How competition between overlapping generations can influence optimal egg-laying strategies in annual social insects

Annual social insects are an integral functional group of organisms, particularly in temperate environments. An emblematic part of their annual cycle is the social phase, during which the colony-founding queen rears workers that later assist her in rearing sexual progeny (gynes and drones). In many annual social insects, such as species of bees, wasps, and other groups, developing larvae are provisioned gradually as they develop (progressive provisioning) leading to multiple larval generations being reared simultaneously. We present a model for how the queen in such cases should optimize her egg-laying rate throughout the social phase depending on number-size trade-offs, colony age-structure, and energy balance. Complementing previous theory on optimal allocation between workers vs. sexuals in annual social insects and on temporal egg-laying patterns in solitary insects, we elucidate how resource competition among overlapping larval generations can influence optimal egg-laying strategies. With model parameters informed by knowledge of a common bumblebee species, the optimal egg-laying schedule consists of two temporally separated early broods followed by a more continuous rearing phase, matching empirical observations. However, eggs should initially be laid continuously at a gradually increasing rate when resources are scarce or mortality risks high and in cases where larvae are fully supplied with resources at the egg-laying stage (mass-provisioning). These factors, alongside sexual:worker body size ratios, further determine the overall trend in egg-laying rates over the colony cycle. Our analysis provides an inroad to study and mechanistically understand variation in colony development strategies within and across species of annual social insects. Supplementary Information The online version contains supplementary material available at 10.1007/s00442-023-05411-z.


Appendix S1: Motivation of baseline parameter values
Parameter values of our model are rounded to facilitate interpretations of graphs and calculations. The parameters w1 and gmax represent the larval demands of energy for growth during the first and second week, respectively (a = 1 and 2). We assume pollen consumption and larval mass undergoes an exponential-like increase with time, in which only a small proportion of the larval weight gain occurs during the first week (Ribeiro et al. 1993;Cnaani and Hefetz 1994). That said, rearing early stage larvae is still not cheap as incubation and temperature regulation are needed (Heinrich 1979) and nectar storage requires frequent replenishment (Sladen 1912). To capture the asynchrony in larval demands between early and late stage larvae by setting 1 = 0.3 and max = 0.3 equating to late stage larvae (a = 2) requiring more than double the energy for growth to early stage larvae (a = 1). We then set the minimal adult worker body mass min = 0.4. For simplicity, we then assume energy costs and minimum body sizes of sexuals are proportional to those of workers, but scaled with their maximum body size by assuming 1 = 1 max and min = min max . Lopez-Vaamonde et al. (2009) estimated the mass of individual gynes and drones to 799 mg and 266 mg respectively, or 3.8 and 1.25 times the mass of a worker (estimated to 210 mg by Holehouse, Hammond, & Bourke [2003]). Considering that a colony typically produces more drones than gynes, our model assumes sexuals on average require twice as much energy than workers and set vmax = 2.
To estimate the contribution of the founder queen ( ) and the workers ( ) to the work afforded by the colony, we consider a stylized B. terrestris egg-laying pattern where a first generation of workers (eggs laid week 1) help to rear a second generation of workers (eggs laid week 3) . Beekman et al (1998) estimated that the first and second brood consisted of 9.7 and 36.7 workers respectively based on data in Duchateau and Velthuis (1988).
Approximating these numbers we assume the queen on her own has the capacity to provide sufficient energy for rearing one generation of 10 full-sized workers, and then together with these is able to rear a second generation of up to 35 full-sized workers.

Appendix S2: Comparison with previous dynamic energy allocation models for annual social insects
In a first section, we here compare the colony growth patterns generated by the optimal egg-laying schedule in our model under mass provisioning with the colony growth pattern described by the dynamic energy allocation model introduced by Macevicz and Oster (1976). Thereafter we compare how the timing of the optimal switch point is affected by model parameters in our model and the Macevicz and Oster (1976) model alongside further studies building thereon.

Colony growth patterns
We focus on colony development in terms of growth of number of workers and sexuals shown in Fig. 3a-d (second row). We will first consider a hypothetical case with unconstrained colony growth and no worker mortality (as in Fig. 3a) and then in turn study the effects of reduced productivity (as in Fig. 3b), worker mortality (as in Fig. 3c) and growth constraints (as in Fig. 3d).
Macevicz and Oster (1976) analysed a model with unconstrained growth defined by: where, t is time, W is the number of workers, Q is the number of sexuals produced by the colony, R(t) is resource availability, b is a energy conversion factor, and v are mortality rates of workers and sexuals, respectively, and u(t) is the fraction (0 ≤ ≤ 1) of colony resources allocated to production of workers as opposed to production of sexuals.
Macevicz and Oster (1976) also considered a model of constrained colony growth by modifying equation S2a to: where b0 corresponds to b above and the parameter b1 reduces the growth rate at high densities, leading to logistic growth of the number of workers (unless the colony switches to produce sexuals or the season ends).
As in the main paper, we assume that resource availability is constant over time by setting R(t) = RM for all t, that sexual mortality v = 0 and that the population use a bang-bang strategy and switches from producing worker to producing sexuals at a time point tM with u = 1 for 0 < t< tM and u = 0 for tM < t < T and. Note that Macevicz and Oster (1976)

