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Resource selection functions: taking space seriously?
CORRESPONDENCE Sexual selection: separating genes from imprinting Owens et al.1 have recently reviewed the importance of imprinting as a mechanism of sexual selection. They paid less attention to how the evolutionary consequences of imprinted and genetically determined aesthetic criteria may differ and interact. When offspring are reared by their genetic parents, the evolutionary consequences of both a genetic and a cultural basis for the transmission of aesthetic partner choice criteria would coincide2. This is because parental fitness is directly dependent on the criteria of partner choice passed onto the offspring whether genetically or through imprinting. There are, however, three types of situation where the evolutionary consequences of culturally and genetically determined criteria of partner choice may diverge: when offspring are reared by individuals other than by their genetic parents. This may happen through (1) extra-pair paternity caused by the promiscuous behaviour of females, which results in paternal care directed towards young fathered by other males, (2) brood parasitism and (3) cooperative breeding. In all these cases, if partner-choice is culturally determined, young may learn from their social parents aesthetic criteria different from the ones learnt by their genetic parents. Whether these differences are detrimental or beneficial to the young and its genetic and social parents may vary with socio-environmental conditions. It is thus difficult to generate predictions about the direction and magnitude of the fitness consequences of learning partner choice from tutors other than their parents. However, it is conceivable that when partner choice is imprinted, the potential discrepancy between the aesthetic criteria of genetic and social parents may affect the evolution of the reproductive strategies I have mentioned (1–3). For example, in a sperm competition scenario, the possibility that foster fathers can imprint daughters of other males with their own sexual preference might influence the tradeoff between parental care and confidence of paternity, and change our current interpretation of ‘misdirected’ paternal care3. Owens et al.1 refer to a crossfostering experiment in which lambs reared by goats developed a preference to copulate with female goats rather than sheep4. It is arguable whether interspecifc parental care is frequent across the animal kingdom. In fact, where interspecific parenting does occur, little evidence suggests that behaviours are determined by foster parents5,6. However, among intraspecific brood parasites, partner choice may be learnt. In cooperative-breeding systems, the time when young are most likely to be imprinted may influence the behaviour of helpers; for example, depending on the fitness consequences derived by helpers from sexually imprinting the helped brood with their aesthetic criteria. Therefore, the ability to condition future partner choice of not-directly related young may provide a further arena where the conflict between cuckolders and cuckolded, parasites and hosts, or helpers and helped is resolved, and a functional explanation for evolutionarily counterintuitive behaviours like adoption7. Finally, when aesthetic criteria for partner choice are imprinted, genes may influence cultural transmission8, determining how signals are TREE vol. 14, no. 10 October 1999
perceived9 and how information is memorized10, thereby creating potential for a complex and possibly conflicting interaction between the genomes of the young and its social parents.
Tommaso Pizzari Evolutionary Ecology Group, Dept of Animal and Plant Sciences, University of Sheffield, Sheffield, UK S10 2TN ([email protected]) References 1 Owens, I.P.F. et al. (1999) Trends Ecol. Evol. 14, 131–132 2 Boyd, R. and Richerson, P.J. (1985) Culture and the Evolutionary Process, University of Chicago Press 3 Wright, J. (1998) in Sperm Competition and Sexual Selection (Birkhead, T.R. and Møller, A.P., eds), pp. 117–145, Academic Press 4 Kendrick, K.M. et al. (1998) Nature 395, 229–230 5 Brooke, M. de L. and Davies, N.B. (1991) Ethology 89, 154–166 6 Teuschl, Y. et al. (1998) Anim. Behav. 56, 1425–1433 7 Avital, E. and Jablonka, E. (1994) Anim. Behav. 48, 1195–1199 8 Dugatkin, L.A. (1998) Behav. Ecol. 9, 323–327 9 Endler, J.A. and Basolo, A. (1998) Trends Ecol. Evol. 13, 415–420 10 Clayton, N.S. et al. (1998) Neuropharmacology 37, 441–453
imprinting a sexual preference that corresponds with its parents’ sexual ornaments. It is also debatable how common it is for offspring to inherit genetically determined preferences that correspond with their parents’ sexual ornaments. Ornaments and preferences do sometimes have a substantial heritable component4, but this is not enough to ensure that all mating pairs have corresponding preferences and ornaments. Are mating preferences sufficiently accurate, and sufficiently emancipated from intrasexual competition, to ensure such perfect correspondence? Is the influence of nongenetic factors on the development of sexual ornaments and preferences as trivial as usually supposed5? The genetic basis of mate recognition systems is contentious, to say the least. Given these outstanding problems, it seems premature to assume that the distinction between cultural and genetic determination of mating preferences is unimportant.
