Acta Biotheor (2014) 62:123–132 DOI 10.1007/s10441-014-9213-0 REGULAR ARTICLE
Reduction of Foraging Work and Cooperative Breeding Hiroshi Toyoizumi • Jeremy Field
Received: 9 July 2013 / Accepted: 4 March 2014 / Published online: 12 March 2014 Springer Science+Business Media Dordrecht 2014
Abstract Using simple stochastic models, we discuss how cooperative breeders, especially wasps and bees, can improve their productivity by reducing foraging work. In a harsh environment, where foraging is the main cause of mortality, such breeders achieve greater productivity by reducing their foraging effort below full capacity, and they may thrive by adopting cooperative breeding. This could prevent the population extinction of cooperative breeders under conditions where a population of lone breeders cannot be maintained. Keywords Cooperation Reduction of foraging work Division of work Cooperative breeding Survivorship insurance Foraging
1 Introduction For many animals with parental care responsibilities, foraging is a risky business (Cant and Field 2001 and the references therein; Field et al. 2007). Parents must leave the relative safety of their nests to seek food for their offspring. Some may be captured by predators, while others could encounter an accident. It is not just the parent’s own life that is at risk, because immature offspring might die without parental care (Field et al. 2000; Shreeves et al. 2003). Although a larger brood size increases the potential reward for parents, it increases the amount of foraging work necessary. Therefore, there is an increased risk that the entire brood will fail when adult carers die. In this paper, we discuss the level of foraging that results in the greatest productivity. H. Toyoizumi (&) Waseda University, Nishi-waseda 1-6-1, Shinjuku, Tokyo 169-8050, Japan e-mail: [email protected] J. Field School of Life Sciences, University of Sussex, Brighton, UK
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A variety of foraging animals breed cooperatively. Often, cooperation involves a hierarchy with a dominant breeder, who foregoes foraging and specializes in egglaying while subordinate helpers forage to feed the dominant’s brood. Various hypotheses have been proposed for the somewhat paradoxical evolution of helping behaviour, whereby a helper (at least temporarily) forfeits its own chance to reproduce and instead helps to rear another individual’s offspring. Helpers are nearly always rearing the offspring of a relative, meaning that copies of the helper’s genes are propagated indirectly (Hamilton 1964). However, as the relative’s offspring rarely carry as large a proportion of the helper’s genes as would the helper’s own offspring, natural selection should favour this behaviour only if helpers compensate by being more productive than they would be nesting alone (Queller 1996). In insects such as wasps and bees, there are various ways in which this could happen, some of which rely on the relatively short lifespan of adults compared with the long development time of their progressively fed immature offspring (Field 2005). The extended parental care (EPC) implicit in progressive feeding means that a mother often dies before her offspring mature (Queller 1994). Therefore, for a potential helper, staying in the natal nest and rearing the part-matured brood of a relative may be more productive than starting a new nest and rearing her own brood: the part-matured offspring are more likely to reach adulthood before the group as a whole fails [headstart hypothesis: (Queller 1989)]. A subtly different idea is that, if a helper dies young, any dependent offspring that she has only part-reared can be brought to adulthood by the other individuals still remaining in the group, whereas for a female nesting independently, an early death means total brood failure [assured fitness return: (Field et al. 2000; Gadagkar 1990; Shreeves et al. 2003)]. If a helper has the chance of inheriting dominant status, it may be worth waiting without immediate fitness return if the expected reproductive success as the dominant breeder is large enough to outweigh the chance of death while waiting in the queue [delayed fitness return: (Kokko and Johnstone 1999; Kokko et al. 2001; Shreeves and Field 2002)]. Further discussion of these hypotheses can be found in previous studies (Field 2008; Heinsohn and Legge 1999; Nonacs et al. 2006; Shen and Reeves 2010; Shen et al. 2011). Most of these previous studies aim to understand cooperative breeding from the evolutionary perspective of rational individual decision making. Here, we analyse it from the perspective of nest or population productivity and survival, i.e. as if group members have identical genetic interests. We test the effect on productivity of a general characteristic of cooperative breeding: reduction of foraging work (RFW) below the full capacity of the group, which is observed in various social insects such as sweat bees (Field 1996; Field et al. 2012) and harvester ants (Gordon 2013). In a social nest, the dominant typically concentrates on breeding, not risking her life by foraging away from the nest, while her helpers forage. Such reduced effort may increase the inclusive fitness of high-ranking individuals (Cant and Field 2001). This results in reduced foraging effort at the group level because the potential foraging ability of all group members (dominants and workers) is not fully utilized. We investigate whether this reduced foraging effort will result in an increase or a decrease in the total number of reproductive individuals dispersing from the group. It should be noted that only the total amount of work carried out by the group is considered to be important in our analysis, rather than how the work is divided among the group.
