arXiv subjects

Maciej Tadej

Publications and source records attributed to Maciej Tadej.

3 recordsLinked to original sources

arXiv2026

Eigenvalues and eigenfunctions of the non-local dispersal with Neumann-type boundary condition and symmetric kernel

In this paper, we investigate the eigenvalue problem for a non-local dispersal operator defined on a bounded spatial domain with Neumann-type boundary conditions. Unlike the classical Laplacian, the non-local operator lacks compactness, which gives rise to an essential (continuous) spectrum and severely complicates the study of its discrete eigenvalues. The main contribution of this work is the rigorous variational construction of a finite or infinite sequence of non-trivial eigenfunctions corresponding to isolated eigenvalues located strictly above the continuous spectrum. Since the existence of these eigenvalues is not generally guaranteed due to the potential collapse of the spectral gap, we establish explicit sufficient geometric conditions linking the domain's size and geometry with the variance of the dispersal kernel that assure the emergence of the principal eigenvalue. Subsequent eigenpairs are constructed inductively via finite-dimensional Galerkin approximations. Utilizing a decomposition of the operator into compact and multiplicative part, we prove the strong convergence of the minimizing sequences.

math.SP↗

arXiv2026

Extinction and persistence criteria in non-local Klausmeier model of vegetation dynamics on flat landscapes

This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system and provide a rigorous analysis of the conditions required for vegetation survival. Our results identify a critical patch size governed by the trade-off between local growth and boundary losses; habitats smaller than this threshold lead to inevitable extinction. Furthermore, we derive a critical maximal biomass density below which the population collapses to a desert state, regardless of the domain size. We determine stability criteria for stationary solutions and describe the emergence of stable, non-trivial biomass distributions. Numerical experiments comparing sub-Gaussian and super-Gaussian kernels confirm that non-local dispersal mechanisms, particularly those with fat tails, enhance ecosystem resilience by allowing vegetation to persist in smaller, fragmented habitats than predicted by classical local diffusion models.

math.AP↗

arXiv2024

Long time behaviour of solutions to non-local and non-linear dispersion problems

This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the $n$-dimensional Euclidean space or a periodic environment modeled by $n$-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.

math.AP↗