Periodic Models

Abstract

Periodic Models 8.1 Introduction Periodic matrix models are often used to study cyclical temporal variation (seasonal or interannual), sometimes as a (perhaps crude) approximation to stochastic models. However, formally periodic models also appear when multiple processes (e.g., demography and dispersal) operate within a single projection interval. The models take the form of periodic matrix products. A familiar example is when population projection over an annual interval is described as a product of seasonal operators. The perturbation analysis of periodic models (Caswell and Trevisan 1994; Lesnoff et al. 2003; Caswell and Shyu 2012) must specify both the vital rates affected by the perturbation and the timing of the perturbation within the cycle. This chapter presents a general approach to the perturbation analysis of both linear and nonlinear periodic models. The results consist of a series of analyses of some of the most commonly encountered periodic models. If the environment is time-invariant on the scale of a chosen projection interval (e.g., from year to year), the result is a periodic matrix population model in which the seasonal product repeats itself. Such a model can be written as n(t + 1) = B p • • • B 2 B 1 n(t) (8.1) Here, B i is the matrix at phase i of the cycle and p is the period. The period is the number of phases in the cycle; i.e., the number of matrices in the periodic matrix product in (8.1). Neither the identities nor the number of stages need be the same from one phase to the next, so the matrices B i may be rectangular rather than square.

Reparations for Black American Descendants of Persons Enslaved in the U.S. and Their Estimated Impact on SARS-CoV-2 Transmission

“…Our next-generation matrix analysis shows that, in a segregated society like the U.S. where SARS-CoV-2 transmission rates are disproportionate across racial groups, small changes in the ratio between β b→b and β w→w can result in large changes in the reproductive ratio for the population (Figure 3a), due mainly to 1) the effects of high assortative mixing structured by racism on the value of c b→b ; and 2) the fact that the expected number of secondary infections generated within high-risk subgroups (i.e., the value g b→b in the next generation matrix—in this case driven by high relative values of c b→b ) comes to dominate R 0 for a population. 29,52…”

Transient amplification enhances the persistence of tropicalising coral populations in marginal high latitude environments

“…Thus, we incorporated this variability into our IPMs by allowing the probabilities of larval settlement and recruit survival to fluctuate within observed boundaries, therefore introducing an element of within–population variability to our models. Using the transient LTRE approach detailed below, we combined a traditional Life Table Response Experiment with a Kitagawa & Keyfitz decomposition (Kitagawa 1955; Keyfitz 1968; Caswell 2019). Briefly, this decomposition approach allowed us to account for within-population variability when evaluating the vital rate mechanisms underlying the differences between the transient dynamics of two populations (Maldonado-Chaparro et al .…”

“…We first paired up tropical and subtropical model variants to evaluate the overall contributions ( C ) of the vital rate patterns of established colonies ( Θ ), larval settlement ( ψ ), and recruit survival ( ⍰ ), towards variation in using a Kitagawa & Keyfitz decomposition. The overall contribution of each component was obtained by averaging the effect on of substituting the tropical and subtropical form of the selected component against a fixed background of the other components (Caswell 2019): …”

“…Next, we decomposed the separate contributions of the vital rates of survival, changes in size, and fragmentation, observed in established colonies, towards variation in . The contribution of each vital rate (C[ θ i ]) corresponds with the change in that vital rate between paired tropical and subtropical models combined with the environmental-specific elasticity matrices of (Caswell 2019): …”

A mathematical framework for evo-devo dynamics

“…We use “derivative-like” notation like this one for various differential operators, motivated by some analogous use by Caswell (2019) (his Eqs. 1.2 and 2.21, which he based on a suggestion by Samuelson 1947).…”