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In this paper, we use simple stochastic models to simulate the life history of cooperative breeders living in an aseasonal environment. We consider nests initiated by single foundresses, who continuously rear offspring until the termination of the nest, such as hover wasps (Stenogastrinae) (Field 2008). We model both lone breeders and cooperative breeders, and use the number of offspring dispersing from the nest as an estimate of productivity. From our model, we conclude that cooperative breeders achieve greater productivity if they do not work to their maximum potential, and instead reduce the total foraging risk below the level that lone breeders must accept. 2 Lone Breeders and Reduction of Foraging Work First, we analyse productivity in a population of lone breeders, i.e. where all nests are initiated by single foundresses and all offspring disperse so that the adult group size never exceeds one. Productivity is evaluated by the number of offspring H that disperse from the nest and potentially initiate new nests. To maintain the population, the expectation of H should be [1. Note that we neglect males, who are not involved in nesting in wasp and bee populations. We assume that a lone breeder rears successive batches of n broods in her nest (n = brood size, see Fig. 1). The immature broods must be fed continuously, and take one time unit to mature to adulthood (maturation cycle). This time scale is used throughout the paper. Because foraging is risky, if a lone breeder rears multiple broods simultaneously (n [ 1), her chance of death increases proportionally: we assume that the chance of death during foraging is random and proportional to the brood size n (Cant and Field 2001; Field et al. 2007). Together with the foragingindependent intrinsic death rate l0, the overall death rate of a lone breeder is l1 ¼ na þ l0 ;
ð1Þ
where a is the foraging risk factor, and the lone breeder’s lifetime is exponentially distributed with mean 1/l1. If a lone breeder dies before a brood matures to adulthood, the nest is terminated and the part-grown brood fails. When a brood matures, all disperse, and the lone breeder initiates a new batch of n brood. This maturation cycle is repeated until the lone breeder’s death at time s. Let M be the number of successful maturation cycles in [0, s). Because the chance of death for a lone breeder is exponentially distributed, M is geometrically distributed as PfM ¼ mg ¼ f1 el1 gfel1 gm :
ð2Þ
As n broods mature at each maturation time, the total number of individuals dispersed from the nest is nel1 : ð3Þ 1 el1 As shown in Fig. 2, E[H] is a convex function of n that attains its maximum at some intermediate brood size n0. Figure 3 shows the region in which a population of lone breeders is viable when they choose the optimal brood size n0 for various intrinsic death rates l0 and foraging risk factors a. E½H ¼ nE½M ¼
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Fig. 1 Maturation cycles and adult lifespan for lone breeders in our model. Thick arrows represent adult life time, and thin arrows represent the time required for brood maturation. Red arrows represent abandoned brood. a illustrates the case where the lone breeder rears only a single immature brood at a time (n = 1) and dies during the 6th maturation cycle: the 6th brood item therefore fails. b illustrates the case where the lone breeder rears batches of 3 broods simultaneously (n = 3) and dies during the 2nd maturation cycle. The second batch of 3 broods therefore fails. (Color figure online)
Fig. 2 The expected number of dispersed individuals E[H] for lone breeders caring for n broods, where the founder’s death rate is l1 = na ? 1/3 for various values of the foraging risk factor a
As the environment becomes harsher (larger a), lone breeders should decrease their brood size to mitigate the increased foraging risk (see Fig. 3). Sometimes, the environment is so harsh that lone breeders cannot afford to rear even a single brood. They then face extinction. Let us consider a brood size that can take the fractional number 1/2 (see Fig. 3b). In this case, lone breeders can be more productive and maintain a viable population in a harsh environment, such as when a [ 0.7.
