- Email: [email protected]
Reply from G.D. Ruxton
CORRESPONDENCE Chaos and scale In a recent issue of TREE,Ruxtonl points out Csilling et a/.‘~ results* concerning the dynamic of a theoretical metapopulation. They observed that a set of linked chaotic populations exhibits neither chaotic temporal variations nor spatial structuring. In other studies, the opposite result was observeds-6. Ruxton argued that this contrast could be explained by the fact that the timescales of reproduction and dispersal are decoupled in Csilling et a/.‘~ model. I think the absence of chaos in Csilling et a/.‘s study could be explained by a particular feature of their model. The modelled metapopulation occupies a landscape composed of a finite number of patches. All local populations have the same chaotic dynamic. Dispersal occurs when the population density exceeds a fixed threshold according to a stepping-stone rule. A fixed fraction of the population emigrates to neighbouring patches. The dispersal process is reiterated until all populations are below the threshold, then reproduction starts again. Since a given population may receive dispersers from other patches, numerous dispersal steps may occur in what Csilling et al. call a ‘migration avalanche’. The individuals dispersing out of the landscape are lost, so that all excess individuals are eliminated. Such a dispersal process is an efficient densitydependent regulating mechanism. It should be interpreted as an inhibition of reproduction in local populations as long as at least one population density exceeds the threshold. Such a mechanism does not seem biologically realistic, since it assumes that reproduction in a population is not only directly influenced by its own density but also by densities in the other populations. May and his collaborators7p8 pointed out a long time ago that chaos occurs when there is a discrepancy between the timescales of reproduction and its density-dependent inhibition. In Csilling et al.‘s model, the local densitydependent mechanisms are outweighed by the dispersal mechanism. As rightly emphasized by Ruxton, the relative timescales of reproduction and dispersal are critical for the dynamic of a metapopulation when local dynamics are taken into account. Two mechanisms of dispersal can be distinguished as extremes along a continuum -the stepping-stone mechanism whereby the dispersers can reach only the neighbouring patches, and the ‘common pool’ of dispersers where the distance between patches is unimportant for the dispersers. This general scheme is further complicated by habitat selection when habitat quality varies among patches, and by the triggering mechanism of dispersal. The ‘common pool’ mechanism can have a strong stabilizing influence both on population and metapopulation dynamics even when local dynamics are strongly chaoticg,lo. This is achieved providing the dispersal rate is sufficiently low. When dispersal rate increases, it ‘homogenizes’ the whole system (as seasonality can doll) so that chaos occurs. Stepping-stone dispersal contributes critically to metapopulation persistenceY however, its stabilizing influence on global dynamics is yet to be demonstrated under realistic biological assumptionsl*. Both TREE
vol.
10,
no.
8 August
1995
dispersal mechanisms could be important for the same species at different spatial or temporal scales.
Emmanuel
Paradis
Institut des Sciences de l’Evolution, CC 64, UniversitC Montpellier II, Place Eugkne Bataillon, F-34095 Montpellier cedex 5, France
References Ruxton, G.D.(1995) Trends Ecol. Evol. 10, 141-142
Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50,1083-1092 Comins, H.N., Hassell, M.P. and May, R.M. (1992)
J. Anim. Ecol. 61, 735-748
Hassell, M.P., Comins, H.N. and May, R.M. (1994)
Nature 370, 209-292
Pascual, M. (1993) Proc. R. Sot. London Ser. B
where individuals moved from a given patch once the density reached a threshold, and continued to move until reaching a patch with a population below the threshold, is equivalent to reducing the strength of the density dependence (nonlinearity) in the model. We do concur with the conclusion that such an effect may be biologically important and is worthy of further study. The sophistication of our models of spatial and temporal interactions is reaching the level where quantitative tests are appropriate. Much greater knowledge of movement dynamics, in particular the effect of density on movement, is likely to be a key factor.
