- Email: [email protected]
The economics of honeybee swarming
Regional Science and Urban Economics 33 (2003) 581–594 www.elsevier.com / locate / econbase
The economics of honeybee swarming Chung-cheng Lin a , *, Tze-wei Chen b , Ching-chong Lai c a
Institute of Economics, Academia Sinica, Nankang, Taipei 115, Taipei, Taiwan b National Taiwan Normal University, Taipei, Taiwan c Sun Yat-Sen Institute for Social Sciences and Philosophy, Academia Sinica, Taiwan Received 6 July 2001; received in revised form 12 August 2002; accepted 27 December 2002
Abstract This paper uses the Anas [Regional Science and Urban Economics 22 (1992) 243] model to provide an economic explanation of the cause of honeybee swarming. We assume that a honeybee is rational in choosing to stay in the old hive or to leave to build a new one. Rationality here refers to a bee’s behavior to maximize the food (honey) that it can share or contribute. We show that all bees will live together when the total number of bees is small. As soon as the total number of bees grows over a threshold, half of the population will immediately swarm to another hive. Contrary to the traditional wisdom that the queen bee leads the swarm, our model demonstrates that swarming may be a collective action of the rational choices of many selfish individual bees. 2003 Elsevier B.V. All rights reserved. Keywords: Honeybee swarming; Rationality JEL classification: A12
1. Introduction The scope of economics is extending everyday. While economic activities of
*Corresponding author. E-mail address: [email protected] (C.-c. Lin). 0166-0462 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0166-0462(03)00004-8
582
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
human beings remain the major subject of modern economics, a few economists have explored the interface and analogies between economics and biology.1 This paper is an application of economics to interpret a biological phenomenon, honeybee swarming, in which one honeybee hive separates into two almost equally sized hives suddenly and quickly. While some biological explanations have been provided to explain this phenomenon, this paper tries to furnish an economic interpretation. Anas (1992) set up a simple economic model to investigate different types of city growth, showing that one type of city growth is a sudden jump from a one-city equilibrium to a symmetric two-city equilibrium. We regard the Anas analytical framework as an analogy to the biological phenomenon of honeybee swarming. The Anas framework is therefore adopted to prove that all bees will live together when the total number of bees is relatively small. As soon as the population grows over a critical value, half of them will immediately swarm to a new hive. We develop a model based upon the assumption that a bee is rational in choosing whether to stay in the old hive or to leave to build a new one. Rationality here refers to a bee’s behavior to maximize the food (honey) that it can potentially share or contribute. This assumption is built on two facts. First, biologists conclude that a honeybee hive is a highly cooperative society, and the range of tasks necessary for supporting the hive are efficiently distributed among working bees (workers). Second, biologists also have discovered that for bees to sustain the survival of the hive and consequently their own, workers can change jobs when the needs of the hive increase. The efficiency distribution of tasks and the autonomous job switching constitute the foundation of the assumption of rationality. Our findings conclude that, contrary to the traditional wisdom that the queen bee leads the swarm, the occurrence of a swarming may be an aggregation of the autonomous choices of many ultimately selfish bees. This conclusion is consistent with the biological finding that honeybee swarming is dominated by worker bees and not by the queen. The remainder of the paper is organized as follows. Section 2 reviews the process of honeybee swarming and the relevant biological explanations. Section 3 argues that honeybees can make conscious choices. A simple economic framework is used to model honeybee swarming in Section 4. Finally, concluding remarks are presented in Section 5.
1 Literature on the interrelationship between economics and biology has been produced recently. Kagel et al. (1975) and Battalio et al. (1981) conducted experiments with laboratory animals to test the validity of some elementary principles of economic theories. Moreover, some scholars, for example Gadgil and Bossert (1970) and Rapport (1971), have made significant use of tools and approaches in economics to explain some biological phenomena. A comprehensive survey is provided by Hirshleifer (1977).
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
583
2. The biological viewpoints of honeybee swarming Biologists have observed that honeybees will start a new colony by swarming when the old hive becomes crowded or when a new queen emerges. The swarming process of most kinds of honeybees can be divided into the following two stages: queen rearing and swarming. According to Winston (1987, pp. 181–190), the rearing of new queens can sometimes begin as early as 1 month before an actual swarming. The process starts when eggs are laid or placed into special areas known as ‘queen cups’. These cups are built by workers and are specifically used for queen rearing. However, the existence of eggs in the queen cups does not guarantee the occurrence of a swarming. Larvae (wormlike grubs that hatch from eggs) in the queen cups are provided with special food (royal jelly) that turns them into queens. At the end of the larvae feeding period, once the first queen cups are sealed, a swarming can occur that day or the day after. The entire process of queen rearing in most cases takes | 8–10 days. About 10 days prior to swarming, workers begin to store honey in their honey stomachs. A few hours before swarming, some workers become ‘excited’ and ‘‘begin running back and forth in waves, buzzing to excite the other workers’’ (Winston, 1987, p. 187). Contrary to the traditional wisdom that the old queen leads the swarm, the entire swarming is dominated by workers. They form the swarm, scout for a new site and force the queen to join the swarm. A swarm happens when the old queen is successfully forced to join the clustering workers and then is relocated to a new site.2 The swarm usually consists of about a little more than half of the adult workers of the colony (Winston and Otis, 1978; Getz et al., 1982).3 Along with detailed documentation on the process of swarming, biologists have also studied its causes. Studies on honeybees and other social insects can be generally categorized into two groups: the colony-level approach and the individual-level approach. The former asserts that the behavior of bees and other social insects is subject to generating the greatest benefit to the colony, not necessarily for themselves (Seeley, 1985). The latter argues that the behavior of all social insects is ‘‘simply statistical summaries of many individuals’ ultimately selfish actions’’ (Seeley, 1985, p. 5). The colony-level approach has dominated the theory of evolution ever since Charles Darwin published his theory of evolution.
2 Sometimes when the workers do not succeed in driving the queen with them, the workers will return to the old hive. This is known as ‘false swarms’. 3 The swarming process of some kinds of honeybees, for example Apis mellifera in South African, can be divided into three stages: queen rearing, prime swarm (swarm), and afterswarms. Afterswarms are near replicas of the prime swarm except that workers are joined by virgin queens rather than the mated queen. The frequency of afterswarms varies according to factors such as weather, food and the number of virgin queens (Winston, 1987, pp. 181–190). We exclude the case of afterswarming in our model.
584
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
However, the individual-level approach is now receiving more attention among scientists. Early studies tended to attribute the cause of swarming to the amount of food in the hive. Morland (1930) argued that swarming was the result of too many young nurse bees that produced too much brood food. In order to reduce the amount of food, queen rearing began, which led to swarming. Another group of scientists believed that the crowding of adult workers and limited space for food rearing resulted in swarming (Demuth, 1921; Winston, 1987). There is, however, no sufficient evidence to support these hypotheses (Winston, 1987). Other investigations (Simpson, 1958; Winston and Taylor, 1980; Lensky and Slabezki, 1981) found a multifactorial relationship between swarming and several within-colony demographic factors such as colony size, brood hive congestion, worker age distribution, and reduced transmission of queen substances, but none of these factors alone are found to initiate swarming (Winston, 1987). Visscher (1993) approached the question of individual behavior during swarming from the perspective of individual interests. Employing the theory of natural selection, he argued that there are ‘conflicts’ among colony members during the process of swarming. The individual choice of whether to stay or to leave can therefore be predicted by evaluating the cost of each potential outcome. For example, on the question of which queen will leave, the conclusion is drawn from examining the probabilities that the colony will survive from the mother queen’s, the virgin (daughter) queen’s and the workers’ perspectives. He also showed that there are sharp differences among the interests of different colony members. The final outcomes are always the results of conflicts.