3 Cooperative Breeders In some foraging insects, the expected adult lifespan is significantly shorter than the brood maturation time, mainly because of risky foraging work. In such a harsh environment, a population of lone breeders has no chance of escaping the threat of extinction, even if individuals rear only a single brood at a time, as seen in Sect. 2.
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Fig. 3 a Region in which a population of lone breeders is viable for various intrinsic death rates l0 and foraging risk factors a (blue regions). The region is divided into sub-regions according to the optimal integer number of broods n0 = 1, 2, 3, …. b Sub-region for n0 = 1/2 is added, assuming that lone breeders could rear 1/2 a brood (pink region). (Color figure online)
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Fig. 4 Dynamics of cooperative breeding. Each arrow represents the life history of cooperative breeders, and thick and thin arrows represent adult and brood lifetimes, respectively. A foundress (thick black arrow) starts her nest with a single brood. The 1st, 3rd, and 4th broods (blue arrows) stay in the natal nest and start cooperative breeding, whereas the 2nd and 5th broods (black arrows) find two adults in the nest and disperse. The last brood (red arrow) fails because of the termination of the nest. (Color figure online)
However, by reducing the total amount of foraging work in the nest, cooperative breeders are able to increase the chance of population survival. In order to analyse cooperative breeding in a minimum setting, we consider a nest with a single individual brood cared for by (at most) two adult cooperative breeders (results are qualitatively similar if we allow more than two adults). On reaching adulthood, a brood remains in the natal nest if there is currently only one adult. Once two adults exist in the nest, cooperative breeding starts and further matured offspring must all disperse until one of the two existing adults dies. At time 0, a cooperative breeder founds her new nest and initiates a single brood (see Fig. 4). A single cooperative breeder behaves exactly like a lone breeder with n = 1, and her overall death rate, as before, is l1 ¼ l0 þ a: ð4Þ If the single cooperative breeder succeeds in feeding her single brood for the unit maturation cycle, the matured brood stays in the nest and cooperates to rear future successive broods. When a single brood is taken care of by two cooperative breeders, the sum of their death rates due to risky foraging work will again be a. If intrinsic mortality is also included, the summed death rate with two cooperative breeders, l2, is obtained by l2 ¼ 2l0 þ a:
ð5Þ
In nature, the division of work in most cooperatively breeding wasps and bees is complete: one adult (dominant) reproduces while the other (helper) forages. Thus, the total foraging work is less than the full foraging capacity of two adults. However, for our purposes, only the total amount of work carried out by the group is important, rather than how the work is divided among the group. If one of the two adults dies, the brood is maintained by the remaining single adult until the next maturation time [if the helper dies, this corresponds to assured fitness returns in the sense of Gadagkar (1990)]. If both adults die before brood maturation, the current single brood is abandoned and the nest is terminated. This process repeats until time s, when the last adult in the nest dies. Note we do not assume that cooperative breeders forage more efficiently than lone breeders—the foraging risk factor a remains the same as in (4) and (5).
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Fig. 5 a Figure 3 redrawn to show that cooperative breeders have greater productivity than lone breeders (yellow region surrounded by a dashed line). b Region of lone breeders with a fractional brood n0 = 1/2 is added. Part of the advantage of breeding cooperatively is that only cooperative breeders can effectively rear a fractional number of broods (pink region). (Color figure online)
By careful observation of the stochastic dynamics of cooperative breeders (see Appendix), the expected number of dispersers is explicitly obtained by
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E½H ¼
el1 e l 2 1 l2
eðl0 þaÞ ¼ 2l þa : e 0 1 a 2l0
ð6Þ
Figure 5 displays the region in which cooperative breeders maintain a viable population (E[H] [ 1), also exceeding the productivity of lone breeders at the same time. Even in a harsh foraging environment, where lone breeders have too small a productivity (E[H] \ 1) for population survival, cooperative breeders can achieve sufficiently high productivity to avoid population extinction (E[H] [ 1). There are two reasons for this outcome. First, two adults nesting together can afford to take a greater summed risk than a lone breeder, because if one of them dies, the other can continue to rear some of the brood (the insurance effect). Second, two cooperative breeders nesting together can reduce risky foraging work more than is possible for a lone breeder: two cooperative breeders can rear just a single brood at a time, effectively rearing 1/2 a brood each—less than the minimum single brood for a lone breeder (the effect of RFW). Figure 5 shows the RFW effect: cooperative breeders have a viable population in a harsh environment, whereas only lone breeders with 1/2 a brood have a viable population.
