laplace operator developed by Dambrine, Kateb and Lamboley.
In this note we present a study of the solutions associated to a particular spatial extension of the Rosenzweig-MacArthur model for predator and prey. The analysis presented here shows that positive steady state solutions emerge via a transcritical bifurcation mechanism, in accordance with the insight obtained from previous numerical and analytical results. In the model under discussion, prey is assumed to move avoiding crowds via a density-dependent diffusion and also incorporates the existence of a refuge zone, where predators cannot consume prey. Saturation in prey consumption is also included through a Holling type II functional response.
We study a spatial (two-dimensional) Rosenzweig-MacArthur model under the following assumptions: $(1)$ prey movement follows a nonlinear diffusion, $(2)$ preys have a refuge zone (sometimes called "protection zone") where predators cannot enter, (3) predators move following linear diffusion. We present a bifurcation analysis for the system that shows the existence of positive solutions at the steady state. We complement the theoretical results with numerical computations and compare our results with those obtained in the case of having linear diffusion for the prey movement. Our results show that both models, with linear and nonlinear diffusion for the prey, have the same bifurcation point and the positive solution curves are virtually the same in a neighborhood of this point, but they get drastically different as the bifurcation parameter approaches to zero.