Principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type

In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed inVo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119, 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.

Automated summary

This paper studies the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. It provides two general sufficient conditions for the existence of the principal eigenvalue and proves that the semigroup generated by solutions of the model exhibits asynchronous exponential growth. It also overcomes the non-monotonicity issue of the principal eigenvalue of a nonlocal Neumann operator and establishes the strong maximum principle for the age-structured operator with nonlocal diffusion.