4 Conclusion We have used a stochastic model to evaluate foraging risk and productivity for different breeding scenarios. This analysis clearly shows the advantage of a RFW as a whole. We also found that RFW may enable cooperative breeders to achieve a higher productivity than lone breeders, and thus maintain a viable population.
Appendix E[H] for Cooperative Breeders Let Hi be the number of dispersers reared in the interval (i, i ? 1). The matured brood from the initial maturation cycle stay in the nest, so H0 = 0. Thus, 1 X Hi : ð7Þ H¼ i¼1
Assume the single cooperative breeder who founded the nest succeeded in rearing the first brood, and let H0 be the total number of dispersers by conditioning to this case. Because the first matured brood stay in the nest, there exist two adults at time 1. Indeed, until the termination of the nest, two cooperative breeders always exist at the start of each maturation cycle (see Fig. 3). In the maturation cycle (1, 2), depending on the survival of the adults, there are three possibilities: (a) two adults survive and the matured brood disperse at time 2, (b) one of the two adults dies and
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the brood stay in the nest, and (c) both adults die and the nest is terminated. As the death of the adults is a Poisson process with intensity l2, we have 0 1 þ H2 þ H3 þ . . . with probability em2 ; 0 H ¼ @ 0 þ H2 þ H3 þ . . . with probability m2 em2 ; ð8Þ 0 with probability 1 em2 m2 em2 : After time 2, dispersal from a nest of two cooperative breeders will be repeated stochastically in the same manner, and H2 ? H3 ? … is stochastically equivalent to H 0 . Thus, taking the expectation on both sides of (8), we have E½H 0 ¼ el2 ð1 þ E½H 0 Þ þ l2 el2 E½H 0 :
ð9Þ
Rearranging this, we have E½Hjthe nest survives up to time 1 ¼ E½H 0 ¼
1
el2 : l2 el2
el2
ð10Þ
Because the death of the founding single breeder is exponentially distributed with the rate l1, we have E½H ¼ el1 E½Hjthe nest survives up to time1 el1 ¼ l : e 2 1 l2
ð11Þ
References Cant MA, Field J (2001) Helping effort and future fitness in cooperative animal societies. Proc R Soc Lond B Biol Sci 268:1959–1964. http://rspb.royalsocietypublishing.org/content/268/1479/1959. abstractN2 Field J (1996) Patterns of provisioning and iteroparity in a solitary halictine bee, Lasioglossum (evylaeus) fratellum (perez), with notes on. (e.) calceatum (scop.) andl. (e.) villosulum (k.). Insectes Soc 43:167–182 Field J (2005) The evolution of progressive provisioning. Behav Ecol 16:770–778. http://beheco. oxfordjournals.org/content/16/4/770.abstract Field J (2008) The ecology and evolution of helping hover wasps. In: Korb JHJ (ed) Ecology of social evolution, chap 4. Springer, Berlin, pp 83–105 Field J, Shreeves G, Sumner S, Casiraghi M (2000) Insurance-based advantage to helpers in a tropical hover wasp. Nature 404:869–871. doi:10.1038/35009097 Field J, Turner E, Fayle T, Foster WA (2007) Costs of egg-laying and ospring provisioning: multifaceted parental investment in a digger wasp. Proc R Soc B Biol Sci 274: 445–451. http://rspb.royals ocietypublishing.org/content/274/1608/445 Field J, Paxton R, Soro A, Craze P, Bridge C (2012) Body size, demography and foraging in a socially plastic sweat bee: a common garden experiment. Behav Ecol Sociobiol 66:743–756. doi:10.1007/ s00265-012-1322-7 Gadagkar R (1990) Evolution of eusociality: the advantage of assured fitness returns. Philos Trans Biol Sci 329:17–25. http://www.jstor.org/stable/76892 Gordon DM (2013) The rewards of restraint in the collective regulation of foraging by harvester ant colonies. Nature 498:91–93. doi:10.1038/nature12137 Hamilton W (1964) The genetical evolution of social behaviour. ii. J Theor Biol 7:17–52 Heinsohn R, Legge S (1999) The cost of helping. Trends Ecol Evol 14:53–57. doi: 10.1016/s01695347(98)01545-6. http://www.sciencedirect.com/science/article
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Kokko H, Johnstone RA (1999) Social queuing in animal societies: a dynamic model of reproductive skew. Proc R Soc Lond B Biol Sci 266:571–578. doi: 10.1098/rspb.1999.0674. Kokko H, Johnstone RA, Clutton-Brock TH (2001) The evolution of cooperative breeding through group augmentation. Proc R Soc Lond B Biol Sci 268:187–196. http://www.journals.royalsoc.ac.uk/ content/en6x5dgrht76ulhw Nonacs P, Liebert A, Starks P (2006) Transactional skew and assured fitness return models fail to predict patterns of cooperation in wasps. Am Nat 167:467–480 Queller D (1989) The evolution of eusociality: reproductive head starts of workers. PNAS 86:3224–3226 Queller D (1994) Extended parental care and the origin of eusociality. Proc R Soc Lond B Biol Sci 256:105–111 Queller D (1996) The origin and maintenance of eusociality: the advantage of extended parental care. Natural history and evolution