Alan Hastings Kevin Higgins Division of Environmental Studies, Institute for Theoretical Dynamics, and Center for Population Biology, University of California, Davis, CA 95616, USA
References 1 Ruxton,G.D. (1995) Trends Ecol. Evol. 10, 141-142
251,1-7
Sol& R.V., Valls, J. and Bascompte, J. (1992) Phys. Lett. A 166,123-128
May, R.M. (1973) Eco/ogy54,315-325 May, R.M., Conway,G.R., Hassell, M.P. and Southwood, T.R.E.(1974) J. Anim. Ecof.43,
Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50. 1083-1092 3 Hastings, A. and Higgins, K. (1994) Science 263, 2
1133-1136
747-770 9
Ruxton, G.D. (1994) Proc. R. Sot. London Ser. 8 256,189-193
10 Stone, L. (1993) Nature 365,617-620 11 Grenfell, B.T., Balker, B.M. and Kleczkowski,A. (1995)
Reply from G.D. Ruxton
Proc. R. Sot. London Ser. B 259,97-103
I2 Mackey, M. and Milton, J. (1995) Physica D 81, 1-17
The discussion of the relationship between spatial and temporal scales and chaos by Ruxtonl emphasizes the importance and complexity of unravelling the role of migration in determining the likelihood of complex dynamics. Ruxton focuses on a recent study by Csilling et a/.*, which presented a simulation model in which movement appeared to eliminate the likelihood of chaotic dynamics. As one explanation for why this finding differs from earlier studies using coupled map lattices, Ruxton suggests that the possibility of movement beyond adjacent cells within a single time step included in the Csilling et al. model is the crucial stabilizing factor. Our recent study3 demonstrated unequivocally that a discrete time model with movement beyond adjacent cells within a single time step not only allowed for the persistence of complex chaotic behavior, but allowed even more complex behavior to emerge than was found in models that did not incorporate spatial dynamics. As we noteds, the numerical analysis of our model corresponded to a study of a coupled map lattice model with long-range coupling. We demonstrated that such a model, using either Ricker equations or logistic equations as the underlying dynamics, led to extraordinarily long, transient dynamics that are likely to be of great ecological importance. As to why the study of Csilling et a/.* produced different results, we suggest a much simpler answer. The form of dispersal they described,
Many previous modelling studies of linked populations assume that reproduction occurs simultaneously in all populations, followed by a dispersal phase. Typically, a fixed fraction of each population moves to each nearest-neighbour population or joins a common pool, which is then distributed evenly between all populations in the system. The model of Hastings and Higgins1 can be considered as intermediate between these two extremes. In this model, the probability of an individual dispersing from one population to another is related to the distance between the populations by a Gaussian distribution. The model of Csilling et a/.* is different again. It assumes that if a population size is above a threshold value then excess individuals move to nearest-neighbour sites. This occurs simultaneously in all the populations. Further rounds of dispersal occur until all populations are below threshold. Hence, like the model of Hastings and Higgins, there is a finite probability of an individual moving to a distant population, which depends on the distance between two populations. However, unlike Hastings and Higgins’ model, the probability is also affected by the sizes of the intermediate populations (along all possible paths) at the time when the individual encounters these populations. Both models make different assumptions about the underlying biology. As I emphasized in my TREEarticles, the time has come for ecologists to consider which (if any) of the various modelling assumptions are relevant to given types of natural systems. This issue aside, Hastings
335
PERSPECTIVES and Higgins’ warning-that the transient behaviour of these models may have more practical significance than the final behaviour should be of great importance to future studies. I agree with Paradis that a model that assumes that reproduction in one population can be inhibited by the action of a far-distant population seems biologically implausible. However, the model of Csilling et al. does not necessarily rest on such an assumption. To me, a more plausible alternative assumption underlying the model of Csilling et al. is that the timescale of dispersal is much quicker than that of reproduction, so that all rounds of dispersal are always comfortably accomplished before the next round of reproduction is triggered.