3. Rational behavior For hundreds of years human beings have wondered about honeybee societies. How does each newly hatched bee ‘know’ what to do? Is it possible that each bee is hatched to serve a particular kind of work such as nursing or foraging? From the experiments of Rosch (reviewed by Selsam, 1986), we learn that each worker bee follows a schedule of duties according to its age. A worker bee lives for only 5 or 6 weeks, and jobs change as the bee grows older. At first it cleans out brood cells for new eggs. After 3 days at this job, it becomes a nurse bee and feeds pollen and honey to older larvae. From the 6th to the 12th day it feeds the younger larvae and the queen with royal jelly secreted by special nursing glands in her head. On the 12th day the same bee becomes a builder of wax cells. From the 16th day on, it begins to receive the nectar and pollen brought to the hive. Around the 20th day it stands guard at the entrance to the hive. From the 3rd week until the end of its life, the bee shifts to the last and longest job: foraging, collecting of nectar and pollen. This schedule is not rigid. As Rosch showed, this schedule changes to suit the
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
585
needs of the colony. If only forager bees are left in a hive, then some of these older bees will grow new nursing glands and start to nurse the brood again, even though they are far past the normal nursing age. Likewise, a colony left without foraging bees can also respond to the emergency by dispatching young bees at their nursing age to forage after a few days. Lindauer et al. (reviewed by Selsam, 1986) asked how does a bee ‘‘find out’’ what job has to be done for the survival of the colony? They found that every bee gathers its own information about the needs of the hive. Every day a bee spends about a third of its time patrolling the hive: It inspects cells and larvae, and looks over the building areas and the storage of food. From such inspection tours new forms of activity start, nursing or building or clearing, as the need arises. In a word, a honeybee is able to willingly adjust its course of action according to either the needs of the hive or its own interests. The goal of this behavior, from the viewpoint of the colony-level approach, seems to be the pursuit of the continuing existence of the hive. From the perspective of the individual-level approach, the goal appears to be its own continuous survival. In economic terms we argue that a honeybee is rational to maximize the benefit of its colony or its own interests by adjusting its daily duties and determining whether to stay or to leave.
4. The model In this section we propose a simple economic framework to model honeybee swarming from both the individual-level and the colony-level perspectives. We first discuss the individual-level approach and then the colony-level approach.
4.1. The individual-level approach For the individual-level approach we assume that bees are selfish. That is, each honeybee cares only about its own interests. The goal of a bee is to maximize the probability of its own survival. The existence of the hive is merely a means, not the goal. The larger its share or amount of honey (food), the more likely it will survive. Therefore, we presume that a honeybee is rational in maximizing the honey that it can potentially consume (i.e. the average product) by determining whether to stay or to leave. The total output (honey) of a hive is assumed to increase initially at an increasing rate, and then at a decreasing rate with the number of bees (or more 4 There are three kinds of bees in a hive: the queen, drones, and workers. The queen is a large female who lays all the eggs. The drones are the male bees. The only function of the drones is to mate with the queen. The workers are small females. They do not lay eggs, but they carry out all maintenance and growth of the colony.
586
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
precisely, workers) in the colony.4 This relationship can be explained as follows. An increase in the total number of bees exerts two different effects on the total product: the cooperative effect and the distance effect. The cooperative effect indicates that the total output rises at an increasing rate with the number of bees in the same hive, because the possibility of locating food increases and thus the time spent on food searching decreases as the number of bees grows. However, the cooperative effect will decline as the bee population grows. On the other hand, the distance effect makes the total output increase at a decreasing rate with an increasing number of bees in the same hive. The reason is that more bees there are, the farther the average distance each bee must fly in order to find a source of food. Obviously, this distance disadvantage in food production will rise as the number of bees grows. As a result, as the number of bees increases the cooperative effect initially prevails and the total output increases at an increasing rate. After a certain point the distance effect will dominate and the total output will increase at a decreasing rate.5 Let F(n) denote the total product of a colony with n bees. This is shown in the upper part of Fig. 1. The lower part is the corresponding average product f(n) ; F(n) /n and marginal product F9(n) ; dF(n) / dn. Both average and marginal product functions have inverted U-shaped curves, increasing initially and then decreasing as the number of bees grows. Let nˆ (n¯ ) denote the number of bees at the maximum of the average (marginal) product, which is determined by the efficiency of labor division among bees and the natural environment around the hive. Both nˆ and n¯ play a critical role in the following discussions. From the perspective of the individual-level approach, we assert that a honeybee maximizes the average product f(n) that it can potentially consume in choosing to stay or to leave. The total number of bees n can either live together in one hive or live separately in two hives. Let n 1 and n 2 represent the number of bees in the old and the new hives, respectively, and n 5 n 1 1 n 2 . Let an overdot denote a derivative with respect to time. For simplicity, building a new hive is assumed to be costless for now. The simplest dynamic migration processes where bees move between two hives can be specified as follows: n~ 1 5 f(n 1 ) 2 f(n 2 ),
(1)
n~ 2 5 f(n 2 ) 2 f(n 1 ).
(2)
These migration processes simply state that the number of bees in a colony 5 This would be true even if we define output net of food consumption in foraging. At some population, additional bees may use more food foraging than they bring in, and so if they were foraging then total output net of foraging cost would fall. Our conclusion still holds in the case where the total output decreases after a certain point.
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
587
Fig. 1. The total, average and marginal outputs.
increases (decreases) when its average product is higher (lower) than that of its rival’s. It remains intact when the average products of both hives are equal.6 Inspection of these equations shows that there are three types of equilibria (n *1 ,n 2* ): two-colony of equal size (symmetric equilibrium), two-colony of unequal size (asymmetric equilibrium), and one-colony equilibrium. The first two types of 6
The purpose of this paper is to find the causes of honeybee swarming rather than the reasons for the growth of the bee population. In other words, we will explore when swarming will happen as the number of bees grows, and thus we do not specify the growth function of bees.
588
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
equilibrium occur when the time derivatives of n 1 and n 2 vanish (n~ 1 5 n~ 2 5 0), and they require f(n *1 ) 5 f(n 2* ). A symmetric equilibrium exists when both colonies have exactly half of the total bees (n *1 5 n 2* 5 n / 2). An asymmetric equilibrium appears when one hive is small and the other hive is large, but both have the same average product (for example, n *1 . nˆ . n *2 ). It is clear that the large hive must ˆ the number of contain a population larger than (and the small hive smaller than) n, bees at which the average product is the maximum, in order to reach an asymmetric equilibrium. The third type of equilibrium happens when one hive contains the entire population and the second hive does not appear, that is, the corner solution (n *1 ,0) or (0,n *2 ). In this case, the time derivatives (1) and (2) do not vanish. Given all possible equilibria, we now investigate the dynamic stability of each possible solution to examine the attainability of each equilibrium. First, we check the stability of the one-colony equilibrium. We explore the case of (n *1 ,0) and ignore (0,n *2 ), because only the case of all bees staying in the old colony is meaningful. As mentioned above, a honeybee hive is a highly cooperative society. The range of tasks from nursing to foraging necessary for supporting the hive must be efficiently coordinated among working bees. It would be sensible to argue that when the population falls below a certain threshold, the hive will not be able to survive. We can therefore check the stability of the corner solution by investigating the case where all bees but one stay in the old hive. From Eqs. (1) and (2), we can obtain: n~ 1 5 f(n 1 5 n 2 1) 2 f(n 2 5 1) . 0,
(3)
n~ 2 5 f(n 2 5 1) 2 f(n 1 5 n 2 1) , 0.