of paper wasps. Oxford University Press, Oxford, pp 218–234 Shen SF, Kern Reeve H (2010) Reproductive skew theory unified: The general bordered tug-of-war model. J Theor Biol 263:1–12. http://www.sciencedirect.com/science/article/B6WMD-4XRYTBM3/2/15051acaa41548cf09231c01f66303a5 Shen SF, Kern Reeve H, Vehrencamp SL (2011) Parental care, cost of reproduction and reproductive skew: a general costly young model. J Theor Biol 284:24–31. http://www.sciencedirect.com/ science/article/pii/S0022519311002803 Shreeves G, Field J (2002) Group size and direct fitness in social queues. Am Nat 159:81–95 Shreeves G, Cant MA, Bolton A, Field J (2003) Insurance-based advantages for subordinate cofoundresses in a temperate paper wasp. Proc R Soc Lond B Biol Sci 270:1617–1622. http://rspb. royalsocietypublishing.org/content/270/1524/1617.abstract
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Reduction of Foraging Work and Cooperative Breeding Hiroshi Toyoizumi • Jeremy Field
Received: 9 July 2013 / Accepted: 4 March 2014 / Published online: 12 March 2014 Springer Science+Business Media Dordrecht 2014
Abstract Using simple stochastic models, we discuss how cooperative breeders, especially wasps and bees, can improve their productivity by reducing foraging work. In a harsh environment, where foraging is the main cause of mortality, such breeders achieve greater productivity by reducing their foraging effort below full capacity, and they may thrive by adopting cooperative breeding. This could prevent the population extinction of cooperative breeders under conditions where a population of lone breeders cannot be maintained. Keywords Cooperation Reduction of foraging work Division of work Cooperative breeding Survivorship insurance Foraging
1 Introduction For many animals with parental care responsibilities, foraging is a risky business (Cant and Field 2001 and the references therein; Field et al. 2007). Parents must leave the relative safety of their nests to seek food for their offspring. Some may be captured by predators, while others could encounter an accident. It is not just the parent’s own life that is at risk, because immature offspring might die without parental care (Field et al. 2000; Shreeves et al. 2003). Although a larger brood size increases the potential reward for parents, it increases the amount of foraging work necessary. Therefore, there is an increased risk that the entire brood will fail when adult carers die. In this paper, we discuss the level of foraging that results in the greatest productivity. H. Toyoizumi (&) Waseda University, Nishi-waseda 1-6-1, Shinjuku, Tokyo 169-8050, Japan e-mail: [email protected] J. Field School of Life Sciences, University of Sussex, Brighton, UK
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A variety of foraging animals breed cooperatively. Often, cooperation involves a hierarchy with a dominant breeder, who foregoes foraging and specializes in egglaying while subordinate helpers forage to feed the dominant’s brood. Various hypotheses have been proposed for the somewhat paradoxical evolution of helping behaviour, whereby a helper (at least temporarily) forfeits its own chance to reproduce and instead helps to rear another individual’s offspring. Helpers are nearly always rearing the offspring of a relative, meaning that copies of the helper’s genes are propagated indirectly (Hamilton 1964). However, as the relative’s offspring rarely carry as large a proportion of the helper’s genes as would the helper’s own offspring, natural selection should favour this behaviour only if helpers compensate by being more productive than they would be nesting alone (Queller 1996). In insects such as wasps and bees, there are various ways in which this could happen, some of which rely on the relatively short lifespan of adults compared with the long development time of their progressively fed immature offspring (Field 2005). The extended parental care (EPC) implicit in progressive feeding means that a mother often dies before her offspring mature (Queller 1994). Therefore, for a potential helper, staying in the natal nest and rearing the part-matured brood of a relative may be more productive than starting a new nest and rearing her own brood: the part-matured offspring are more likely to reach adulthood before the group as a whole fails [headstart hypothesis: (Queller 1989)]. A subtly different idea is that, if a helper dies young, any dependent offspring that she has only part-reared can be brought to adulthood by the other individuals still remaining in the group, whereas for a female nesting independently, an early death means total brood failure [assured fitness return: (Field et al. 2000; Gadagkar 1990; Shreeves et al. 2003)]. If a helper has the chance of inheriting dominant status, it may be worth waiting without