Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate \(\lambda \), or a stable-stage distribution \(\psi \). In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector \(\psi \) corresponding to its spectral radius \(\lambda \), this \(\lambda \) is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population \(\varphi _0\), there is a \(c>0\) such that \(T^n\varphi _0/\lambda ^n \rightarrow c \cdot \psi \).

生态学家最近使用积分投影模型 (IPM) 来研究鱼类和其他动物，它们在其一生中继续生长。这种动物不能缩小，因为它们有骨骼；这样做的数学结果是积分投影算子T的核是无界的，并且算子不是紧致的。据我们所知，在 IPM 上所做的所有理论工作都假设算子是紧凑的，特别是有一个有界内核。先验地，不清楚这些 IPM 是否具有渐近增长率\(\lambda \)或稳定阶段分布\(\psi \). 在紧凑算子的情况下，这些量分别是它的谱半径和相关的特征向量。在生物学上合理的假设下，我们证明了这些 IPM 中的非紧凑算子与其紧凑对应物共享一些重要特征：算子T具有唯一的正特征向量\(\psi \)对应于其谱半径\(\lambda \)，此\(\lambda \)严格大于所有其他谱值的模数的上限值，并且对于任何非负初始总体\(\varphi _0\)，存在\(c>0\)使得\ (T^n\varphi _0/\lambda ^n \rightarrow c \cdot \psi \)。