(4)
Evidently, the total and average output of a very small hive will be very low. To the extreme, we might expect that the survival of a hive cannot be maintained if the number of bees is too small. Accordingly, it is reasonable to argue that f(n 2 5 1) 5 0 and f(n 1 5 n 2 1) . 0 for a suitable size of bees in the old hive. The sign of Eq. (3) hence is positive and that of Eq. (4) is negative. These signs indicate that the average product of the old hive with almost all bees is bigger than that of the ‘new’ hive with only one bee. In other words, it is unlikely that a very small number of bees will migrate and successfully build a new hive. Consequently, the one-colony equilibrium (n *1 5 n,0) is stable at a suitable size of the bee population.7 We secondly examine the stability of the symmetric equilibrium where n *1 5 n *2 5 n / 2. Eqs. (1) and (2) are mutual dependent, because of the constraint 7 We use f(n 2 5 1) rather than f(n 2 5 0) to investigate the stability of the one-colony equilibrium (n 1 5 n, n 2 5 0), because f(n 2 5 0); the average output at n 2 5 0, is undefined, since it is 0 / 0.
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
589
n 5 n 1 1 n 2 .8 We can hence investigate the stability by checking only one of two equations. From Eq. (1) we have: dn~ ]1 dn 1
U
≠f(n *1 ) ≠f(n *2 5 n 2 n *1 ) 5 ]] n 1* 5n / 2 2 ]]]]] n 2* 5n / 2 ≠n 1 ≠n 1 n n ˆ 5 2f 9 ] . ( , )0; f 9( ] ) . ( , )0; n , ( . )2n. 2 2
SD
U
(5)
ˆ the sign of Eq. (5) is positive, because f 9(n / 2) . 0, and the When n , 2n, ˆ the sign of Eq. (5) is negative, symmetric equilibrium is unstable. When n . 2n, because f 9(n / 2) , 0, and the symmetric equilibrium is stable. The reasoning is ˆ the average products of both hives are equal quite straightforward. When n , 2n, and increase with the number of bees. A bee moving from the old hive to the new one will obtain a higher average product there and simultaneously lower the average product of the old colony. This makes the new hive more attractive and induces more bees to move in. The stability of symmetric equilibrium thus cannot ˆ the average products of both hives are be retained. By contrast, when n . 2n, equal, but decrease with the number of bees. A bee moving to the new hive will raise the average product of the old one and result in a lower average product of the new colony. The rational choice of the bee is obviously to stay in the old hive. The symmetric equilibrium accordingly becomes a stable solution as the bee ˆ population grows over the critical value 2n. We finally examine the stability of the asymmetric solution. As mentioned above, the asymmetric equilibrium (n *1 . nˆ . n *2 ) can only exist when the number of bees in the larger hive is more than the maximum value of the average product (nˆ ). Since the distance effect will become relatively mild as bees travel far, the inverse U-shaped average product curve may decline relatively mildly after reaching its maximum value. Under this situation, we will find that the asymmetric equilibrium may appear only when the total number of bees is more than the ˆ In other words, no asymmetric solution will exist when n , 2n. ˆ critical value 2n. Following Anas (1992), the phase diagrams drawn in Fig. 2 are used to show the above argument. The phase diagram of Fig. 2(a) illustrates that when n , nˆ there are two types of equilibrium: one-colony and symmetric solutions. As shown above, the one-colony equilibrium is stable and the symmetric equilibrium is ˆ the one-colony and symmetric unstable. Fig. 2(b) shows that when nˆ , n , 2n, equilibria are again stable and unstable, respectively. Fig. 2(c) shows that when
8
Given n 5 n 1 1 n 2 , mutual dependence between Eqs. (1) and (2) can be proved by substituting n 2 5 n 2 n 1 and n~ 2 5 2 n~ 1 into Eqs. (1) and (2); that is: n~ 1 5 f(n 1 ) 2 f(n 2 n 1 ), 2 n~ 1 5 f(n 2 n 1 ) 2 f(n 1 ). Therefore, these two equations are mutually dependent.
590
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
Fig. 2. The phase diagrams.
ˆ the one-colony and symmetric equilibria are both stable. Since two n . 2n, adjacent equilibria cannot simultaneously be stable (or unstable), this implies that the asymmetric equilibria are unstable. The one-colony equilibrium for any suitable number of bees and the symmetric equilibrium for n . 2nˆ are accordingly stable and attainable solutions. When the honeybee society expands, there are two potential outcomes. One is that all of the bees live in a hive no matter how large the population is. The other is that all bees live collectively in the old hive when the total number of bees is under the critical
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
591
ˆ but they swarm into two equally sized hives when the total number just value 2n, passes the critical value. It is obvious that the average product of each of the two colonies with the half ˆ or mathematipopulation is more than that of the all-in-one colony when n . 2n, ˆ This is because the average product is a cally, f(n / 2) . f(n) for all n . 2n. ˆ This decreasing function when the number of bees in the hive is larger than n. result implies that half of the bees will leave the old colony when the total number of bees is large enough. As a consequence, when an old hive becomes ‘crowded’, honeybees will start a new colony by swarming. The purpose of the swarm is to pursue the larger average product which each of them can consume. In what follows, each of the two hives may follow the same growth pattern, and as a result, there are numerous honeybee hives in the world.
4.2. The colony-level approach Now we turn to the colony-level approach. Before proceeding, we first discuss why a bee maximizes the probability of the survival of its colony and not just itself. At first glance, it seems to be a behavior of pure altruism. However, biologists (for example, Dawkins, 1976) have argued that altruism is a result of purely selfish (non-altruistic) behavior on the part of the genetic material that is the true medium for natural selection. If a man is carrying a certain gene, then his child is carrying 50% of his same gene. The gene’s survival probability is enhanced if he behaves in a way that improves the survival prospects of his children. Now suppose that some particular gene has the effect of making him altruistic toward his children. The gene will then gain an evolutionary advantage and tend to propagate. The same reasoning applies to honeybees. Because worker bees are sterile, they have to rely on the queen for reproduction. As a consequence, if their objective is to maximize reproductive success, as Darwinian theory suggests, then individual self-interest will not be relevant. Thus, a bee will maximize the probability of the survival of the queen and the colony. At the individual-level approach, we assume that a bee is an average product maximizer. On the other hand, what does a bee maximize from the viewpoint of the colony-level approach? It seems reasonably to believe that the more the total food, the higher the probability the hive(s) may survive. To maximize the total output, a bee should stay in the hive where the marginal product is larger. In other words, a marginal product-maximizing bee is to pursue the highest probability of successful reproduction and / or the greatest benefit of the colony. Supposing that the new queen and the old queen are equally closely related to the worker, then successful reproduction for both is equally important and independent of whether the worker remains or leaves. The mathematical representation of the colony-level approach is very similar to that of the individual-level approach. It could be reproduced by replacing the average product, f(n), with the marginal product, F9(n). Readers may notice that our conclusion is based on the
592
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
inverted U-shaped curve of the average product function in the individual-level approach. Our conclusion is still valid at the colony-level approach, because the marginal product function has the same inverted U-shaped curve. However, now 2n¯ instead of 2nˆ is the critical value over which a swarming will occur. By using the explanation of the cooperation effect and the distance effect, we have shown that the total output initially increases at an increasing rate, and then after a certain point, increases at a decreasing rate. The production function hence is an S-shaped curve. Under this S-shaped production function, it is well known that the average and marginal product curves are inverse U-shaped, with the latter going through the maximum point of the former (i.e. nˆ . n¯ ). As a result, the individual level time for forming a new hive is later than the colony level time for forming a new hive (i.e. 2nˆ . 2n¯ ).
