DOI: 10.5061/dryad.v41ns1rxr
title: Incorporating effects of age on energy dynamics predicts non-linear maternal allocation patterns in iteroparous animals

description: Iteroparous parents face a trade-off between allocating current resources
 to reproduction versus maximizing survival to produce further offspring.
 Optimal allocation varies across age, and follows a hump-shaped pattern
 across diverse taxa, including mammals, birds and invertebrates. This
 non-linear allocation pattern lacks a general theoretical explanation,
 potentially because most studies focus on offspring number rather than
 quality and do not incorporate uncertainty or age-dependence in energy
 intake or costs. Here, we develop a life history model of maternal
 allocation in iteroparous animals. We identify the optimal allocation
 strategy in response to stochasticity when energetic costs, feeding
 success, energy intake, and environmentally-driven mortality risk are
 age-dependent. As a case study, we use tsetse, a viviparous insect that
 produces one offspring per reproductive attempt and relies on an uncertain
 food supply of vertebrate blood. Diverse scenarios generate a hump-shaped
 allocation: when energetic costs and energy intake increase with age; and
 also when energy intake decreases, and energetic costs increase or
 decrease. Feeding success and mortality risk have little influence on
 age-dependence in allocation. We conclude that ubiquitous evidence for
 age-dependence in these influential traits can explain the prevalence of
 non-linear maternal allocation across diverse taxonomic groups.
description: Here, we investigate how stochasticity and age-dependence in
 energy dynamics influence maternal allocation in iteroparous females. We
 develop a state-dependent model to calculate the optimal maternal
 allocation strategy with respect to maternal age and energy reserves,
 focusing on allocation in a single offspring at a time. We introduce
 stochasticity in energetic costs– in terms of the amount of energy
 required to forage successfully and individual differences in metabolism –
 and in feeding success. We systematically assess how allocation is
 influenced by age-dependence in energetic costs, feeding success, energy
 intake per successful feeding attempt, and environmentally-driven
 mortality. First, using stochastic dynamic programming,
 we calculate the optimal amount of reserves M that mothers allocate to
 each offspring depending on their own reserves R and age A. The optimal
 life history strategy is then the set of allocation decisions M(R, A) over
 the whole lifespan which maximizes the total reproductive success of
 distant descendants.  Second, we simulated the life
 histories of 1000 mothers following the optimisation strategy and the
 reserves at the start of adulthood R1, the distribution of which was
 determined, the distribution of which was determined using an iterative
 procedure as described . For each individual, we calculated maternal
 allocation Mt, maternal reserves Rt, and relative allocation Mt⁄Rt  at
 each time period t. The relative allocation helps us to understand how
 resources are partitioned between mother and offspring.
 Third, we consider how the optimal strategy varies when there is
 age-dependence in resource acquisition, energetic costs and survival.
 Specifically, we include varying scenarios with an age-dependent increase
 or a decrease with age in energetic costs (c_t), feeding success (q_t),
 energy intake per successful feeding attempt (y_t), and
 environmentally-driven extrinsic mortality rate (d_t) (Table 2). We
 consider the age-dependence of parameters one at a time or in pairs,
 altering the slope, intercept, or asymptote of the age-dependence (linear
 or asymptotic function). Our aim is to identify whether the observed
 reproductive senescence can arise from optimal maternal allocation. As
 such, we do not impose a decline in selection in later life as all
 offspring are equally valuable at all ages (for a given maternal
 allocation), and there are no mutations. For each scenario, we run the
 backward iteration process with these age-dependent functions, obtain the
 allocation strategy, and simulate the life history of 1000 individuals
 based on the novel strategy. We then fit quadratic and linear models to
 the reproduction of these 1000 individuals using the lme function, nlme
 package in R. For these models, the response variable is the maternal
 allocation Mt and explanatory variables are the time period t and t2 (for
 the quadratic fit only), with individual identity as a random term. We use
 likelihood ratio tests to compare linear and quadratic models using the
 anova function (package nlme) with the maximum-likelihood method. If the
 comparison is significant (p-value &lt;0.05), we considered the
 quadratic model to have a better fit, otherwise the linear model is
 considered more parsimonious. We were particularly interested in
 identifying scenarios where the fit was quadratic with a negative
 quadratic term. For each scenario, the pseudo R2 conditional value
 (proportion of variance explained by the fixed and random terms,
 accounting for individual identity) is calculated to assess the
 goodness-of-ﬁt of the lme model, on a scale from 0 to 1, using the
 “r.squared” function, package gabtool.  All
 calculations and coding are done in R.