Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

Abstract

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.

References in corpus (9)

• A Generalized Krein-Rutman Theorem
• Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon
• Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator
• Existence and qualitative properties of travelling waves for an epidemiological model with mutations
• A homogenization approach for the motion of motor proteins
• When can a population spreading across sink habitats persist ?
• Pulsating fronts for Fisher-KPP systems with mutations as models in evolutionary epidemiology
• Propagation dynamics of solutions to spatially periodic reaction-diffusion systems with hybrid nonlinearity
• Nonconcavity of the Spectral Radius in Levinger's Theorem