4.3. Discussion Based on the Anas (1992) analysis, we have so far provided an economic explanation as to why and how honeybee behaviors can cause swarming. It is worthwhile pointing out the ‘mass-migration’ process that we adopt is not completely the same as the atomistic defection assumption of Anas. In line with the atomistic defection process in Anas (1992) and Krugman (1991, 1998) (which they use to motivate the formation of new cities / agglomerations in a laissez-faire environment), the new hive will be started by a single defecting bee when it realizes that its average product will increase. Others will quickly follow and a cumulative process will cause half of the bee population to move to the new hive. Nevertheless, as mentioned above, one bee is not sufficient to maintain the survival of a new hive; instead, a large number of bees must move together. We thus adopt a slightly different process in which swarming begins when some bees become aware that their average product will increase by leaving the old hive. These bees will exhibit this message by roaming and using other signals of herd behaviors of swarming. A large number of bees will swarm together and others will follow to start the new hive. In the above analysis, we have shown that, no matter whether bees are average product or marginal product maximizers, all will live together when the total number of bees is small. As soon as the population of a hive grows over a critical value, exactly half of the population will move to a new hive. However, as reviewed above, some biologists find that the swarm usually consists of about a little more than half of the adult workers of the colony. This implies that our simple model does not take into account certain factors that affect the number of swarming bees. One possible explanation can be provided for the more-than-half result. Although our conclusion is based on the assumption that the new hive can be built at no cost, this assumption is obviously unrealistic. When the construction cost of the new hive is included, the hive itself becomes a sizable capital asset,
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
593
which will remain with those bees that do not swarm. From an individual rationality standpoint, the average output should be equalized if more than half of the bees remain with the hive. The colony-level approach, on the contrary, yields a different result. If the objective is to maximize total output, then it is sensible to have more than half the workers join the new hive. Some of them will be employed in construction work, which reduces the number of foragers. Total output is maximized with equal numbers of foragers in each hive, which requires more workers in the new hive. As a result, the colony-level approach is consistent with the more-than-half findings.9
5. Conclusion People have labeled the mother bee a ‘queen’ ever since ancient times. Our ancestors believed that the mother bee was the ruler of the colony: the queen coordinated and dictated the admirable efficiency of labor division and the collective action of swarming. Nevertheless, biologists have observed that swarming is dominated by workers. In this paper we applied an economic model to explain the autonomous and collective behavior of honeybee swarming. We assumed that a bee is selfish and it chooses whether to stay in the old colony or to leave according to the maximum probability of its own survival or its reproduction. We showed that all bees will live together when the colony is small. As soon as the population grows over a threshold, the colony will suddenly break into two almost equally sized hives. We also justify that the swarming may not be dictated by the queen, but may be an aggregation of the collective decisions of many ultimately selfish bees. Behaviors of social insects may have important implications for human society. Bees as average product maximizers bear a close similarity to the goal of labor-managed firms. As the objective of a labor-managed firm is to maximize the profit that each of its members can share, this framework may be applied to the study of the growth of such firms. The model may also be potentially applied to investigate the growth of political parties. The economists’ way of thinking can be productively applied to a wide range of activities both inside and outside of the marketplace. It may work just as well in explaining biological phenomena as human activities. In this paper we applied the technique of economic analysis to formally model and analyze the biological phenomenon of honeybee swarming. The purpose of this attempt is not only to 9
Additionally, in the foregoing discussions, we suppose that the new queen (perhaps, the typical worker’s sister) and the old queen (the typical worker’s mother) are equally closely related to the worker. If the new queen is more or less related to the old, then the worker should weigh the interests of the two new hives accordingly. Consequently, the other explanation for the more-than-half results may be due to the possibility that workers are more related to the old queen.
594
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
provide helpful insight into the research of honeybee swarming. It is also confirms the common belief among some of our colleagues in the economics profession that economics is a great instrument to understand the natural world.
Acknowledgements We are greatly indebted to an anonymous referee for very useful comments and suggestions. All remaining errors are our responsibility.
References Anas, A., 1992. On the birth and growth of cities: laissez-faire and planning compared. Regional Science and Urban Economics 22, 243–258. Battalio, R., Green, L., Kagel, J.H., 1981. Income-leisure tradeoffs of animal workers. American Economic Review 71, 621–632. Dawkins, R., 1976. The Selfish Gene. Oxford University Press, New York. Demuth, G.S., 1921. Swarm control. Farmers’ Bulletin, U.S. Department of Agriculture 1198, 1–28. Gadgil, M., Bossert, W.H., 1970. Life-historical consequences of natural selection. American Naturalist 104, 1–24. Getz, W.M., Bruckner, D., Parisian, T.R., 1982. Kin structure and the swarming behavior of honey bee, Apis mellifera. Behavioral Ecology and Sociobiology 10, 265–270. Hirshleifer, J., 1977. Economics from a biological viewpoint. Journal of Law and Economics 20, 1–52. Kagel, J.H., Battalio, R., Rachlin, H., Green, L., Basmann, R., Klemm, W.R., 1975. Experimental studies of consumer demand behavior using laboratory animals. Economic Inquiry 13, 22–38. Krugman, P., 1991. Increasing returns and economic geography. Journal of Political Economy 99, 483–499. Krugman, P., 1998. What’s new about the new economic geography? Oxford Review of Economic Policy 14, 7–17. Lensky, Y., Slabezki, Y., 1981. The inhibiting effect of queen bee (Apis mellifera L.) foot-print pheromone on the construction of swarming queen cups. Journal of Insect Physiology 66, 185–193. Morland, D., 1930. On the causes of swarming in the honeybee: an examination of brood food theory. Annals of Applied Biology 17, 137–147. Rapport, D.J., 1971. An optimization model of food selection. American Naturalist 105, 575–587. Seeley, T.D., 1985. Honeybee Ecology—A Study of Adaptation in Social Life. Princeton University Press, Princeton, NJ. Selsam, M.E., 1986. In: Bees. The New Book of Knowledge, Vol. 3. Grolier Danbury, Connecticut, pp. 117–124. Simpson, J., 1958. The factors which cause colonies of Apis mellifera. Journal of Insect Physiology 5, 77–95. Visscher, P.K., 1993. A theoretical analysis of individual interests and intracolony conflict during swarming of honey bee colonies. Journal of Theoretical Biology 165, 191–212. Winston, M.L., 1987. The Biology of the Honey Bee. Harvard University Press, Cambridge, MA. Winston, M.L., Otis, G.W., 1978. Ages of bees in swarming and afterswarms of Africanized honeybee. Journal of Apicultural Research 17, 123–129. Winston, M.L., Taylor, O.R., 1980. Factors preceding queen rearing in the Africanized honeybee (Apis mellifera) in South Africa. Insects Sociaux 27, 289–304.
