Study of the optimal harvesting control for an elliptic problem with slow diffusion

In this paper our aim is to present a survey of known results of an optimal control problem with concave non-quadratic cost functional and a state equation arising from population dynamics. First, we study in detail the state equation, and then we show existence and uniqueness of optimal control and also a numerical approximation of the optimal control.Ministerio de Ciencia y Tecnologí

Algunos modelos de la dinámica de poblaciones

En este trabajo presentamos el estudio teórico de algunos modelos de ecuaciones en derivadas parciales que aparecen en la modelización de la dinámica de poblacione

A logistic equation with degenerate diffusion and Robin boundary condition

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.Ministerio de Educación y Cienci

Degenerate competition problem

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as uniqueness of positive solution. In order to obtain the results we employ monotonicity methods, singular spectral theory and a fixed point index.Ministerio de Ciencia y Tecnologi

Some remarks on the comparison principle in Kirchhoff equations

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y Competitivida

Biodiversity and vulnerability in a 3D mutualistic system

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.Fondo Europeo de Desarrollo Regional MTM2011-22411Ministerio de Economía y Competitividad (España) MTM2011-22411Ministerio de Ciencia e Innovación (España) MTM2009-12367Fondo Europeo de Desarrollo Regional MTM2009-1236

Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term

The main goal of this work is to study the existence and uniqueness of positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a-priori estimates. To prove uniqueness, the sweeping method of Serrin is employed.Ministerio de Ciencia y TecnologíaJunta de Andalucí

Cooperative systems with any number of species

In this paper we study the positive solutions of a cooperative system of any number of equations which considers the case of the slow diffusion and includes the Lotka-Volterra model. We determine conditions of existence of global solution and blow-up in finite time in term of the value of the spectral radius of a certain nonnegative matrix associated to the system. The results generalize the ones known for the particular case of two equations and we justify them by using the specific properties of nonnegative matrices which translate the cooperative character of the system.Comisión Interministerial de Ciencia y Tecnologí

On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem

This paper concerns with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation method to obtain our results.Ministerio de Ciencia y Tecnologí