A logistic equation with degenerate diffusion and Robin boundary conditions
Antonio Suárez
Communications on Pure & Applied Analysis 2008, 7(5): 1255-1267 doi: 10.3934/cpaa.2008.7.1255
In this paper we study a model for a species confined in a bounded region. This species diffuses slowly, follows a logistic law in the habitat and there is a flux of population across the boundary of the habitat.
Basically, we give some theoretical results of the model depending on some parameters which appear in the model.
keywords: Logistic equation bifurcation methods. degenerate diffusion Robin boundary conditions
Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient
Giovany M. Figueiredo Tarcyana S. Figueiredo-Sousa Cristian Morales-Rodrigo Antonio Suárez
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-23 doi: 10.3934/dcdsb.2018311

In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.

keywords: Bifurcation method variable local and nonlocal diffusion coefficient
On a chemotaxis model with competitive terms arising in angiogenesis
Manuel Delgado Inmaculada Gayte Cristian Morales-Rodrigo Antonio Suárez
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 177-202 doi: 10.3934/dcdss.2020010

In this paper we study an anti-angiogenic therapy model that deactivates the tumor angiogenic factors. The model consists of four parabolic equations and considers the chemotaxis and a logistic law for the endothelial cells and several boundary conditions, some of them are non homogeneous. We study the parabolic problem, proving the existence of a unique global positive solution for positive initial conditions, and the stationary problem, justifying the existence of one real number, an eigenvalue of a certain problem, which determines if the semi-trivial solutions are stable or unstable and the existence of a coexistence state.

keywords: Chemotaxis cross-diffusion anti-angiogenic therapy angiogenesis bifurcation
Biodiversity and vulnerability in a 3D mutualistic system
Giovanny Guerrero José Antonio Langa Antonio Suárez
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4107-4126 doi: 10.3934/dcds.2014.34.4107
In this paper we study a three dimensional mutualistic model of two plants in competition and a pollinator with cooperative relation with plants. We compare the dynamical properties of this system with the associated one under absence of the pollinator. We observe how cooperation is a common fact to increase biodiversity, which it is known that, generically, holds for general mutualistic dynamical systems in Ecology as introduced in [4]. We also give mathematical evidence on how a cooperative species induces an increased biodiversity, even if the species is push to extinction. For this fact, we propose a necessary change in the model formulation which could explain this kind of phenomenon.
keywords: increasing biodiversity 3D mutualistic model global stability. permanence vulnerability
Some eigenvalue problems with non-local boundary conditions and applications
Rafael Abreu Cristian Morales-Rodrigo Antonio Suárez
Communications on Pure & Applied Analysis 2014, 13(6): 2465-2474 doi: 10.3934/cpaa.2014.13.2465
In this paper we study an elliptic eigenvalue problem with non-local boundary condition. We prove the existence of the principal eigenvalue and its main properties. As consequence, we show the existence and uniqueness of positive solution of a nonlinear problem arising from population dynamics.
keywords: Non-local boundary condition eigenvalue problem.
Anti-angiogenic therapy based on the binding to receptors
Manuel Delgado Cristian Morales-Rodrigo Antonio Suárez
Discrete & Continuous Dynamical Systems - A 2012, 32(11): 3871-3894 doi: 10.3934/dcds.2012.32.3871
This paper deals with a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to specific receptors of the endothelial cells. We study the time-dependent problem as well as the stationary problem associated to it.
keywords: Chemotaxis anti-angiogenic therapy bifurcation.

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