The economics of honeybee swarming Chung-cheng Lin a , *, Tze-wei Chen b , Ching-chong Lai c a
Institute of Economics, Academia Sinica, Nankang, Taipei 115, Taipei, Taiwan b National Taiwan Normal University, Taipei, Taiwan c Sun Yat-Sen Institute for Social Sciences and Philosophy, Academia Sinica, Taiwan Received 6 July 2001; received in revised form 12 August 2002; accepted 27 December 2002
Abstract This paper uses the Anas [Regional Science and Urban Economics 22 (1992) 243] model to provide an economic explanation of the cause of honeybee swarming. We assume that a honeybee is rational in choosing to stay in the old hive or to leave to build a new one. Rationality here refers to a bee’s behavior to maximize the food (honey) that it can share or contribute. We show that all bees will live together when the total number of bees is small. As soon as the total number of bees grows over a threshold, half of the population will immediately swarm to another hive. Contrary to the traditional wisdom that the queen bee leads the swarm, our model demonstrates that swarming may be a collective action of the rational choices of many selfish individual bees. 2003 Elsevier B.V. All rights reserved. Keywords: Honeybee swarming; Rationality JEL classification: A12
1. Introduction The scope of economics is extending everyday. While economic activities of
*Corresponding author. E-mail address: [email protected] (C.-c. Lin). 0166-0462 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0166-0462(03)00004-8
582
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
human beings remain the major subject of modern economics, a few economists have explored the interface and analogies between economics and biology.1 This paper is an application of economics to interpret a biological phenomenon, honeybee swarming, in which one honeybee hive separates into two almost equally sized hives suddenly and quickly. While some biological explanations have been provided to explain this phenomenon, this paper tries to furnish an economic interpretation. Anas (1992) set up a simple economic model to investigate different types of city growth, showing that one type of city growth is a sudden jump from a one-city equilibrium to a symmetric two-city equilibrium. We regard the Anas analytical framework as an analogy to the biological phenomenon of honeybee swarming. The Anas framework is therefore adopted to prove that all bees will live together when the total number of bees is relatively small. As soon as the population grows over a critical value, half of them will immediately swarm to a new hive. We develop a model based upon the assumption that a bee is rational in choosing whether to stay in the old hive or to leave to build a new one. Rationality here refers to a bee’s behavior to maximize the food (honey) that it can potentially share or contribute. This assumption is built on two facts. First, biologists conclude that a honeybee hive is a highly cooperative society, and the range of tasks necessary for supporting the hive are efficiently distributed among working bees (workers). Second, biologists also have discovered that for bees to sustain the survival of the hive and consequently their own, workers can change jobs when the needs of the hive increase. The efficiency distribution of tasks and the autonomous job switching constitute the foundation of the assumption of rationality. Our findings conclude that, contrary to the traditional wisdom that the queen bee leads the swarm, the occurrence of a swarming may be an aggregation of the autonomous choices of many ultimately selfish bees. This conclusion is consistent with the biological finding that honeybee swarming is dominated by worker bees and not by the queen. The remainder of the paper is organized as follows. Section 2 reviews the process of honeybee swarming and the relevant biological explanations. Section 3 argues that honeybees can make conscious choices. A simple economic framework is used to model honeybee swarming in Section 4. Finally, concluding remarks are presented in Section 5.
1 Literature on the interrelationship between economics and biology has been produced recently. Kagel et al. (1975) and Battalio et al. (1981) conducted experiments with laboratory animals to test the validity of some elementary principles of economic theories. Moreover, some scholars, for example Gadgil and Bossert (1970) and Rapport (1971), have made significant use of tools and approaches in economics to explain some biological phenomena. A comprehensive survey is provided by Hirshleifer (1977).
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
583
2. The biological viewpoints of honeybee swarming Biologists have observed that honeybees will start a new colony by swarming when the old hive becomes crowded or when a new queen emerges. The swarming process of most kinds of honeybees can be divided into the following two stages: queen rearing and swarming. According to Winston (1987, pp. 181–190), the rearing of new queens can sometimes begin as early as 1 month before an actual swarming. The process starts when eggs are laid or placed into special areas known as ‘queen cups’. These cups are built by workers and are specifically used for queen rearing. However, the existence of eggs in the queen cups does not guarantee the occurrence of a swarming. Larvae (wormlike grubs that hatch from eggs) in the queen cups are provided with special food (royal jelly) that turns them into queens. At the end of the larvae feeding period, once the first queen cups are sealed, a swarming can occur that day or the day after. The entire process of queen rearing in most cases takes | 8–10 days. About 10 days prior to swarming, workers begin to store honey in their honey stomachs. A few hours before swarming, some workers become ‘excited’ and ‘‘begin running back and forth in waves, buzzing to excite the other workers’’ (Winston, 1987, p. 187). Contrary to the traditional wisdom that the old queen leads the swarm, the entire swarming is dominated by workers. They form the swarm, scout for a new site and force the queen to join the swarm. A swarm happens when the old queen is successfully forced to join the clustering workers and then is relocated to a new site.2 The swarm usually consists of about a little more than half of the adult workers of the colony (Winston and Otis, 1978; Getz et al., 1982).3 Along with detailed documentation on the process of swarming, biologists have also studied its causes. Studies on honeybees and other social insects can be generally categorized into two groups: the colony-level approach and the individual-level approach. The former asserts that the behavior of bees and other social insects is subject to generating the greatest benefit to the colony, not necessarily for themselves (Seeley, 1985). The latter argues that the behavior of all social insects is ‘‘simply statistical summaries of many individuals’ ultimately selfish actions’’ (Seeley, 1985, p. 5). The colony-level approach has dominated the theory of evolution ever since Charles Darwin published his theory of evolution.
2 Sometimes when the workers do not succeed in driving the queen with them, the workers will return to the old hive. This is known as ‘false swarms’. 3 The swarming process of some kinds of honeybees, for example Apis mellifera in South African, can be divided into three stages: queen rearing, prime swarm (swarm), and afterswarms. Afterswarms are near replicas of the prime swarm except that workers are joined by virgin queens rather than the mated queen. The frequency of afterswarms varies according to factors such as weather, food and the number of virgin queens (Winston, 1987, pp. 181–190). We exclude the case of afterswarming in our model.
584
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
However, the individual-level approach is now receiving more attention among scientists. Early studies tended to attribute the cause of swarming to the amount of food in the hive. Morland (1930) argued that swarming was the result of too many young nurse bees that produced too much brood food. In order to reduce the amount of food, queen rearing began, which led to swarming. Another group of scientists believed that the crowding of adult workers and limited space for food rearing resulted in swarming (Demuth, 1921; Winston, 1987). There is, however, no sufficient evidence to support these hypotheses (Winston, 1987). Other investigations (Simpson, 1958; Winston and Taylor, 1980; Lensky and Slabezki, 1981) found a multifactorial relationship between swarming and several within-colony demographic factors such as colony size, brood hive congestion, worker age distribution, and reduced transmission of queen substances, but none of these factors alone are found to initiate swarming (Winston, 1987). Visscher (1993) approached the question of individual behavior during swarming from the perspective of individual interests. Employing the theory of natural selection, he argued that there are ‘conflicts’ among colony members during the process of swarming. The individual choice of whether to stay or to leave can therefore be predicted by evaluating the cost of each potential outcome. For example, on the question of which queen will leave, the conclusion is drawn from examining the probabilities that the colony will survive from the mother queen’s, the virgin (daughter) queen’s and the workers’ perspectives. He also showed that there are sharp differences among the interests of different colony members. The final outcomes are always the results of conflicts.
