arXiv subjects

Xiandong Lin

Publications and source records attributed to Xiandong Lin.

4 recordsLinked to original sources

arXiv2026

Threshold dynamics of a time-periodic nonlocal dispersal SIS epidemic model with saturated incidence function and Neumann boundary conditions

In this paper, we consider a time-periodic nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with saturated incidence and Neumann boundary conditions in a spatiotemporally heterogeneous environment. First, we define the basic reproduction number for the model, which depends on dispersal rates, total population size and saturation parameters, and is quite different from those of models incorporating standard or bilinear incidence. Then, we establish its variational characterization and investigate the impacts of those parameters on it. Next, we explore the existence, uniqueness and global attractivity of the equilibria. We also analyze the asymptotic behaviors of endemic steady states for small saturation parameters, and for both small and large diffusion rates. Our results show that the saturation effect enables the total population size to have a significant impact on the disease dynamics. Furthermore, as the saturation parameter tends to zero, the basic reproduction number and the endemic equilibrium reduce to those of the standard-incidence model, respectively. Finally, we support our findings by numerical simulations.

math.AP↗

arXiv2026

Principal spectral theory and variational characterizations for nonlocal coupled cooperative systems and applications

This paper investigates the principal spectral theory of a nonlocal dispersal operator with coupled diffusion and aims to establish a variational characterization of the spectral bound for the case where the system is not strongly coupled. In this setting, a key difficulty arises since the principal eigenfunction may have components that are identically zero, rendering existing generalized eigenvalue methods inapplicable. To overcome this, we reorder the components of the operator using a permutation matrix, thereby decomposing it into suitable suboperators, and characterize the spectral bound of the original operator in terms of the spectral bounds of these suboperators. Building on this principal spectral theory, we provide a variational characterization of the basic reproduction ratio for nonlocal dispersal systems and analyze the dynamical behavior of a class of multi-genotype stem cell regeneration models with epigenetic transitions, both in the presence and absence of gene mutations, without assuming the existence of a principal eigenvalue. Furthermore, we investigate the threshold dynamics when the system is not strongly coupled.

math.AP↗

arXiv2026

Variational characterizations of weighted eigenvalue and basic reproduction ratio for nonlocal dispersal systems and application

The basic reproduction ratio is a crucial threshold parameter in infectious disease models. In nonlocal dispersal systems, its variational characterization is challenging due to the possible absence of a principal eigenvalue caused by non-compactness. In this paper, we aim to establish such a characterization even when the principal eigenvalue does not exist. To this end, we first study the spectral bound of a class of nonlocal dispersal operators, establishing a Collatz-Wielandt characterization as well as a Rayleigh-Ritz characterization when the operator is self-adjoint. Using this, we characterize the unique parameter value at which the spectral bound equals zero, covering both non-degenerate and partially degenerate cases, and subsequently obtain an explicit expression for the basic reproduction ratio. To demonstrate the utility of our theoretical framework, we apply it to a nonlocal dispersal SIS epidemic model with saturated incidence rate. The analysis shows that, in the degenerate case of the saturation coefficient, the limiting behavior of the basic reproduction ratio as the total population tends to zero is strikingly different from that in local diffusion case.

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arXiv2025

Global Dynamics of Nonlocal Diffusion Systems on Time-Varying Domains

We propose a class of nonlocal diffusion systems on time-varying domains, and fully characterize their asymptotic dynamics in the asymptotically fixed, time-periodic and unbounded cases. The kernel is not necessarily symmetric or compactly supported, provoking anisotropic diffusion or convective effects. Due to the nonlocal diffusion on time-varying domains in our systems, some significant challenges arise, such as the lack of regularizing effects of the semigroup generated by the nonlocal operator, as well as the time-dependent inherent coupling structure in kernel. By investigating a general nonautonomous nonlocal diffusion system in the space of bounded and measurable functions, we establish a comprehensive and unified framework to rigorously examine the threshold dynamics of the original system on asymptotically fixed and time-periodic domains. In the case of an asymptotically unbounded domain, we introduce a key auxiliary function to separate vanishing coefficients from nonlocal diffusions. This enables us to construct appropriate sub-solutions and derive the global threshold dynamics via the comparison principle. The findings may be of independent interest and the developed techniques, which do not rely on the existence of the principal eigenvalue, are expected to find further applications in the related nonlocal diffusion problems. We also conduct numerical simulations based on a practical model to illustrate our analytical results.

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