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.

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 <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.