Because the (st)age structure of a population may rarely be stable, studies of transient population dynamics and population momentum are becoming ever more popular. Yet, studies of "population momentum" are restricted in the sense that they describe the inertia of population size resulting from a demographic transition to the stationary population growth rate. Although rarely mentioned, inertia in population size is a general phenomenon and can be produced by any demographic transition or perturbation. Because population size is of central importance in demography, conservation, and management, formulas relating the sensitivity of population inertia to changes in underlying vital rates and population structure could provide much-needed insight into the dynamics of populations with unstable (st)age structure. Here, we derive such formulas, which are readily computable, and provide examples of their potential use in studies of life history and applied arenas of population study.
Quantification and understanding of demographic variation across intra‐ and inter‐annual temporal scales can benefit from the development of theoretical models of evolution and applied conservation of species. We used long‐term survey data for northern bobwhites (Colinus virginianus) collected at the northern and southern extent of its geographic range to develop matrix population models which would allow investigation of intra‐ and inter‐annual patterns in bobwhite population dynamics. We first evaluated intra‐annual patterns in the importance of a seasonal demographic rate to asymptotic population growth rate with prospective perturbation analysis (elasticity analysis). We then conducted retrospective analysis (life table response experiments) of inter‐annual patterns in the contribution of observed changes in demography to the observed change in population growth rate. Survival in the earliest age class during the nonbreeding season had the greatest potential influence in both the northern and southern populations. Examination of inter‐annual variation in demography indicated that variation in nonbreeding season survival in the earliest age class contributed the most to observed changes in population growth rate in the northern population. In contrast, changes in fertility in the earliest age class in the southern population had the greatest influence on changes in population growth rate. Prospective elasticity analyses highlight the similarities in bobwhite demography throughout different parts of its geographic range, while retrospective life table response experiments revealed important patterns in the temporal differences of bobwhite life history at the northern and southern extent of its geographic range.
The simple restricted modules for the restricted simple Cartan-type Lie algebras fall into two classes. There is a large class consisting of modules induced from simple modules for the subalgebra of nonnegative components. Then there is the small class of exceptional simple modules. Explicit constructions of these exceptional modules and formulas for their dimensions were obtained by Shen [9] for the Witt and special algebras and by Holmes [3] for the contact algebra (for sufficiently large characteristic). In this paper, we take up the case of the hamiltonian algebra. Already in [9] Shen has made considerable progress in this case, especially in considering the map δ of Section 3 below. However, we feel that some proofs are incomplete and some statements are incorrect. In particular, we believe that the formula given for the dimensions of the exceptional modules is inaccurate. We provide a new formula under the assumption of sufficiently large characteristic (see 5.10). For more detailed comments comparing Shen's results with ours, see the remarks following 3.2, 4.8, 5.8, and 5.10.
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