I feel that Csilling et al. have made an important step in adopting the assumption that reproduction and dispersal occur on different timescales. However, they (and Hastings and Higgins) follow convention in assuming that reproduction and migration occur in separate time compartments, all reproduction being completed before dispersal occurs and vice versa. While this is realistic for some species, in others reproduction and migration events will not occur in such an ordered way. There has been at least one published attempt to model the latter cased, although the model formulation used has since been criticized as unbiological5. However, future studies should shed light on how the assumption of temporal ordering affects model predictions6.
G.D. Ruxton
In biological markets, two classes of traders exchange commodities to their mutual benefit. Characteristics of markets are: competition within trader classes by contest or outbidding; preference for partners offering the highest value; and conflicts over the exchange value of commodities. BIological markets are currently studied under at least three different headings: sexual selection, intraspeclflc cooperation and interspecific mutualism. The time is ripe for the development of game theoretic models that describe the common core of biologlcal markets and integrate existing knowledge from the separate fields.
Ronald No& and Peter Hammerstein are at the Max-PlanckInstitut fiir Verhaltensphysiologie, Seewiesen, 82319 Starnberg, Germany.
corpionfly females accept or reject a nuptial gift of a certain size depending on the number of males that they are likely to encounter’. Caterpillars of lycaenid butterflies adjust the amount of nectar offered in reaction to the number of ants protecting them*. Male pied kingfishers bring food to unrelated nestlings, probably to enhance their chance to mate with the mother in the future, and increase the amount in reaction to an increase in the number of helpers3. The above examples stem from sexual selection, interspecific mutualism, and
S
33 6
o 1995, Elsevier
Science
Ltd
References 1 Hastings, 2 3 4 5 6
A. and Higgins, K. (1992) Science 263, 1133-1136 Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50,1083-1092 Ruxton, G.D. (1995) Trends Ecol. Evol. 10, 141-142 Bascompte, J. and Sol& R.V. (1994) 1. Anim. Ecol. 63,256-264 Hassell, M.P., Miramontes, O., Rohani, P. and May, R.M. J. Anim. Ecol. (in press) Ruxton, G.D. J. Anim. Ecol. (in press)
Box 1. Market jargon
Biological markets Ronald No4 Peter Hammerstein
Biomathematics &Statistics Scotland, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Edinburgh, UK EH9 352
intraspecific cooperation, respectively. These fields are traditionally studied separately, in spite of some obvious analogies: (1) individuals exchange commodities to their mutual benefit; (2) the exchange value of commodities is a source of conflict; (3) behavioural mechanisms such as partner-searching, partner choice, and contest among competitors determine the composition of trading pairs or groups. We call situations with these characteristics ‘biological markets’ because of the analogy with human markets. As far as we are aware, the market mechanism was not recognized as a common evolutionary mechanism of sexual selection, cooperation and mutualism until last yea+. This does not mean that market mechanisms have not been implicitly or explicitly recognized in each discipline separately. The existence of mating markets is assumed throughout the literature on sexual selection. A ‘market effect’ was described for intraspecific cooperation5. Selection through partner choice has also been postulated” and shown7 for obligate mutualisms between pollinators and plants. The idea of market selection (Box 1) applies to traits, such as providing nuptial gifts, helping unrelated young, and nectar production, that would not evolve under natural selection in the absence of op portunities for the formation of mutually beneficial partnerships. Market selection is best recognized by its effect on easily quantified chosen traits, but is expected to determine the degree of choosiness as well. Trading may take place on the basis of an honest signal that is correlated with the access to a commodity, instead of being based on the commodity itself.
