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Constructing a demographic model to explore the process of colony failure: the hypothesis: We hypothesise that colony failure occurs when the death rate of bees in the colony is unsustainable. At this point normal social dynamics break down, it becomes impossible for the colony to maintain a viable population, and the colony will fail. We hypothesise that any factor that causes an elevated forager death rate will reduce the strength of social inhibition, resulting in a precocious onset of foraging behaviour in young bees [5]. Because foraging is high-risk [2], precocious foraging shortens overall bee lifespan. Precocious foragers are also less effective and weaker than foragers that have made the behavioural transition at the normal age [25], [26]. Consequently, as the mean age of the foraging force decreases forager death rates increase further, which accelerates the population decline. A precocious onset of foraging reduces the population of hive bees engaged in brood care. This reduces colony brood rearing capacity, and the population crashes. A similar hypothesis has been proposed to explain the impact of Nosema ceranae on colonies [15], but we argue this hypothesis is applicable to any factor that chronically elevates forager bee death rates. We explore this hypothesis using the following simple mathematical model.
A mathematical model allows us to explore the effects of different factors and forces on the population of the hive in a quantitative way. Such a model has the potential to make predictions for the outcome of various manipulations, and to allow a preliminary exploration of the problem before investing in experimental work. We construct a simple compartment model for the worker bee population of the hive (Fig. 1). Our model only considers the population of female workers since males (drones) do not contribute to colony work. Let H be the number of bees working in the hive and F the number of bees who work outside the hive, referred to here as foragers. We assume that all adult worker bees can be classed either as hive bees or as foragers, and that there is no overlap between these two behavioural classes [1], [4]. Hence the total number of adult worker bees in the colony is N = H+F.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g001 Elements of honey bee social dynamics considered by our model.Eggs laid by the queen are reared as brood that eclose three weeks later as adult bees. Adult bees work in the hive initially before becoming foragers. Our model considers the death rate of adult bees within the hive to be negligible, but forager death rate is a parameter varied in our simulations. We assume the amount of brood reared is influenced by the size of the colony (number of hive and forager bees) and that the rate at which bees transition from hive bees to forager bees is influenced by the number of foragers to represent the effect of social inhibition.
Our model does not consider the impact of brood diseases on colony failure, however we believe our approach is still useful because many cases of colony failure and CCD are not caused by brood diseases [21], [22], [23]. Hive bees eclose from pupae and mature into foragers. Death rates of adult hive bees in a healthy colony are extremely low as the environment is protected and stable. We assume that the death rate of hive bees is negligible. Workers are recruited to the forager class from the hive bee class and die at a rate m. Let t be the time measured in days. Then we can represent this process as a differential equation model:(1)(2)The function E(H,F) describes the way that eclosion depends on the number of hive bees and foragers. The recruitment rate function R(H,F) models the effect of social inhibition on the recruitment rate. It is known that the number of eggs reared in a colony (and hence the eclosion rate) is related to the number of bees in the hive. Big colonies raise more brood [27], [28], [29]. The nature of this dependence is not known, however. We assume that the maximum rate of eclosion is equivalent to the queen's laying rate L and that the eclosion rate approaches this maximum as N (the number of workers in the hive) increases. In the absence of other information we use the simplest function that increases from zero for no workers and tends to L as N becomes very large:(3)Here w determines the rate at which E(H,F) approaches L as N gets large. Figure 2 shows E(H,F) as a function of N for a range of values of w.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g002 Plot of the eclosion function E(h,F) = LN/(w+N) where N = H+F for different values of w.The solid line has w = 4000; the dashed line, w = 10 000 and the dash-dot line, w = 27 000. We write the recruitment function as(4)The first term represents the maximum rate that hive bees will become foragers when there are no foragers present in the colony. The second term represents social inhibition and, in particular, how the presence of foragers reduces the rate of recruitment of hive bees to foragers. We have assumed that social inhibition is directly proportional to the fraction of the total number of adult bees that are foragers, such that a high fraction of foragers in the hive results in low recruitment. In the absence of any foragers new workers will become foragers at a minimum of four days after eclosing [30], so an appropriate choice for the rate of uninhibited transition to foraging is = 0.25. We chose = 0.75 since this factor implies that a reversion of foragers to hive bees would only occur if more than one third of the hive are foragers. We also chose L = 2000 as the daily laying rate of the queen [31] and w = 27,000.