Optimal timing for switching to reproduction
Based on their model and for the case of unconstrained, exponential colony growth (eq. S2a). Macevicz and Oster (1976) calculated that the optimal switch point occurs earlier when worker mortality increases and when worker productivity decreases. In the scenario of mass-provisioning in our model we see these same effects in Fig. 3b and 3c (when compared to Fig. 3a) where we similarly consider a situation with unconstrained growth (d = 0). In our model we find that growth constraints (d>0) advances the optimal switch time ( Fig. 3d). This is in line with our simulations of the Macevicz and Oster model where the constrained model (Fig. S1d) has an earlier switch than the unconstrained model (Fig. S1a).
Increasing degree of growth constraints similarly advances the optimal switch point in In the scenario of progressive provisioning with parameters representing the B. terrestris life history as the starting point (Fig. 4), the optimal switch occurs later when worker mortality increase (Fig. 5a) and when worker productivity decreases (Fig. 5b). These responses are opposite to those predicted by Macevicz and Oster (1976) for unconstrained growth and shown in Fig. S1b,c. This difference occurs because we assume that B. terrestris is subject to growth constraints (d > 0), a factor that can modulate in which direction the optimal switch time responds to variation in model parameters. For example, using a model for annual plants analogous to the Macevicz and Oster model with logistic growth (eq. S3), Lindh et al (2015) showed that decreasing productivity can delay the optimal switch to reproductive growth if the degree of growth constraint (corresponding to b1) is sufficiently strong, similar to here (Fig. 5a). Increasing the worker mortality can delay the optimal switch point under stronger growth constraints in the Macevicz and Oster model (eq. S3) as well (as in Fig. 5b). We note, lastly, that the effects of less constrained growth work contributed by queen = r_r_queen_productivity % (r_w) work contributed per worker = r_worker_productivity % (s_a) survival of adults = surv_a; % (d) degree of growth constraints = d_constraints % (t_s) switch point = switch_time % (e_a,t) relative efficiency of a worker = e_effectivity_per_worker_v (vector) % (E_t) total amount of work afforded by colony = E_total_work_afforded % (R_t) colony energy income per week = R_colony_energy_income_t % (w_max_weight_workers) maximum (optimal) weight of worker = w_max_weight_workers__weight_sexuals % (v_max) maximal (optimal)weight of sexual = v_max_weight_sexuals % (w_min_weight_workers) minimal weight of worker to be functional = w_min_weight_workers % (v_min) minimal weight of sexual to be functional = v_min % (w_1) weight/cost of 1st stage worker larvae= w_1_cost % (v_1) weight/cost of 1st stage sexual larvae= v_1_cost % (g_max) maximum energy demand of 2nd stage worker larvae = g_max % (h_max) maximum energy demand of 2nd stage sexual larvae = h_max % (c_t) egg_laying rate = c_EGGNUM_v (vector) % (n_a,t) number of worker adults = nv (vector) % (w_a,t) weight of worker adults = wv (vector) % (m_a,t) number of sexual adults = nsv (vector) % (v_a,t) weight of sexual adults = wsv (vector) %NOTES ABOUT THE NUMERICAL IMPLEMENTATION % %In relation to the model definition the following steps of the numerical %implementation are worth noting: % % %A. REPRESENTATION OF EGG-LAYING RATES % % %For numerical reasons during the optimization process, egg-laying rates are %in some cases expressed as "relative egg-laying effort", rather than "number of eggs". % The relative egg-laying effort is expressed by a vector "U" %i.e. the fraction of available energy used for egg-laying per week. %taking values between 0-1. % %In some analyses it is however necessary to calculate egg-laying %rates as "number of eggs per week", as in the model definition. These %rates are denoted using the variable "c_EGG_NUM_v" %so the implementation switches between these two %ways of representing egg-laying rate when appropriate. % % %CONSTRAINED OPTIMIZATION %During the search for an optimal egg-laying schedule,the optimizer function (fminsearch) %updates the candidate egg-laying schedule iteratively after having evaluated its fitness. %It may then occur that the candidate egg-laying schedule takes on values %that are biologically unrealistic, such as negative values or laying of more eggs %than there are resource for. This is here addressed in two ways: % % %1)The relative egg-laying effort is expressed by a parameter U %ensures % %2) Penalty-functions add a fitness cost to egg-laying schedules %that are not within biologically realistic bounds, for example %due to rounding errors or approximations. % %max/min-functions (Heavyside-functions) can cause numerical problems, since %the optimizer function (fminsearch) works best with smooth objective functions %Therefore max-functions in the model definitions have here %been replaced by soft min/max function where the sharpness is controlled by a parameter K %the soft min/max functions are denoted softJmax() and softJmin() %respectively % %Parameter values of g_max = 0 and w1 = 1 can cause numerical problems, and have %been implemented as g_max = 0.000001 and w1 =0.999999. % % % % function main_simulator clear all close all %choose which figure to show by setting the simulation_type %Simulation_type 1 %simulation with predetermined egg laying scheduel, %example with constant egg-laying rate with switch point at t=6 %( Figure 2AB in the manuscript) %Simulation_type 2 %simulation of an optimal egg-laying schedule %with fitness consequences of small deviations from the %example with the optimal egg-laying schedule for B. terrestris settings %( Figure 4AB in the manuscript) %Simulation_type 3 %simulation to predict optimal egg-laying schedule %( Figure 3ABCDE and Figures 5ABCDE in the manuscript) %(to switch between reproducing Figure 3ABCDE or Figure 5ABCDE