The following terms have been inspired either by sexual selection jargon or by idiom commonly used in connection with human markets. Mallcet selectlon: selection of traits that maximize fitness in biological markets, such as the ability to compete with members of the same class, the ability to attract trading partners, the ability to sample alternative partners efficiently. Commodity: a benefit that members of one trader class can offer to members of another trader class. Alternatives are: ‘service’ for an intangible commodity (e.g. an alarm call) and ‘reward’ for a tangible commodity (e.g. nectar). Trader class: all traders that offer the same kind of commodity. Members of the same class may belong to different species, sexes, etc. Any entity that can ‘choose’ a strategy independently qualifies as a trader. Some traders have to compete with abiotic sources of commodities. Trader-class ratio: the relative number of traders belonging to each class. In parallel with sexual selection, an operational class ratio can be defined. Partner choice: the preference for a partner based on the value of the commodity offered. The value can be relative or absolute, depen& ing on the sampling strategy of the choosing individual. Outbidding competition: (used in contrast to contest competition) a trader ‘outbids’ another trader belonging to his own class, if he secures the favours of a member of the other class by making a better offer.
Market models Here, we limit ourselves to two-sided markets with unrelated traders, in which the members of each class compete for the favours of the members of the other class by outbidding each other. The best strategy for a trader depends on the behaviour of the members of both his own and the opposite class. Game theory is especially suited to model this two-layered frequency dependence. The important concepts in evolutionary game theory are the evolutionarily stable strategy (ESS) for intraspecific interactions and the coevolutionarily stable strategy TREE
vol.
IO.
no.
8 August
I995
vol.
10,
no.
8 August
1995
dispersal mechanisms could be important for the same species at different spatial or temporal scales.
Emmanuel
Paradis
Institut des Sciences de l’Evolution, CC 64, UniversitC Montpellier II, Place Eugkne Bataillon, F-34095 Montpellier cedex 5, France
References Ruxton, G.D.(1995) Trends Ecol. Evol. 10, 141-142
Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50,1083-1092 Comins, H.N., Hassell, M.P. and May, R.M. (1992)
J. Anim. Ecol. 61, 735-748
Hassell, M.P., Comins, H.N. and May, R.M. (1994)
Nature 370, 209-292
Pascual, M. (1993) Proc. R. Sot. London Ser. B
where individuals moved from a given patch once the density reached a threshold, and continued to move until reaching a patch with a population below the threshold, is equivalent to reducing the strength of the density dependence (nonlinearity) in the model. We do concur with the conclusion that such an effect may be biologically important and is worthy of further study. The sophistication of our models of spatial and temporal interactions is reaching the level where quantitative tests are appropriate. Much greater knowledge of movement dynamics, in particular the effect of density on movement, is likely to be a key factor.
Alan Hastings Kevin Higgins Division of Environmental Studies, Institute for Theoretical Dynamics, and Center for Population Biology, University of California, Davis, CA 95616, USA
References 1 Ruxton,G.D. (1995) Trends Ecol. Evol. 10, 141-142
251,1-7
Sol& R.V., Valls, J. and Bascompte, J. (1992) Phys. Lett. A 166,123-128
May, R.M. (1973) Eco/ogy54,315-325 May, R.M., Conway,G.R., Hassell, M.P. and Southwood, T.R.E.(1974) J. Anim. Ecof.43,
Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50. 1083-1092 3 Hastings, A. and Higgins, K. (1994) Science 263, 2
1133-1136
747-770 9
Ruxton, G.D. (1994) Proc. R. Sot. London Ser. 8 256,189-193
10 Stone, L. (1993) Nature 365,617-620 11 Grenfell, B.T., Balker, B.M. and Kleczkowski,A. (1995)
Reply from G.D. Ruxton
Proc. R. Sot. London Ser. B 259,97-103
I2 Mackey, M. and Milton, J. (1995) Physica D 81, 1-17
