Julián López-Gómez

Spectral Theory and Nonlinear Functional Analysis

INTRODUCTION General Assumptions and Basic Concepts Some New Results Historical Remarks BIFURCATION FROM SIMPLE EIGENVALUES Simple Eigenvalues and Transversality The Theorem of M.G. Crandall and P.H. Rabinowitz Local Bifurcation Diagrams The Exchange Stability Principle Applications FIRST GENERAL BIFURCATION RESULTS Lyapunov-Schmidt Reductions The theorem of J. Ize The Global Alternative of P.H. Rabinowitz The Theorem of D. Westreich THE ALGEBRAIC MULTIPLICITY Motivating the Concept of Transversality Transversal Eigenvalues Algebraic Eigenvalues Analytic Families Simple Degenerate Eigenvalues FUNDAMENTAL PROPERTIES OF THE MULTIPLICITY The Multiplicity of R.J. Magnus Relations between c and m The Fundamental Theorem The Classical Algebraic Multiplicity Finite Dimensional Characterizations The Parity of the Crossing Number GLOBAL BIFURCATION THEORY Preliminaries Local Bifurcation Global Behavior of the Bounded Components Unilateral Global Bifurcation Unilateral Bifurcation for Positive Operators APPLICATIONS Positive Solutions o Semilinear Elliptic Problems Coexistence States for Elliptic Systems Examples A Further Application REFERENCES INDEX

Coexistence states and global attractivity for some convective diffusive competing species models

In this paper we analyze the dynamics of a general competing species model with diffusion and convection. Regarding the interaction coefficients between the species as continuation parameters, we obtain an almost complete description of the structure and stability of the continuum of coexistence states. We show that any asymptotically stable coexistence state lies in a global curve of stable coexistence states and that Hopf bifurcations or secondary bifurcations only may occur from unstable coexistence states. We also characterize whether a semitrivial coexistence state or a coexistence state is a global attractor. The techniques developed in this work can be applied to obtain generic properties of general monotone dynamical systems.

Metasolutions: Malthus versus Verhulst in Population Dynamics. A Dream of Volterra

Abstract This paper analyzes how the study of the interplay within the same habitat between the most classical laws of Population Dynamics is originating the modern theory of nonlinear parabolic differential equations, where metasolutions are imperative for ascertaining the dynamics in the regimes where they cannot be described with classical solutions.

Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems

In this work we analyze the existence, multiplicity and stability of positive solutions for a class of indefinite superlinear elliptic boundary value problems. The main contribution of this paper consists in the change of mind inherent to the fact of adding the superlinear amplitude e as an unfolding parameter. This change of mind allows us to unify many previous results obtained separately in the literature, it helps us to realize the global structure of the set of positive steady states, and provides us with a great variety of new general results. Our techniques can be applied to much more general equations and systems.

Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model

We analyse the dynamics of a prototype model for competing species with diffusion coefficients ( d 1 d 2 ) in a heterogeneous environment Ω. When diffusion is switched off, at each point x ∊ Ω we have a pair of ODEs: the kinetic . If for some x ∊ Ω kinetic has a unique stable coexistence state, we show that there exist such that for every the RD-model is persistent , in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there exist x u , x v ∊ Ω such that the semitrivial coexistence states ( u , 0) and (0, v ) of the kinetic are globally asymptotically stable at x = x u and x = x v , respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients. Singular perturbation techniques, monotone schemes, fixed point index, global analysis of persistence curves , global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems. We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model has three or more coexistence states.

Coexistence in a simple food chain with diffusion

We show the global existence of classical positive solutions in each component of a Lotka-Volterra system with diffusion and logistic growing conditions. We are mainly interested in the search of coexistence states solving the associated elliptic problem under homogeneous Dirichlet boundary conditions.

Optimal multiplicity in local bifurcation theory I. Generalized generic eigenvalues

Abstract In this paper we introduce a class of eigenvalues for a family of operators depending on a real parameter. For this class of eigenvalues we define a multiplicity concept which is the best in local bifurcation theory. In Part II we shall reduce the general situation to the one considered here.

Linear second order elliptic operators

The Minimum Principle Classifying Supersolutions of Linear Elliptic Problems The Theorems of Stampacchia and Lax - Milgram Existence of Weak Solutions Regularity of Weak Solutions The Krein - Rutman Theorem The Strong Maximum Principle Properties of the Principal Eigenvalue.

Semiclassical analysis of general second order elliptic operators on bounded domains

In this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator, not necessarily selfadjoint, on a bounded domain. Our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for C?? perturbations of the Laplacian.

Multiparameter local bifurcation based on the linear part

Abstract In this paper we give a result in multiparameter local bifurcation theory. This result is a generalization of the Hopf bifurcation theorem and of a previous result by J. K. Hale. Our result can be applied to Fredholm operators with arbitrary index.