Ian Owens Dept of Zoology and Entomology, University of Queensland, Brisbane, Australia ([email protected])
Candy Rowe Dept of Psychology, University of Newcastle, Newcastle, UK NE1 7RU ([email protected])
Adrian Thomas
Reply from I.P.F. Owens, C. Rowe and A.L.R. Thomas We are wary of Pizzari’s assumption that ‘when offspring are reared by their genetic parents, the evolutionary consequences of both a genetic and a cultural basis for the transmission of aesthetic partner choice criteria would coincide’1. For the evolutionary consequences of cultural and genetic transmission of mating preferences to coincide, both mechanisms must result in offspring acquiring preferences that correspond with their parents’ sexual ornaments. In cases of cultural transmission this must occur through a form of sexual imprinting in which offspring of both sexes simply imprint on their parents’ sexual ornaments. In cases of genetic determination this must occur through, first, offspring inheriting the genetic preferences of their parents, and, second, those genetically determined preferences corresponding with the sexual ornaments of the parents. Under these conditions, offspring end up with mating preferences that correspond with their parents’ sexual signals, irrespective of the mechanism by which they acquire their sexual preferences. Such conditions might, however, be uncommon, for reasons described below. As we stressed in our earlier note2, recent empirical work shows that offspring do not simply imprint on their parents’ sexual ornaments. Rather, sexual imprinting is almost always ‘asymmetrical’ and ‘sex-biased’3. Offspring prefer mates that have even more extreme sexual signals than those of their parents, and sons imprint more on their mothers than daughters do on their fathers. It is unclear, therefore, how common it is for an offspring to acquire through
Dept of Zoology, University of Oxford, Oxford, UK OX1 3PS ([email protected]) References 1 Pizzari, T. (1999) Trends Ecol. Evol. 14, 399 2 Owens, I.P.F., Rowe, C. and Thomas, A.L.R. (1999) Trends Ecol. Evol. 14, 131–132 3 ten Cate, C. and Vos, D.R. (1999) Adv. Stud. Behav. 28, 1–31 4 Andersson, M. (1994) Sexual Selection, Princeton University Press 5 Griffith, S.C., Owens, I.P.F. and Burke, T. (1999) Nature 400, 358–360
Resource selection functions: taking space seriously? In a recent TREE review of resource selection functions, Boyce and McDonald1 outline important advances in the modelling of population density or presence/absence in the spatial domain as a function of habitat coverage. This methodological approach is common to other areas of ecology dealing with spatial data. Independence of the response variable across sites is the first major assumption of such models. As the authors emphasize, the second major assumption is that the important resources (explanatory variables) are included in the model2. Unfortunately, because we usually do not know what these are, we may often be forced into doing some hypothesis testing. This is where violation of the first assumption can introduce systematic errors
0169-5347/99/$ – see front matter © 1999 Elsevier Science Ltd. All rights reserved.