immediate fitness return if the expected reproductive success as the dominant breeder is large enough to outweigh the chance of death while waiting in the queue [delayed fitness return: (Kokko and Johnstone 1999; Kokko et al. 2001; Shreeves and Field 2002)]. Further discussion of these hypotheses can be found in previous studies (Field 2008; Heinsohn and Legge 1999; Nonacs et al. 2006; Shen and Reeves 2010; Shen et al. 2011). Most of these previous studies aim to understand cooperative breeding from the evolutionary perspective of rational individual decision making. Here, we analyse it from the perspective of nest or population productivity and survival, i.e. as if group members have identical genetic interests. We test the effect on productivity of a general characteristic of cooperative breeding: reduction of foraging work (RFW) below the full capacity of the group, which is observed in various social insects such as sweat bees (Field 1996; Field et al. 2012) and harvester ants (Gordon 2013). In a social nest, the dominant typically concentrates on breeding, not risking her life by foraging away from the nest, while her helpers forage. Such reduced effort may increase the inclusive fitness of high-ranking individuals (Cant and Field 2001). This results in reduced foraging effort at the group level because the potential foraging ability of all group members (dominants and workers) is not fully utilized. We investigate whether this reduced foraging effort will result in an increase or a decrease in the total number of reproductive individuals dispersing from the group. It should be noted that only the total amount of work carried out by the group is considered to be important in our analysis, rather than how the work is divided among the group.
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In this paper, we use simple stochastic models to simulate the life history of cooperative breeders living in an aseasonal environment. We consider nests initiated by single foundresses, who continuously rear offspring until the termination of the nest, such as hover wasps (Stenogastrinae) (Field 2008). We model both lone breeders and cooperative breeders, and use the number of offspring dispersing from the nest as an estimate of productivity. From our model, we conclude that cooperative breeders achieve greater productivity if they do not work to their maximum potential, and instead reduce the total foraging risk below the level that lone breeders must accept. 2 Lone Breeders and Reduction of Foraging Work First, we analyse productivity in a population of lone breeders, i.e. where all nests are initiated by single foundresses and all offspring disperse so that the adult group size never exceeds one. Productivity is evaluated by the number of offspring H that disperse from the nest and potentially initiate new nests. To maintain the population, the expectation of H should be [1. Note that we neglect males, who are not involved in nesting in wasp and bee populations. We assume that a lone breeder rears successive batches of n broods in her nest (n = brood size, see Fig. 1). The immature broods must be fed continuously, and take one time unit to mature to adulthood (maturation cycle). This time scale is used throughout the paper. Because foraging is risky, if a lone breeder rears multiple broods simultaneously (n [ 1), her chance of death increases proportionally: we assume that the chance of death during foraging is random and proportional to the brood size n (Cant and Field 2001; Field et al. 2007). Together with the foragingindependent intrinsic death rate l0, the overall death rate of a lone breeder is l1 ¼ na þ l0 ;
ð1Þ
where a is the foraging risk factor, and the lone breeder’s lifetime is exponentially distributed with mean 1/l1. If a lone breeder dies before a brood matures to adulthood, the nest is terminated and the part-grown brood fails. When a brood matures, all disperse, and the lone breeder initiates a new batch of n brood. This maturation cycle is repeated until the lone breeder’s death at time s. Let M be the number of successful maturation cycles in [0, s). Because the chance of death for a lone breeder is exponentially distributed, M is geometrically distributed as PfM ¼ mg ¼ f1 el1 gfel1 gm :
ð2Þ
As n broods mature at each maturation time, the total number of individuals dispersed from the nest is nel1 : ð3Þ 1 el1 As shown in Fig. 2, E[H] is a convex function of n that attains its maximum at some intermediate brood size n0. Figure 3 shows the region in which a population of lone breeders is viable when they choose the optimal brood size n0 for various intrinsic death rates l0 and foraging risk factors a. E½H ¼ nE½M ¼