The equations (1) and (2) with the functions (3) and (4) were analysed using standard linear stability analysis and phase plane analysis [32]. The model has a globally stable steady state (H0,F0) where(5)when(6)Otherwise the state with no adult bees is an attractor and the hive population goes to zero. Figure 3 shows phase plane solutions for a low death rate, m = 0.24, when the populations tend to a positive steady state, and a higher death rate m = 0.40, when the population goes extinct. In each case the solution rapidly approaches the line F = JH so that the ratio of hive bee numbers to forager numbers is close to being constant. The population size adjusts more slowly to either a positive steady state or to zero. Figure 4 shows the decline of a doomed population as a function of time (dotted line). If the foragers become less able and more likely to die as they get younger then the decline will be more rapid (solid line).
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g003 Phase plane diagrams of solutions to the model for different values of m.Each line on the diagrams represents a solution trajectory, giving the number of foragers F and the number of hive bees H. As time t increases the solutions change along the trajectory in the direction of the arrows. In (a) m = 0.24 and the populations tend to a stable equilibrium population, marked by a dot. In (b), m = 0.40 there is no nonzero equilibrium and the hive populations collapses to zero. Parameter values are L = 2000, = 0.25, = 0.75 and w = 27 000.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g004 The effect of inefficient precocious foraging on population decline.This plot shows the time course of colony decline when all foragers perform equally well (dashed line) and when precocious foragers die faster than mature foragers (solid line). The effect of precocious foraging is modeled by replacing the death rate m by m = ml R2/(2+R2) whenever R<0 where R is the recruitment rate of foragers given in eqn (4). Parameter values are L = 2000, = 0.25, = .75, w = 27 000, ml = 0.6 and 2 = 0.059.
Figure 5 is a bifurcation diagram, which shows that for low values of the forager death rate m there are large numbers of bees in the colony, but once m passes a critical value the colony population cannot support itself and the colony fails.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g005 The dependence of the colony population at equilibrium on the death rate of foragers.For this set of parameter values, when the death rate m exceeds 0.355, the only stable equilibrium population is zero. Parameter values are the same as Figure 3.
Figure 6 shows how the average age at commencement of foraging and the average age at death depend on the forager death rate m. The model predicts that at a higher death rate the forager population will be smaller and also made up of younger bees.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0018491.g006 The average age of adult worker bees (dashed line) and the average age of onset of foraging (solid line) as a function of forager death rate.Parameter values are the same as Figure 3. We compared results from the model to experimental observations of Rueppell et al [33]. We used the observed flightspan [the number of days bees were observed foraging 33], to estimate the death rate of foragers since m is the reciprocal of flightspan. With these values of m we used the model to calculate the average age of onset of foraging (AAOF) and the lifespan of worker bees for each colony and compared these model values to observed results. These observed and calculated results are shown in Table 1. Even with the somewhat rough estimates of parameters, the model matches the observational data well for average age at onset of foraging, although it is slightly high for worker lifespan. Nevertheless, given that the model is a very simple representation of honey bee demographics, the results are encouraging.
Table data removed from full text. Table identifier and caption: 10.1371/journal.pone.0018491.t001 Comparison of experimental data and model results for average age of onset of foraging (AAOF) and lifespan. Experimental data is from Rueppell et al [33] and model results were obtained by running the model for 40 days (approximately the observational period used by Rueppell et al). At the start of each model run H = 9000 for large colonies and 4500 for small colonies and F = 0. The parameters were L = 2000, w = 27000, = 0.25 and = 0.75.
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