399
CORRESPONDENCE
Jack J. Lennon Centre for Biodiversity and Conservation, School of Biology, University of Leeds, Leeds, Yorkshire, UK LS2 9JT ([email protected]) References 1 Boyce, M.S. and McDonald, L.L. (1999) Trends Ecol. Evol. 14, 268–272 2 Manly, B.F.J., McDonald, L.L. and Thomas, D.L. (1993) Resource Selection by Animals: Statistical Design and Analysis for Field Studies, Chapman & Hall
400
3 Mladenoff, D.J. et al. (1995) Conserv. Biol. 9, 279–294 4 Akcakaya, H.R. and Atwood, J.L. (1997) Conserv. Biol. 11, 422–434 5 Osborne, P.E. and Tigar, P.J. (1992) J. Appl. Ecol. 29, 55–62 6 Buckland, S.T. and Elston, D.A. (1993) J. Appl. Ecol. 30, 478–495 7 Clifford, P., Richardson, S. and Hemon, D. (1989) Biometrics 45, 123–134 8 Lennon, J.J. Ecography (in press) 9 Lennon, J.J. and Turner, J.R.G. (1995) J. Anim. Ecol. 64, 370–392 10 Cerioli, A. (1997) Biometrics 53, 619–628 11 Haining, R. (1990) Spatial Data Analysis in the Social and Environmental Sciences, Cambridge University Press
Synchronicity in population systems: cause and consequence mixed In his letter on two recent TREE articles1,2 on synchronous temporal and spatial dynamics of populations, Jansen3 correctly identifies two causes of synchrony to be the Moran effect and dispersal of individuals. The Moran effect4–7 is a global random disturbance affecting populations sharing a common density-dependent structure that will bring them into synchrony. The Moran effect can itself have a spatial structure8. It is composed of local effects, affecting only the focal population, and of distance-dependent spatially autocorrelated effects. At the limit, the latter will be perfectly global and correlated with disturbances. Dispersal among spatially separate populations will also lead to increased synchrony independent of the Moran effect8. The degree of distance dependence in the dispersal will affect the degree of synchrony among populations8. It is relatively straightforward to show that population synchrony can be generated in several ways8–10 and that perfect synchrony (‘phaselocking’) is a special outcome9, not a cause. In the simplest case, one can assume that the Moran effect is composed of local and global disturbances, and that dispersal serves to couple population subunits together. The two components, noise and dispersal, affecting spatial population dynamics can be set on and off in different combinations7,8. We have used a setting of 25 population subunits randomly located in a coordinate space9. The local population renewal followed the Moran–Ricker second-order nonlinear process9,10. Without any effort, such a renewal process, when simulated in spatially explicit space, will yield rich dynamics. In such a system there can be ‘phase locking’ periods for a substantially long time (Fig. 1a). But this is not all: there can be periods when ‘phase-locked’ populations drift out of phase (Fig. 1b); drastic changes in amplitude may emerge; and even disappearance and reappearance of the local cyclic dynamics can be witnessed (Fig. 1c). The question of interest – which we share with Jansen – is: what is needed to achieve temporal dynamics where a pair (or more) of populations fluctuates initially in phase, drifts away from phase, and returns back into phase? We have
0169-5347/99/$ – see front matter © 1999 Elsevier Science Ltd. All rights reserved.
also asked whether cyclic dynamics can be lost altogether9. With dispersal and disturbance combined in four different ways, we can draw the following conclusions7–10. Dispersal alone is capable of retaining all the features in our list: phase locking, drifting out of phase, regaining of synchrony, loss of cyclic dynamics and its reappearance. Adding Moran noise would change some of the fine details in the dynamics, but not the major elements10. We shall summarize by saying that to observe temporal phase locking in dynamics of populations, dispersal alone, or noise alone, are the sufficient elements. When the list of desired features is extended to include loss and gain of cyclic dynamics9, dispersal among population subunits is called for. In this connection, it is of interest to note that phase locking and drifting out of phase are indeed observed in the Canadian lynx dynamics9 and in the Fenoscandian voles11,12. We also note with great interest that the vole dynamics have lost the cyclic feature in most parts of northern Fennoscandia during the past decade11,13. We shall correct Jansen’s inaccuracy by saying that ‘phase locking’ is not a cause of synchronicity in population systems, but a result. The possible causes behind phase locking are dispersal, or – in more general terms – interaction among local units, Moran noise or their combination.
Trends in Ecology & Evolution
into our understanding of the relative importance of resources. In many cases circumstances dictate that we cannot apply a sampling strategy2 to include locations sufficiently separate to make the response variable independent between sites. Moreover, a great deal of data on distribution and abundance are already available that clearly are not independent between sites. For example, several of the studies the authors cite3–5 are likely to lack this independence. This has several consequences. When, as is usually the case, classic ‘spaceless’ multivariate methods6 are used to build a model from a set of potential explanatory variables, if there is any spatial autocorrelation present in the response variable, then the spatially smoother (more autocorrelated) explanatory variables are detected as ‘significant’ much more frequently7,8. The magnitudes of the coefficients of these variables are inflated8: any attempt to read biological meaning into such a model is problematical at best. However, if the purpose is purely spatial interpolation of the response variable, then the explanatory variables can serve as surrogates for the truly important factors9. When extrapolation is involved, either outside the spatial arena of the sampled sites, outside the explanatory variable envelope, or in time, then the problem of a lack of causal connection is, of course, more serious. But, in either case, not knowing if such an explanatory variable has any real biological meaning seems rather unsatisfactory to say the least. Contrary to what the authors say, it is not autocorrelation of resource (explanatory) variables but that of the response variable that is the main problem: if the response variable is not autocorrelated then spatial structure in the explanatory variables is unimportant7,8. They correctly note that spatial autocorrelation can make comparing alternative models difficult. However, the bias in favour of spatially smoother explanatory factors is an additional feature that is not just an inconvenience but a source of confusion throughout much of spatial ecology. We need to take spatial structure more seriously if the link between landscape and habitat ecology and population biology is not to be in several important respects illusory. A more widespread awareness of the consequences of spatial structure for statistical inference would help, as would the routine application of currently available spatially explicit statistical methods where possible7,8,10,11. But perhaps most importantly, we need the urgent development of sufficiently powerful and flexible statistical methods (e.g. generalized models with spatial errors) to match the current expansion in volume and quality of GIS (geographic information system) manipulated data.