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Fig. 1 Maturation cycles and adult lifespan for lone breeders in our model. Thick arrows represent adult life time, and thin arrows represent the time required for brood maturation. Red arrows represent abandoned brood. a illustrates the case where the lone breeder rears only a single immature brood at a time (n = 1) and dies during the 6th maturation cycle: the 6th brood item therefore fails. b illustrates the case where the lone breeder rears batches of 3 broods simultaneously (n = 3) and dies during the 2nd maturation cycle. The second batch of 3 broods therefore fails. (Color figure online)
Fig. 2 The expected number of dispersed individuals E[H] for lone breeders caring for n broods, where the founder’s death rate is l1 = na ? 1/3 for various values of the foraging risk factor a
As the environment becomes harsher (larger a), lone breeders should decrease their brood size to mitigate the increased foraging risk (see Fig. 3). Sometimes, the environment is so harsh that lone breeders cannot afford to rear even a single brood. They then face extinction. Let us consider a brood size that can take the fractional number 1/2 (see Fig. 3b). In this case, lone breeders can be more productive and maintain a viable population in a harsh environment, such as when a [ 0.7.
3 Cooperative Breeders In some foraging insects, the expected adult lifespan is significantly shorter than the brood maturation time, mainly because of risky foraging work. In such a harsh environment, a population of lone breeders has no chance of escaping the threat of extinction, even if individuals rear only a single brood at a time, as seen in Sect. 2.
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Fig. 3 a Region in which a population of lone breeders is viable for various intrinsic death rates l0 and foraging risk factors a (blue regions). The region is divided into sub-regions according to the optimal integer number of broods n0 = 1, 2, 3, …. b Sub-region for n0 = 1/2 is added, assuming that lone breeders could rear 1/2 a brood (pink region). (Color figure online)
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Fig. 4 Dynamics of cooperative breeding. Each arrow represents the life history of cooperative breeders, and thick and thin arrows represent adult and brood lifetimes, respectively. A foundress (thick black arrow) starts her nest with a single brood. The 1st, 3rd, and 4th broods (blue arrows) stay in the natal nest and start cooperative breeding, whereas the 2nd and 5th broods (black arrows) find two adults in the nest and disperse. The last brood (red arrow) fails because of the termination of the nest. (Color figure online)
However, by reducing the total amount of foraging work in the nest, cooperative breeders are able to increase the chance of population survival. In order to analyse cooperative breeding in a minimum setting, we consider a nest with a single individual brood cared for by (at most) two adult cooperative breeders (results are qualitatively similar if we allow more than two adults). On reaching adulthood, a brood remains in the natal nest if there is currently only one adult. Once two adults exist in the nest, cooperative breeding starts and further matured offspring must all disperse until one of the two existing adults dies. At time 0, a cooperative breeder founds her new nest and initiates a single brood (see Fig. 4). A single cooperative breeder behaves exactly like a lone breeder with n = 1, and her overall death rate, as before, is l1 ¼ l0 þ a: ð4Þ If the single cooperative breeder succeeds in feeding her single brood for the unit maturation cycle, the matured brood stays in the nest and cooperates to rear future successive broods. When a single brood is taken care of by two cooperative breeders, the sum of their death rates due to risky foraging work will again be a. If intrinsic mortality is also included, the summed death rate with two cooperative breeders, l2, is obtained by l2 ¼ 2l0 þ a:
ð5Þ
In nature, the division of work in most cooperatively breeding wasps and bees is complete: one adult (dominant) reproduces while the other (helper) forages. Thus, the total foraging work is less than the full foraging capacity of two adults. However, for our purposes, only the total amount of work carried out by the group is important, rather than how the work is divided among the group. If one of the two adults dies, the brood is maintained by the remaining single adult until the next maturation time [if the helper dies, this corresponds to assured fitness returns in the sense of Gadagkar (1990)]. If both adults die before brood maturation, the current single brood is abandoned and the nest is terminated. This process repeats until time s, when the last adult in the nest dies. Note we do not assume that cooperative breeders forage more efficiently than lone breeders—the foraging risk factor a remains the same as in (4) and (5).
