Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.08387

Wave Propagation Dynamics via Lattice Difference Equations

Eddy Kwessi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop and analyze a lattice difference equation (LDE) framework to model the spatial dynamics of invasion in populations. This framework extends beyond classical integro-difference and reaction-diffusion models by incorporating spatial discreteness and habitat fragmentation more faithfully, making it well-suited for urban and patchy landscapes. We characterize the local stability of equilibria, and demonstrate the existence of traveling wave solutions. A key focus is on how dispersal kernels--ranging from Gaussian to Cauchy--interact with the Allee effect to influence wave formation, propagation speed, and invasion success. Our numerical simulations reveal that long-tailed kernels can overcome the Allee threshold through seeding effects, significantly accelerating wave fronts. These findings have direct implications for vector control strategies, informing optimal release thresholds and spatial targeting in heterogeneous environments. Furthermore, we derive a stochastic characterization of outbreak size as a Poisson-binomial distribution, offering probabilistic insight into local infection burden and paving the way to a novel minimization criteria of release strategies based on bi-modality. Our results provide a mathematically grounded basis for predicting spatial spread and optimizing release strategies for {\it Wolbachia}-based vector control programs. This work bridges theory and application, providing both analytical insights and computational tools for understanding spatial epidemiology in discrete habitats.

Explore related subjects

math.DS

Keep this discovery

BibTeXRIS

Eddy Kwessi. 2025-08-11. Wave Propagation Dynamics via Lattice Difference Equations. https://arxiv.org/abs/2508.08387

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