Fig. 1. An example of temporal dynamics of a pair of populations selected from among 25 subpopulations in an explicit space obeying Moran–Ricker dynamics with 9–10 year periodicity (for details of the model, see Ref. 9). The populations are linked with dispersing individuals (10% of the current population size) and are affected with local and global noise. Note that at periods (a) the two populations are phase-locked, (b) they drift away from the perfect synchrony and (c) tend to gain it again. The amplitude of the deterministic fluctuations can vary, even to the extent that the periodic fluctuations appear to wane away.
TREE vol. 14, no. 10 October 1999
perceived9 and how information is memorized10, thereby creating potential for a complex and possibly conflicting interaction between the genomes of the young and its social parents.
Tommaso Pizzari Evolutionary Ecology Group, Dept of Animal and Plant Sciences, University of Sheffield, Sheffield, UK S10 2TN ([email protected]) References 1 Owens, I.P.F. et al. (1999) Trends Ecol. Evol. 14, 131–132 2 Boyd, R. and Richerson, P.J. (1985) Culture and the Evolutionary Process, University of Chicago Press 3 Wright, J. (1998) in Sperm Competition and Sexual Selection (Birkhead, T.R. and Møller, A.P., eds), pp. 117–145, Academic Press 4 Kendrick, K.M. et al. (1998) Nature 395, 229–230 5 Brooke, M. de L. and Davies, N.B. (1991) Ethology 89, 154–166 6 Teuschl, Y. et al. (1998) Anim. Behav. 56, 1425–1433 7 Avital, E. and Jablonka, E. (1994) Anim. Behav. 48, 1195–1199 8 Dugatkin, L.A. (1998) Behav. Ecol. 9, 323–327 9 Endler, J.A. and Basolo, A. (1998) Trends Ecol. Evol. 13, 415–420 10 Clayton, N.S. et al. (1998) Neuropharmacology 37, 441–453
imprinting a sexual preference that corresponds with its parents’ sexual ornaments. It is also debatable how common it is for offspring to inherit genetically determined preferences that correspond with their parents’ sexual ornaments. Ornaments and preferences do sometimes have a substantial heritable component4, but this is not enough to ensure that all mating pairs have corresponding preferences and ornaments. Are mating preferences sufficiently accurate, and sufficiently emancipated from intrasexual competition, to ensure such perfect correspondence? Is the influence of nongenetic factors on the development of sexual ornaments and preferences as trivial as usually supposed5? The genetic basis of mate recognition systems is contentious, to say the least. Given these outstanding problems, it seems premature to assume that the distinction between cultural and genetic determination of mating preferences is unimportant.