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Fig. 5 a Figure 3 redrawn to show that cooperative breeders have greater productivity than lone breeders (yellow region surrounded by a dashed line). b Region of lone breeders with a fractional brood n0 = 1/2 is added. Part of the advantage of breeding cooperatively is that only cooperative breeders can effectively rear a fractional number of broods (pink region). (Color figure online)
By careful observation of the stochastic dynamics of cooperative breeders (see Appendix), the expected number of dispersers is explicitly obtained by
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E½H ¼
el1 e l 2 1 l2
eðl0 þaÞ ¼ 2l þa : e 0 1 a 2l0
ð6Þ
Figure 5 displays the region in which cooperative breeders maintain a viable population (E[H] [ 1), also exceeding the productivity of lone breeders at the same time. Even in a harsh foraging environment, where lone breeders have too small a productivity (E[H] \ 1) for population survival, cooperative breeders can achieve sufficiently high productivity to avoid population extinction (E[H] [ 1). There are two reasons for this outcome. First, two adults nesting together can afford to take a greater summed risk than a lone breeder, because if one of them dies, the other can continue to rear some of the brood (the insurance effect). Second, two cooperative breeders nesting together can reduce risky foraging work more than is possible for a lone breeder: two cooperative breeders can rear just a single brood at a time, effectively rearing 1/2 a brood each—less than the minimum single brood for a lone breeder (the effect of RFW). Figure 5 shows the RFW effect: cooperative breeders have a viable population in a harsh environment, whereas only lone breeders with 1/2 a brood have a viable population.
4 Conclusion We have used a stochastic model to evaluate foraging risk and productivity for different breeding scenarios. This analysis clearly shows the advantage of a RFW as a whole. We also found that RFW may enable cooperative breeders to achieve a higher productivity than lone breeders, and thus maintain a viable population.
Appendix E[H] for Cooperative Breeders Let Hi be the number of dispersers reared in the interval (i, i ? 1). The matured brood from the initial maturation cycle stay in the nest, so H0 = 0. Thus, 1 X Hi : ð7Þ H¼ i¼1
Assume the single cooperative breeder who founded the nest succeeded in rearing the first brood, and let H0 be the total number of dispersers by conditioning to this case. Because the first matured brood stay in the nest, there exist two adults at time 1. Indeed, until the termination of the nest, two cooperative breeders always exist at the start of each maturation cycle (see Fig. 3). In the maturation cycle (1, 2), depending on the survival of the adults, there are three possibilities: (a) two adults survive and the matured brood disperse at time 2, (b) one of the two adults dies and
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the brood stay in the nest, and (c) both adults die and the nest is terminated. As the death of the adults is a Poisson process with intensity l2, we have 0 1 þ H2 þ H3 þ . . . with probability em2 ; 0 H ¼ @ 0 þ H2 þ H3 þ . . . with probability m2 em2 ; ð8Þ 0 with probability 1 em2 m2 em2 : After time 2, dispersal from a nest of two cooperative breeders will be repeated stochastically in the same manner, and H2 ? H3 ? … is stochastically equivalent to H 0 . Thus, taking the expectation on both sides of (8), we have E½H 0 ¼ el2 ð1 þ E½H 0 Þ þ l2 el2 E½H 0 :
ð9Þ
Rearranging this, we have E½Hjthe nest survives up to time 1 ¼ E½H 0 ¼
1
el2 : l2 el2
el2
ð10Þ
Because the death of the founding single breeder is exponentially distributed with the rate l1, we have E½H ¼ el1 E½Hjthe nest survives up to time1 el1 ¼ l : e 2 1 l2
ð11Þ
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