Skewed distributions of lifetime reproductive success: beyond mean and variance

Abstract

Lifetime reproductive performance is quantified here by the LRS (lifetime reproductive success), the random number of offspring an individual produces over its lifetime. Many field studies find that distributions of LRS among individuals are non‐normal, zero‐inflated and highly skewed. These results beg the question, what is the distribution of LRS predicted by demographic models when the only source of randomness is demographic stochasticity? Here we present the first exact analysis of the probability distribution of LRS for species described by age + stage models; our analysis starts with estimated vital rates. We illustrate with three examples: the Hadza, human hunter‐foragers (age‐only), the evergreen tree Tsuga canadensis (stage‐only) and Roe deer, Capreolus capreolus (age + stage). For each we obtain the exact distribution of LRS, but also calculate and discuss the first three moments. Our results point to important questions about how such LRS distributions affect natural selection, and life history evolution.

Relative contributions of fixed and dynamic heterogeneity to variation in lifetime reproductive success in kestrels (Falco tinnunculus)

“…We used the LRS probability distribution to calculate the Gini index, a measure of the inequality of LRS between individuals. A Gini index close to zero indicates similar probabilities of producing each LRS, while a Gini index close to one means that there is a wide variation in the probability of producing a certain LRS and is what we expect for a highly skewed LRS distribution (Tuljapurkar et al, 2020).…”

“…The observed (black bars), simulated (white bars) and theoretical (red line) distributions of lifetime reproductive success (LRS) for both females (left) and males (right). The theoretical LRS distribution was obtained by summing the probability values of the possible LRS in each LRS bin from the LRS probability distribution, calculated with the methods from Tuljapurkar et al (2020) and assuming an immature survival of one. For both the simulated and theoretical LRS distribution we used the same population parameters for each individual (no fixed heterogeneity).…”

“…The stage‐specific probability distributions can be combined with the survival and switching probability to calculate probability distributions of producing a certain number of offspring at each age. Convolutions of these age‐specific probability distributions (weighted by survivorship and mortality) then yield the probability distribution of all possible values of the LRS (see Tuljapurkar et al, 2020 for details). From this probability distribution we can calculate the mean and variance of the LRS, which are similar to the mean and variance calculated with the analytical equations from van Daalen and Caswell (2017).…”

“…The total variation in LRS is caused by this dynamic and fixed heterogeneity (note that some studies call these patterns individual stochasticity and heterogeneity, respectively, for example, Caswell, 2009, 2011; van Daalen & Caswell, 2017). Several studies have shown that dynamic heterogeneity constitutes a major part of the total variation in LRS (Caswell, 2011; Jenouvrier, Aubry, Barbraud, Weimerskirch, & Caswell, 2018; Orzack, Steiner, Tuljapurkar, & Thompson, 2011; Snyder & Ellner, 2016, 2018; Steiner & Tuljapurkar, 2012; Steiner, Tuljapurkar, & Orzack, 2010; Tuljapurkar et al, 2009; Tuljapurkar, Zuo, Coulson, Horvitz, & Gaillard, 2020; van Daalen & Caswell, 2017). In this study, we test whether dynamic heterogeneity is also the dominant driver of LRS variation in a Dutch population of the Common Kestrel ( Falco tinnunculus ).…”

Taming fitness: Organism‐environment interdependencies preclude long‐term fitness forecasting

“…To this day, fitness is defined in different and even inconsistent ways. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] In light of this, it is not at all uncommon for prominent evolutionary biologists to concede that "Unfortunately, fitness is difficult to define more specifically so that it can be measured and understood more clearly." [16] Routine concessions like this suggest that little has changed since Stephen Stearns proposed the following satirical definition: "Fitness: something that everyone understands but no one can define precisely."…”