Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

许多研究人员已经研究了随机扩散的年龄结构人口模型。尽管与随机扩散过程相比，非局部扩散过程更适用于许多生物和物理问题，但关于具有非局部扩散的年龄结构人口模型的理论结果却很少。在本文中，我们的目标是发展具有非局部扩散的年龄结构人口动态的基本理论。特别是，我们研究了与具有非局部扩散的年龄结构模型相关的线性算子半群，并使用其无穷小发生器的光谱特性来确定零稳态的稳定性。结果表明： (i) 非局部扩散的年龄结构模型的半群结构本质上由出生率和死亡率均独立于空间变量；(ii) 当出生率和死亡率都取决于空间变量时，渐近行为可以由无穷小发生器的谱界符号确定；(iii) 当出生率和死亡率都取决于空间变量和时间时，可以建立弱解和比较原则；(iv) 上述结果可以推广到年龄结构模型。此外，我们在前两种情况（i）和（ii）中将我们的结果与具有拉普拉斯扩散的年龄结构模型进行了比较。