3. Rational behavior For hundreds of years human beings have wondered about honeybee societies. How does each newly hatched bee ‘know’ what to do? Is it possible that each bee is hatched to serve a particular kind of work such as nursing or foraging? From the experiments of Rosch (reviewed by Selsam, 1986), we learn that each worker bee follows a schedule of duties according to its age. A worker bee lives for only 5 or 6 weeks, and jobs change as the bee grows older. At first it cleans out brood cells for new eggs. After 3 days at this job, it becomes a nurse bee and feeds pollen and honey to older larvae. From the 6th to the 12th day it feeds the younger larvae and the queen with royal jelly secreted by special nursing glands in her head. On the 12th day the same bee becomes a builder of wax cells. From the 16th day on, it begins to receive the nectar and pollen brought to the hive. Around the 20th day it stands guard at the entrance to the hive. From the 3rd week until the end of its life, the bee shifts to the last and longest job: foraging, collecting of nectar and pollen. This schedule is not rigid. As Rosch showed, this schedule changes to suit the
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
585
needs of the colony. If only forager bees are left in a hive, then some of these older bees will grow new nursing glands and start to nurse the brood again, even though they are far past the normal nursing age. Likewise, a colony left without foraging bees can also respond to the emergency by dispatching young bees at their nursing age to forage after a few days. Lindauer et al. (reviewed by Selsam, 1986) asked how does a bee ‘‘find out’’ what job has to be done for the survival of the colony? They found that every bee gathers its own information about the needs of the hive. Every day a bee spends about a third of its time patrolling the hive: It inspects cells and larvae, and looks over the building areas and the storage of food. From such inspection tours new forms of activity start, nursing or building or clearing, as the need arises. In a word, a honeybee is able to willingly adjust its course of action according to either the needs of the hive or its own interests. The goal of this behavior, from the viewpoint of the colony-level approach, seems to be the pursuit of the continuing existence of the hive. From the perspective of the individual-level approach, the goal appears to be its own continuous survival. In economic terms we argue that a honeybee is rational to maximize the benefit of its colony or its own interests by adjusting its daily duties and determining whether to stay or to leave.
4. The model In this section we propose a simple economic framework to model honeybee swarming from both the individual-level and the colony-level perspectives. We first discuss the individual-level approach and then the colony-level approach.
4.1. The individual-level approach For the individual-level approach we assume that bees are selfish. That is, each honeybee cares only about its own interests. The goal of a bee is to maximize the probability of its own survival. The existence of the hive is merely a means, not the goal. The larger its share or amount of honey (food), the more likely it will survive. Therefore, we presume that a honeybee is rational in maximizing the honey that it can potentially consume (i.e. the average product) by determining whether to stay or to leave. The total output (honey) of a hive is assumed to increase initially at an increasing rate, and then at a decreasing rate with the number of bees (or more 4 There are three kinds of bees in a hive: the queen, drones, and workers. The queen is a large female who lays all the eggs. The drones are the male bees. The only function of the drones is to mate with the queen. The workers are small females. They do not lay eggs, but they carry out all maintenance and growth of the colony.
586
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
precisely, workers) in the colony.4 This relationship can be explained as follows. An increase in the total number of bees exerts two different effects on the total product: the cooperative effect and the distance effect. The cooperative effect indicates that the total output rises at an increasing rate with the number of bees in the same hive, because the possibility of locating food increases and thus the time spent on food searching decreases as the number of bees grows. However, the cooperative effect will decline as the bee population grows. On the other hand, the distance effect makes the total output increase at a decreasing rate with an increasing number of bees in the same hive. The reason is that more bees there are, the farther the average distance each bee must fly in order to find a source of food. Obviously, this distance disadvantage in food production will rise as the number of bees grows. As a result, as the number of bees increases the cooperative effect initially prevails and the total output increases at an increasing rate. After a certain point the distance effect will dominate and the total output will increase at a decreasing rate.5 Let F(n) denote the total product of a colony with n bees. This is shown in the upper part of Fig. 1. The lower part is the corresponding average product f(n) ; F(n) /n and marginal product F9(n) ; dF(n) / dn. Both average and marginal product functions have inverted U-shaped curves, increasing initially and then decreasing as the number of bees grows. Let nˆ (n¯ ) denote the number of bees at the maximum of the average (marginal) product, which is determined by the efficiency of labor division among bees and the natural environment around the hive. Both nˆ and n¯ play a critical role in the following discussions. From the perspective of the individual-level approach, we assert that a honeybee maximizes the average product f(n) that it can potentially consume in choosing to stay or to leave. The total number of bees n can either live together in one hive or live separately in two hives. Let n 1 and n 2 represent the number of bees in the old and the new hives, respectively, and n 5 n 1 1 n 2 . Let an overdot denote a derivative with respect to time. For simplicity, building a new hive is assumed to be costless for now. The simplest dynamic migration processes where bees move between two hives can be specified as follows: n~ 1 5 f(n 1 ) 2 f(n 2 ),
(1)
n~ 2 5 f(n 2 ) 2 f(n 1 ).
(2)
These migration processes simply state that the number of bees in a colony 5 This would be true even if we define output net of food consumption in foraging. At some population, additional bees may use more food foraging than they bring in, and so if they were foraging then total output net of foraging cost would fall. Our conclusion still holds in the case where the total output decreases after a certain point.
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
587
Fig. 1. The total, average and marginal outputs.
increases (decreases) when its average product is higher (lower) than that of its rival’s. It remains intact when the average products of both hives are equal.6 Inspection of these equations shows that there are three types of equilibria (n *1 ,n 2* ): two-colony of equal size (symmetric equilibrium), two-colony of unequal size (asymmetric equilibrium), and one-colony equilibrium. The first two types of 6
The purpose of this paper is to find the causes of honeybee swarming rather than the reasons for the growth of the bee population. In other words, we will explore when swarming will happen as the number of bees grows, and thus we do not specify the growth function of bees.
588
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
equilibrium occur when the time derivatives of n 1 and n 2 vanish (n~ 1 5 n~ 2 5 0), and they require f(n *1 ) 5 f(n 2* ). A symmetric equilibrium exists when both colonies have exactly half of the total bees (n *1 5 n 2* 5 n / 2). An asymmetric equilibrium appears when one hive is small and the other hive is large, but both have the same average product (for example, n *1 . nˆ . n *2 ). It is clear that the large hive must ˆ the number of contain a population larger than (and the small hive smaller than) n, bees at which the average product is the maximum, in order to reach an asymmetric equilibrium. The third type of equilibrium happens when one hive contains the entire population and the second hive does not appear, that is, the corner solution (n *1 ,0) or (0,n *2 ). In this case, the time derivatives (1) and (2) do not vanish. Given all possible equilibria, we now investigate the dynamic stability of each possible solution to examine the attainability of each equilibrium. First, we check the stability of the one-colony equilibrium. We explore the case of (n *1 ,0) and ignore (0,n *2 ), because only the case of all bees staying in the old colony is meaningful. As mentioned above, a honeybee hive is a highly cooperative society. The range of tasks from nursing to foraging necessary for supporting the hive must be efficiently coordinated among working bees. It would be sensible to argue that when the population falls below a certain threshold, the hive will not be able to survive. We can therefore check the stability of the corner solution by investigating the case where all bees but one stay in the old hive. From Eqs. (1) and (2), we can obtain: n~ 1 5 f(n 1 5 n 2 1) 2 f(n 2 5 1) . 0,
(3)
n~ 2 5 f(n 2 5 1) 2 f(n 1 5 n 2 1) , 0.