Ian Owens Dept of Zoology and Entomology, University of Queensland, Brisbane, Australia ([email protected])
Candy Rowe Dept of Psychology, University of Newcastle, Newcastle, UK NE1 7RU ([email protected])
Adrian Thomas
Reply from I.P.F. Owens, C. Rowe and A.L.R. Thomas We are wary of Pizzari’s assumption that ‘when offspring are reared by their genetic parents, the evolutionary consequences of both a genetic and a cultural basis for the transmission of aesthetic partner choice criteria would coincide’1. For the evolutionary consequences of cultural and genetic transmission of mating preferences to coincide, both mechanisms must result in offspring acquiring preferences that correspond with their parents’ sexual ornaments. In cases of cultural transmission this must occur through a form of sexual imprinting in which offspring of both sexes simply imprint on their parents’ sexual ornaments. In cases of genetic determination this must occur through, first, offspring inheriting the genetic preferences of their parents, and, second, those genetically determined preferences corresponding with the sexual ornaments of the parents. Under these conditions, offspring end up with mating preferences that correspond with their parents’ sexual signals, irrespective of the mechanism by which they acquire their sexual preferences. Such conditions might, however, be uncommon, for reasons described below. As we stressed in our earlier note2, recent empirical work shows that offspring do not simply imprint on their parents’ sexual ornaments. Rather, sexual imprinting is almost always ‘asymmetrical’ and ‘sex-biased’3. Offspring prefer mates that have even more extreme sexual signals than those of their parents, and sons imprint more on their mothers than daughters do on their fathers. It is unclear, therefore, how common it is for an offspring to acquire through
Dept of Zoology, University of Oxford, Oxford, UK OX1 3PS ([email protected]) References 1 Pizzari, T. (1999) Trends Ecol. Evol. 14, 399 2 Owens, I.P.F., Rowe, C. and Thomas, A.L.R. (1999) Trends Ecol. Evol. 14, 131–132 3 ten Cate, C. and Vos, D.R. (1999) Adv. Stud. Behav. 28, 1–31 4 Andersson, M. (1994) Sexual Selection, Princeton University Press 5 Griffith, S.C., Owens, I.P.F. and Burke, T. (1999) Nature 400, 358–360
Resource selection functions: taking space seriously? In a recent TREE review of resource selection functions, Boyce and McDonald1 outline important advances in the modelling of population density or presence/absence in the spatial domain as a function of habitat coverage. This methodological approach is common to other areas of ecology dealing with spatial data. Independence of the response variable across sites is the first major assumption of such models. As the authors emphasize, the second major assumption is that the important resources (explanatory variables) are included in the model2. Unfortunately, because we usually do not know what these are, we may often be forced into doing some hypothesis testing. This is where violation of the first assumption can introduce systematic errors
0169-5347/99/$ – see front matter © 1999 Elsevier Science Ltd. All rights reserved.
399
CORRESPONDENCE
Jack J. Lennon Centre for Biodiversity and Conservation, School of Biology, University of Leeds, Leeds, Yorkshire, UK LS2 9JT ([email protected]) References 1 Boyce, M.S. and McDonald, L.L. (1999) Trends Ecol. Evol. 14, 268–272 2 Manly, B.F.J., McDonald, L.L. and Thomas, D.L. (1993) Resource Selection by Animals: Statistical Design and Analysis for Field Studies, Chapman & Hall
400
3 Mladenoff, D.J. et al. (1995) Conserv. Biol. 9, 279–294 4 Akcakaya, H.R. and Atwood, J.L. (1997) Conserv. Biol. 11, 422–434 5 Osborne, P.E. and Tigar, P.J. (1992) J. Appl. Ecol. 29, 55–62 6 Buckland, S.T. and Elston, D.A. (1993) J. Appl. Ecol. 30, 478–495 7 Clifford, P., Richardson, S. and Hemon, D. (1989) Biometrics 45, 123–134 8 Lennon, J.J. Ecography (in press) 9 Lennon, J.J. and Turner, J.R.G. (1995) J. Anim. Ecol. 64, 370–392 10 Cerioli, A. (1997) Biometrics 53, 619–628 11 Haining, R. (1990) Spatial Data Analysis in the Social and Environmental Sciences, Cambridge University Press
Synchronicity in population systems: cause and consequence mixed In his letter on two recent TREE articles1,2 on synchronous temporal and spatial dynamics of populations, Jansen3 correctly identifies two causes of synchrony to be the Moran effect and dispersal of individuals. The Moran effect4–7 is a global random disturbance affecting populations sharing a common density-dependent structure that will bring them into synchrony. The Moran effect can itself have a spatial structure8. It is composed of local effects, affecting only the focal population, and of distance-dependent spatially autocorrelated effects. At the limit, the latter will be perfectly global and correlated with disturbances. Dispersal among spatially separate populations will also lead to increased synchrony independent of the Moran effect8. The degree of distance dependence in the dispersal will affect the degree of synchrony among populations8. It is relatively straightforward to show that population synchrony can be generated in several ways8–10 and that perfect synchrony (‘phaselocking’) is a special outcome9, not a cause. In the simplest case, one can assume that the Moran effect is composed of local and global disturbances, and that dispersal serves to couple population subunits together. The two components, noise and dispersal, affecting spatial population dynamics can be set on and off in different combinations7,8. We have used a setting of 25 population subunits randomly located in a coordinate space9. The local population renewal followed the Moran–Ricker second-order nonlinear process9,10. Without any effort, such a renewal process, when simulated in spatially explicit space, will yield rich dynamics. In such a system there can be ‘phase locking’ periods for a substantially long time (Fig. 1a). But this is not all: there can be periods when ‘phase-locked’ populations drift out of phase (Fig. 1b); drastic changes in amplitude may emerge; and even disappearance and reappearance of the local cyclic dynamics can be witnessed (Fig. 1c). The question of interest – which we share with Jansen – is: what is needed to achieve temporal dynamics where a pair (or more) of populations fluctuates initially in phase, drifts away from phase, and returns back into phase? We have
0169-5347/99/$ – see front matter © 1999 Elsevier Science Ltd. All rights reserved.