The discussion of the relationship between spatial and temporal scales and chaos by Ruxtonl emphasizes the importance and complexity of unravelling the role of migration in determining the likelihood of complex dynamics. Ruxton focuses on a recent study by Csilling et a/.*, which presented a simulation model in which movement appeared to eliminate the likelihood of chaotic dynamics. As one explanation for why this finding differs from earlier studies using coupled map lattices, Ruxton suggests that the possibility of movement beyond adjacent cells within a single time step included in the Csilling et al. model is the crucial stabilizing factor. Our recent study3 demonstrated unequivocally that a discrete time model with movement beyond adjacent cells within a single time step not only allowed for the persistence of complex chaotic behavior, but allowed even more complex behavior to emerge than was found in models that did not incorporate spatial dynamics. As we noteds, the numerical analysis of our model corresponded to a study of a coupled map lattice model with long-range coupling. We demonstrated that such a model, using either Ricker equations or logistic equations as the underlying dynamics, led to extraordinarily long, transient dynamics that are likely to be of great ecological importance. As to why the study of Csilling et a/.* produced different results, we suggest a much simpler answer. The form of dispersal they described,
Many previous modelling studies of linked populations assume that reproduction occurs simultaneously in all populations, followed by a dispersal phase. Typically, a fixed fraction of each population moves to each nearest-neighbour population or joins a common pool, which is then distributed evenly between all populations in the system. The model of Hastings and Higgins1 can be considered as intermediate between these two extremes. In this model, the probability of an individual dispersing from one population to another is related to the distance between the populations by a Gaussian distribution. The model of Csilling et a/.* is different again. It assumes that if a population size is above a threshold value then excess individuals move to nearest-neighbour sites. This occurs simultaneously in all the populations. Further rounds of dispersal occur until all populations are below threshold. Hence, like the model of Hastings and Higgins, there is a finite probability of an individual moving to a distant population, which depends on the distance between two populations. However, unlike Hastings and Higgins’ model, the probability is also affected by the sizes of the intermediate populations (along all possible paths) at the time when the individual encounters these populations. Both models make different assumptions about the underlying biology. As I emphasized in my TREEarticles, the time has come for ecologists to consider which (if any) of the various modelling assumptions are relevant to given types of natural systems. This issue aside, Hastings
335
PERSPECTIVES and Higgins’ warning-that the transient behaviour of these models may have more practical significance than the final behaviour should be of great importance to future studies. I agree with Paradis that a model that assumes that reproduction in one population can be inhibited by the action of a far-distant population seems biologically implausible. However, the model of Csilling et al. does not necessarily rest on such an assumption. To me, a more plausible alternative assumption underlying the model of Csilling et al. is that the timescale of dispersal is much quicker than that of reproduction, so that all rounds of dispersal are always comfortably accomplished before the next round of reproduction is triggered.
I feel that Csilling et al. have made an important step in adopting the assumption that reproduction and dispersal occur on different timescales. However, they (and Hastings and Higgins) follow convention in assuming that reproduction and migration occur in separate time compartments, all reproduction being completed before dispersal occurs and vice versa. While this is realistic for some species, in others reproduction and migration events will not occur in such an ordered way. There has been at least one published attempt to model the latter cased, although the model formulation used has since been criticized as unbiological5. However, future studies should shed light on how the assumption of temporal ordering affects model predictions6.
G.D. Ruxton
In biological markets, two classes of traders exchange commodities to their mutual benefit. Characteristics of markets are: competition within trader classes by contest or outbidding; preference for partners offering the highest value; and conflicts over the exchange value of commodities. BIological markets are currently studied under at least three different headings: sexual selection, intraspeclflc cooperation and interspecific mutualism. The time is ripe for the development of game theoretic models that describe the common core of biologlcal markets and integrate existing knowledge from the separate fields.