(4)
Evidently, the total and average output of a very small hive will be very low. To the extreme, we might expect that the survival of a hive cannot be maintained if the number of bees is too small. Accordingly, it is reasonable to argue that f(n 2 5 1) 5 0 and f(n 1 5 n 2 1) . 0 for a suitable size of bees in the old hive. The sign of Eq. (3) hence is positive and that of Eq. (4) is negative. These signs indicate that the average product of the old hive with almost all bees is bigger than that of the ‘new’ hive with only one bee. In other words, it is unlikely that a very small number of bees will migrate and successfully build a new hive. Consequently, the one-colony equilibrium (n *1 5 n,0) is stable at a suitable size of the bee population.7 We secondly examine the stability of the symmetric equilibrium where n *1 5 n *2 5 n / 2. Eqs. (1) and (2) are mutual dependent, because of the constraint 7 We use f(n 2 5 1) rather than f(n 2 5 0) to investigate the stability of the one-colony equilibrium (n 1 5 n, n 2 5 0), because f(n 2 5 0); the average output at n 2 5 0, is undefined, since it is 0 / 0.
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
589
n 5 n 1 1 n 2 .8 We can hence investigate the stability by checking only one of two equations. From Eq. (1) we have: dn~ ]1 dn 1
U
≠f(n *1 ) ≠f(n *2 5 n 2 n *1 ) 5 ]] n 1* 5n / 2 2 ]]]]] n 2* 5n / 2 ≠n 1 ≠n 1 n n ˆ 5 2f 9 ] . ( , )0; f 9( ] ) . ( , )0; n , ( . )2n. 2 2
SD
U
(5)
ˆ the sign of Eq. (5) is positive, because f 9(n / 2) . 0, and the When n , 2n, ˆ the sign of Eq. (5) is negative, symmetric equilibrium is unstable. When n . 2n, because f 9(n / 2) , 0, and the symmetric equilibrium is stable. The reasoning is ˆ the average products of both hives are equal quite straightforward. When n , 2n, and increase with the number of bees. A bee moving from the old hive to the new one will obtain a higher average product there and simultaneously lower the average product of the old colony. This makes the new hive more attractive and induces more bees to move in. The stability of symmetric equilibrium thus cannot ˆ the average products of both hives are be retained. By contrast, when n . 2n, equal, but decrease with the number of bees. A bee moving to the new hive will raise the average product of the old one and result in a lower average product of the new colony. The rational choice of the bee is obviously to stay in the old hive. The symmetric equilibrium accordingly becomes a stable solution as the bee ˆ population grows over the critical value 2n. We finally examine the stability of the asymmetric solution. As mentioned above, the asymmetric equilibrium (n *1 . nˆ . n *2 ) can only exist when the number of bees in the larger hive is more than the maximum value of the average product (nˆ ). Since the distance effect will become relatively mild as bees travel far, the inverse U-shaped average product curve may decline relatively mildly after reaching its maximum value. Under this situation, we will find that the asymmetric equilibrium may appear only when the total number of bees is more than the ˆ In other words, no asymmetric solution will exist when n , 2n. ˆ critical value 2n. Following Anas (1992), the phase diagrams drawn in Fig. 2 are used to show the above argument. The phase diagram of Fig. 2(a) illustrates that when n , nˆ there are two types of equilibrium: one-colony and symmetric solutions. As shown above, the one-colony equilibrium is stable and the symmetric equilibrium is ˆ the one-colony and symmetric unstable. Fig. 2(b) shows that when nˆ , n , 2n, equilibria are again stable and unstable, respectively. Fig. 2(c) shows that when
8
Given n 5 n 1 1 n 2 , mutual dependence between Eqs. (1) and (2) can be proved by substituting n 2 5 n 2 n 1 and n~ 2 5 2 n~ 1 into Eqs. (1) and (2); that is: n~ 1 5 f(n 1 ) 2 f(n 2 n 1 ), 2 n~ 1 5 f(n 2 n 1 ) 2 f(n 1 ). Therefore, these two equations are mutually dependent.
590
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
Fig. 2. The phase diagrams.
ˆ the one-colony and symmetric equilibria are both stable. Since two n . 2n, adjacent equilibria cannot simultaneously be stable (or unstable), this implies that the asymmetric equilibria are unstable. The one-colony equilibrium for any suitable number of bees and the symmetric equilibrium for n . 2nˆ are accordingly stable and attainable solutions. When the honeybee society expands, there are two potential outcomes. One is that all of the bees live in a hive no matter how large the population is. The other is that all bees live collectively in the old hive when the total number of bees is under the critical
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
591
ˆ but they swarm into two equally sized hives when the total number just value 2n, passes the critical value. It is obvious that the average product of each of the two colonies with the half ˆ or mathematipopulation is more than that of the all-in-one colony when n . 2n, ˆ This is because the average product is a cally, f(n / 2) . f(n) for all n . 2n. ˆ This decreasing function when the number of bees in the hive is larger than n. result implies that half of the bees will leave the old colony when the total number of bees is large enough. As a consequence, when an old hive becomes ‘crowded’, honeybees will start a new colony by swarming. The purpose of the swarm is to pursue the larger average product which each of them can consume. In what follows, each of the two hives may follow the same growth pattern, and as a result, there are numerous honeybee hives in the world.
4.2. The colony-level approach Now we turn to the colony-level approach. Before proceeding, we first discuss why a bee maximizes the probability of the survival of its colony and not just itself. At first glance, it seems to be a behavior of pure altruism. However, biologists (for example, Dawkins, 1976) have argued that altruism is a result of purely selfish (non-altruistic) behavior on the part of the genetic material that is the true medium for natural selection. If a man is carrying a certain gene, then his child is carrying 50% of his same gene. The gene’s survival probability is enhanced if he behaves in a way that improves the survival prospects of his children. Now suppose that some particular gene has the effect of making him altruistic toward his children. The gene will then gain an evolutionary advantage and tend to propagate. The same reasoning applies to honeybees. Because worker bees are sterile, they have to rely on the queen for reproduction. As a consequence, if their objective is to maximize reproductive success, as Darwinian theory suggests, then individual self-interest will not be relevant. Thus, a bee will maximize the probability of the survival of the queen and the colony. At the individual-level approach, we assume that a bee is an average product maximizer. On the other hand, what does a bee maximize from the viewpoint of the colony-level approach? It seems reasonably to believe that the more the total food, the higher the probability the hive(s) may survive. To maximize the total output, a bee should stay in the hive where the marginal product is larger. In other words, a marginal product-maximizing bee is to pursue the highest probability of successful reproduction and / or the greatest benefit of the colony. Supposing that the new queen and the old queen are equally closely related to the worker, then successful reproduction for both is equally important and independent of whether the worker remains or leaves. The mathematical representation of the colony-level approach is very similar to that of the individual-level approach. It could be reproduced by replacing the average product, f(n), with the marginal product, F9(n). Readers may notice that our conclusion is based on the
592
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
inverted U-shaped curve of the average product function in the individual-level approach. Our conclusion is still valid at the colony-level approach, because the marginal product function has the same inverted U-shaped curve. However, now 2n¯ instead of 2nˆ is the critical value over which a swarming will occur. By using the explanation of the cooperation effect and the distance effect, we have shown that the total output initially increases at an increasing rate, and then after a certain point, increases at a decreasing rate. The production function hence is an S-shaped curve. Under this S-shaped production function, it is well known that the average and marginal product curves are inverse U-shaped, with the latter going through the maximum point of the former (i.e. nˆ . n¯ ). As a result, the individual level time for forming a new hive is later than the colony level time for forming a new hive (i.e. 2nˆ . 2n¯ ).