also asked whether cyclic dynamics can be lost altogether9. With dispersal and disturbance combined in four different ways, we can draw the following conclusions7–10. Dispersal alone is capable of retaining all the features in our list: phase locking, drifting out of phase, regaining of synchrony, loss of cyclic dynamics and its reappearance. Adding Moran noise would change some of the fine details in the dynamics, but not the major elements10. We shall summarize by saying that to observe temporal phase locking in dynamics of populations, dispersal alone, or noise alone, are the sufficient elements. When the list of desired features is extended to include loss and gain of cyclic dynamics9, dispersal among population subunits is called for. In this connection, it is of interest to note that phase locking and drifting out of phase are indeed observed in the Canadian lynx dynamics9 and in the Fenoscandian voles11,12. We also note with great interest that the vole dynamics have lost the cyclic feature in most parts of northern Fennoscandia during the past decade11,13. We shall correct Jansen’s inaccuracy by saying that ‘phase locking’ is not a cause of synchronicity in population systems, but a result. The possible causes behind phase locking are dispersal, or – in more general terms – interaction among local units, Moran noise or their combination.
Trends in Ecology & Evolution
into our understanding of the relative importance of resources. In many cases circumstances dictate that we cannot apply a sampling strategy2 to include locations sufficiently separate to make the response variable independent between sites. Moreover, a great deal of data on distribution and abundance are already available that clearly are not independent between sites. For example, several of the studies the authors cite3–5 are likely to lack this independence. This has several consequences. When, as is usually the case, classic ‘spaceless’ multivariate methods6 are used to build a model from a set of potential explanatory variables, if there is any spatial autocorrelation present in the response variable, then the spatially smoother (more autocorrelated) explanatory variables are detected as ‘significant’ much more frequently7,8. The magnitudes of the coefficients of these variables are inflated8: any attempt to read biological meaning into such a model is problematical at best. However, if the purpose is purely spatial interpolation of the response variable, then the explanatory variables can serve as surrogates for the truly important factors9. When extrapolation is involved, either outside the spatial arena of the sampled sites, outside the explanatory variable envelope, or in time, then the problem of a lack of causal connection is, of course, more serious. But, in either case, not knowing if such an explanatory variable has any real biological meaning seems rather unsatisfactory to say the least. Contrary to what the authors say, it is not autocorrelation of resource (explanatory) variables but that of the response variable that is the main problem: if the response variable is not autocorrelated then spatial structure in the explanatory variables is unimportant7,8. They correctly note that spatial autocorrelation can make comparing alternative models difficult. However, the bias in favour of spatially smoother explanatory factors is an additional feature that is not just an inconvenience but a source of confusion throughout much of spatial ecology. We need to take spatial structure more seriously if the link between landscape and habitat ecology and population biology is not to be in several important respects illusory. A more widespread awareness of the consequences of spatial structure for statistical inference would help, as would the routine application of currently available spatially explicit statistical methods where possible7,8,10,11. But perhaps most importantly, we need the urgent development of sufficiently powerful and flexible statistical methods (e.g. generalized models with spatial errors) to match the current expansion in volume and quality of GIS (geographic information system) manipulated data.
Fig. 1. An example of temporal dynamics of a pair of populations selected from among 25 subpopulations in an explicit space obeying Moran–Ricker dynamics with 9–10 year periodicity (for details of the model, see Ref. 9). The populations are linked with dispersing individuals (10% of the current population size) and are affected with local and global noise. Note that at periods (a) the two populations are phase-locked, (b) they drift away from the perfect synchrony and (c) tend to gain it again. The amplitude of the deterministic fluctuations can vary, even to the extent that the periodic fluctuations appear to wane away.
TREE vol. 14, no. 10 October 1999