Ronald No& and Peter Hammerstein are at the Max-PlanckInstitut fiir Verhaltensphysiologie, Seewiesen, 82319 Starnberg, Germany.
corpionfly females accept or reject a nuptial gift of a certain size depending on the number of males that they are likely to encounter’. Caterpillars of lycaenid butterflies adjust the amount of nectar offered in reaction to the number of ants protecting them*. Male pied kingfishers bring food to unrelated nestlings, probably to enhance their chance to mate with the mother in the future, and increase the amount in reaction to an increase in the number of helpers3. The above examples stem from sexual selection, interspecific mutualism, and
S
33 6
o 1995, Elsevier
Science
Ltd
References 1 Hastings, 2 3 4 5 6
A. and Higgins, K. (1992) Science 263, 1133-1136 Csilling, A., Janosi, I.M., Pasztor, G. and Scheuring, I. (1994) Phys. Rev. E50,1083-1092 Ruxton, G.D. (1995) Trends Ecol. Evol. 10, 141-142 Bascompte, J. and Sol& R.V. (1994) 1. Anim. Ecol. 63,256-264 Hassell, M.P., Miramontes, O., Rohani, P. and May, R.M. J. Anim. Ecol. (in press) Ruxton, G.D. J. Anim. Ecol. (in press)
Box 1. Market jargon
Biological markets Ronald No4 Peter Hammerstein
Biomathematics &Statistics Scotland, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Edinburgh, UK EH9 352
intraspecific cooperation, respectively. These fields are traditionally studied separately, in spite of some obvious analogies: (1) individuals exchange commodities to their mutual benefit; (2) the exchange value of commodities is a source of conflict; (3) behavioural mechanisms such as partner-searching, partner choice, and contest among competitors determine the composition of trading pairs or groups. We call situations with these characteristics ‘biological markets’ because of the analogy with human markets. As far as we are aware, the market mechanism was not recognized as a common evolutionary mechanism of sexual selection, cooperation and mutualism until last yea+. This does not mean that market mechanisms have not been implicitly or explicitly recognized in each discipline separately. The existence of mating markets is assumed throughout the literature on sexual selection. A ‘market effect’ was described for intraspecific cooperation5. Selection through partner choice has also been postulated” and shown7 for obligate mutualisms between pollinators and plants. The idea of market selection (Box 1) applies to traits, such as providing nuptial gifts, helping unrelated young, and nectar production, that would not evolve under natural selection in the absence of op portunities for the formation of mutually beneficial partnerships. Market selection is best recognized by its effect on easily quantified chosen traits, but is expected to determine the degree of choosiness as well. Trading may take place on the basis of an honest signal that is correlated with the access to a commodity, instead of being based on the commodity itself.
The following terms have been inspired either by sexual selection jargon or by idiom commonly used in connection with human markets. Mallcet selectlon: selection of traits that maximize fitness in biological markets, such as the ability to compete with members of the same class, the ability to attract trading partners, the ability to sample alternative partners efficiently. Commodity: a benefit that members of one trader class can offer to members of another trader class. Alternatives are: ‘service’ for an intangible commodity (e.g. an alarm call) and ‘reward’ for a tangible commodity (e.g. nectar). Trader class: all traders that offer the same kind of commodity. Members of the same class may belong to different species, sexes, etc. Any entity that can ‘choose’ a strategy independently qualifies as a trader. Some traders have to compete with abiotic sources of commodities. Trader-class ratio: the relative number of traders belonging to each class. In parallel with sexual selection, an operational class ratio can be defined. Partner choice: the preference for a partner based on the value of the commodity offered. The value can be relative or absolute, depen& ing on the sampling strategy of the choosing individual. Outbidding competition: (used in contrast to contest competition) a trader ‘outbids’ another trader belonging to his own class, if he secures the favours of a member of the other class by making a better offer.
Market models Here, we limit ourselves to two-sided markets with unrelated traders, in which the members of each class compete for the favours of the members of the other class by outbidding each other. The best strategy for a trader depends on the behaviour of the members of both his own and the opposite class. Game theory is especially suited to model this two-layered frequency dependence. The important concepts in evolutionary game theory are the evolutionarily stable strategy (ESS) for intraspecific interactions and the coevolutionarily stable strategy TREE
vol.
IO.
no.
8 August
I995