4.3. Discussion Based on the Anas (1992) analysis, we have so far provided an economic explanation as to why and how honeybee behaviors can cause swarming. It is worthwhile pointing out the ‘mass-migration’ process that we adopt is not completely the same as the atomistic defection assumption of Anas. In line with the atomistic defection process in Anas (1992) and Krugman (1991, 1998) (which they use to motivate the formation of new cities / agglomerations in a laissez-faire environment), the new hive will be started by a single defecting bee when it realizes that its average product will increase. Others will quickly follow and a cumulative process will cause half of the bee population to move to the new hive. Nevertheless, as mentioned above, one bee is not sufficient to maintain the survival of a new hive; instead, a large number of bees must move together. We thus adopt a slightly different process in which swarming begins when some bees become aware that their average product will increase by leaving the old hive. These bees will exhibit this message by roaming and using other signals of herd behaviors of swarming. A large number of bees will swarm together and others will follow to start the new hive. In the above analysis, we have shown that, no matter whether bees are average product or marginal product maximizers, all will live together when the total number of bees is small. As soon as the population of a hive grows over a critical value, exactly half of the population will move to a new hive. However, as reviewed above, some biologists find that the swarm usually consists of about a little more than half of the adult workers of the colony. This implies that our simple model does not take into account certain factors that affect the number of swarming bees. One possible explanation can be provided for the more-than-half result. Although our conclusion is based on the assumption that the new hive can be built at no cost, this assumption is obviously unrealistic. When the construction cost of the new hive is included, the hive itself becomes a sizable capital asset,
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
593
which will remain with those bees that do not swarm. From an individual rationality standpoint, the average output should be equalized if more than half of the bees remain with the hive. The colony-level approach, on the contrary, yields a different result. If the objective is to maximize total output, then it is sensible to have more than half the workers join the new hive. Some of them will be employed in construction work, which reduces the number of foragers. Total output is maximized with equal numbers of foragers in each hive, which requires more workers in the new hive. As a result, the colony-level approach is consistent with the more-than-half findings.9
5. Conclusion People have labeled the mother bee a ‘queen’ ever since ancient times. Our ancestors believed that the mother bee was the ruler of the colony: the queen coordinated and dictated the admirable efficiency of labor division and the collective action of swarming. Nevertheless, biologists have observed that swarming is dominated by workers. In this paper we applied an economic model to explain the autonomous and collective behavior of honeybee swarming. We assumed that a bee is selfish and it chooses whether to stay in the old colony or to leave according to the maximum probability of its own survival or its reproduction. We showed that all bees will live together when the colony is small. As soon as the population grows over a threshold, the colony will suddenly break into two almost equally sized hives. We also justify that the swarming may not be dictated by the queen, but may be an aggregation of the collective decisions of many ultimately selfish bees. Behaviors of social insects may have important implications for human society. Bees as average product maximizers bear a close similarity to the goal of labor-managed firms. As the objective of a labor-managed firm is to maximize the profit that each of its members can share, this framework may be applied to the study of the growth of such firms. The model may also be potentially applied to investigate the growth of political parties. The economists’ way of thinking can be productively applied to a wide range of activities both inside and outside of the marketplace. It may work just as well in explaining biological phenomena as human activities. In this paper we applied the technique of economic analysis to formally model and analyze the biological phenomenon of honeybee swarming. The purpose of this attempt is not only to 9
Additionally, in the foregoing discussions, we suppose that the new queen (perhaps, the typical worker’s sister) and the old queen (the typical worker’s mother) are equally closely related to the worker. If the new queen is more or less related to the old, then the worker should weigh the interests of the two new hives accordingly. Consequently, the other explanation for the more-than-half results may be due to the possibility that workers are more related to the old queen.
594
C.-c. Lin et al. / Regional Science and Urban Economics 33 (2003) 581–594
provide helpful insight into the research of honeybee swarming. It is also confirms the common belief among some of our colleagues in the economics profession that economics is a great instrument to understand the natural world.
Acknowledgements We are greatly indebted to an anonymous referee for very useful comments and suggestions. All remaining errors are our responsibility.
References Anas, A., 1992. On the birth and growth of cities: laissez-faire and planning compared. Regional Science and Urban Economics 22, 243–258. Battalio, R., Green, L., Kagel, J.H., 1981. Income-leisure tradeoffs of animal workers. American Economic Review 71, 621–632. Dawkins, R., 1976. The Selfish Gene. Oxford University Press, New York. Demuth, G.S., 1921. Swarm control. Farmers’ Bulletin, U.S. Department of Agriculture 1198, 1–28. Gadgil, M., Bossert, W.H., 1970. Life-historical consequences of natural selection. American Naturalist 104, 1–24. Getz, W.M., Bruckner, D., Parisian, T.R., 1982. Kin structure and the swarming behavior of honey bee, Apis mellifera. Behavioral Ecology and Sociobiology 10, 265–270. Hirshleifer, J., 1977. Economics from a biological viewpoint. Journal of Law and Economics 20, 1–52. Kagel, J.H., Battalio, R., Rachlin, H., Green, L., Basmann, R., Klemm, W.R., 1975. Experimental studies of consumer demand behavior using laboratory animals. Economic Inquiry 13, 22–38. Krugman, P., 1991. Increasing returns and economic geography. Journal of Political Economy 99, 483–499. Krugman, P., 1998. What’s new about the new economic geography? Oxford Review of Economic Policy 14, 7–17. Lensky, Y., Slabezki, Y., 1981. The inhibiting effect of queen bee (Apis mellifera L.) foot-print pheromone on the construction of swarming queen cups. Journal of Insect Physiology 66, 185–193. Morland, D., 1930. On the causes of swarming in the honeybee: an examination of brood food theory. Annals of Applied Biology 17, 137–147. Rapport, D.J., 1971. An optimization model of food selection. American Naturalist 105, 575–587. Seeley, T.D., 1985. Honeybee Ecology—A Study of Adaptation in Social Life. Princeton University Press, Princeton, NJ. Selsam, M.E., 1986. In: Bees. The New Book of Knowledge, Vol. 3. Grolier Danbury, Connecticut, pp. 117–124. Simpson, J., 1958. The factors which cause colonies of Apis mellifera. Journal of Insect Physiology 5, 77–95. Visscher, P.K., 1993. A theoretical analysis of individual interests and intracolony conflict during swarming of honey bee colonies. Journal of Theoretical Biology 165, 191–212. Winston, M.L., 1987. The Biology of the Honey Bee. Harvard University Press, Cambridge, MA. Winston, M.L., Otis, G.W., 1978. Ages of bees in swarming and afterswarms of Africanized honeybee. Journal of Apicultural Research 17, 123–129. Winston, M.L., Taylor, O.R., 1980. Factors preceding queen rearing in the Africanized honeybee (Apis mellifera) in South Africa. Insects Sociaux 27, 289–304.