Dispersal and transient dynamics in metapopulations - IEB

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202 F. A. Labra, N. A. Lagos and P. A. Marquet(2000) point out that supertransients required a particularcombination of k, b and r, to emerge but did not explicitlytest for the effect of dispersal and also restricted theiranalysis to a normal dispersal kernel. Our analysis showsthat, contrary to Ruxton & DoebeliÔs (1996) findings,supertransients can emerge for high values of the dispersalparameter r, depending on the size of the system (Figs 3and 4). However, these supertransients may be highlysensitive to initial conditions (Fig. 5), strengthening ourconclusion that they are not the rule in simple ecologicalmodels.Our results also point out the importance of boundaryconditions and system size for the emergence of transientdynamics. It may be expected that loss of individualsthrough dissipative boundaries would result in a lowering ofthe local growth rates and hence in the likelihood ofobserving supertransients. For a small system size, however,supertransients were more common precisely under thisscenario. Thus, loss of individuals through dispersal may stillallow supertransients to emerge. However, system size,boundary conditions and initial conditions interact incomplex ways such that for a large system size (M ¼ 50)transient dynamics tend to be less affected by initialconditions under periodic boundaries, while the reverse istrue when the system is smaller (M ¼ 10, F. Labra,unpublished results). In a recent paper (Hastings 2001), itis reported that supertransients occur more frequently whenthe initial condition includes empty patches. The inclusionof empty sites in our simulations agree with this result.However, the observed increase in transients only occursunder dissipative conditions, and only for supertransientsbut not for mesotransients. To assess the relative importanceof each of these factors is beyond the scope of thepresent paper and deserves further investigation. However,this complex dependency strengthens our point; theemergence of transient dynamic behaviour in simpleecological models is not a common phenomenon butoccurs for a restricted set of conditions. Saravia et al. (2000),using the same model we used, reached a similar conclusionafter relaxing the assumption of global synchrony inreproduction and dispersal. This implies that the set ofconditions that lead to transient dynamics are likely to beeven more restricted than we report here. Although for thesake of comparability we have followed previous authors instudying a one-dimensional chain of interacting populations(e.g. Hastings & Higgins 1994; Ruxton & Doebeli 1996;Saravia et al. 2000) it is likely that system dimensionality willalso have an effect on the emergence of transient behaviour.However, further investigations on two-dimensional systemsare necessary.Most previous studies have not reported an exhaustiveexploration of the parameters that may result in transientdynamics. We found that under both dispersal kernels,supertransient dynamics occur in very low frequency in spiteof strong local nonlinearity. From this perspective, andconsidering that its emergence also depends, as in our case,on the initial conditions used in the simulations (see alsoRuxton & Doebeli 1996) and the value of the parameters k,b and r, it can be argued that supertransients, contrary toHastings & Higgins (1994) and Hastings (1998, 2001), mightnot be a common feature of spatial models, let alone realecological systems and especially so, if chaotic dynamics arenot common in nature (see Rai & Schaffer 2001; below).The range of dispersal we have explored here ranges fromlocal coupling, when r is very low, to nearly global couplingwhen r is very high and periodic boundaries are imposed onthe system. In agreement with Hastings (1998), who used asystem of size M ¼ 10, we found that supertransients wereharder to find when dispersal conditions tended to besimilar to global coupling (large r-values). However, this istrue only for a small system size (M ¼ 10) (compare Figs 3and 4) and not all small and intermediate values of r resultin supertransients, and of those that do, several are sensitiveto initial conditions.It should be noted, however, that we observed supertransientsmuch more frequently than Saravia et al. (2000)did. This discrepancy arises because their study encompassesparameter combinations that span regions of stable,cyclic and chaotic dynamics. However, supertransients ariseonly in the latter regime and for strong nonlinearity(Kaneko 1989, 1990; Kaneko & Tsuda 2000). Simulationscarried out with a different set of parameters (a ¼ 0.5,b ¼ 30 and k ¼ 2), which produce chaotic dynamics, failedto produce either of the two complex behaviours understudy when simulations were carried out for a large systemsize (M ¼ 50). On the other hand, for a second parametersetting (M ¼ 50, a ¼ 0.5, b ¼ 15 and k ¼ 7), the occurrenceof supertransients increased under both kernels to 8.2and 19.5% for normal dispersal (periodic and dissipativeboundary conditions) while for Laplace kernel they occurredin 19.5 and 6.6% (periodic and dissipative boundaryconditions, respectively). If supertransients occur only forfully developed spatiotemporal chaos, then the questionshould be how common are they in this regime. As wediscussed above, they are not very common.Mesotransients, on the other hand, were seen in lowfrequency, and emerged over a narrower range of dispersalparameter values, especially for the large system. For thetwo additional parameter settings we explored, mesotransientswere completely absent in the first setting, while forthe second parameter setting they were found in
Transient dynamics in metapopulations 203intermittent chaos (Doebeli 1994; Cavalieri & Koçak 1995;Gavrilets & Hastings 1995; Strogatz 1997) and spatiotemporalintermittency (Kaneko 1985, 1989). We are not awareof any formal attempt to ascertain if mesotransient dynamicsare in fact intermittent chaos or spatiotemporal intermittency.If it turns out to be the same to either of them, thenits prevalence and relevance in spatial ecological modelsmight be determined more clearly.In conclusion, we show that supertransient and mesotransientdynamics are not likely to be either common orwidespread in ecological systems being heavily affected bydispersal intensity, boundary conditions, system size andinitial conditions.ACKNOWLEDGEMENTSF.L. and N.L. acknowledge a doctoral scholarship providedby Conicyt. F.L. would like to thank I. Labra, I. Rodriguezand C. Huerta for their support and encouragement. P.M.acknowledges the financial support of grants FONDECYT1990144 and FONDAP-FONDECYT 1501–0001. J. Velasco-Hernández,provided cogent criticisms, and suggestionsto an early draft of this manuscript. We speciallythank Michael Hochberg, Mercedes Pascual and fiveanonymous reviewers for their constructive criticisms andfor making us to realize that supertransients, far frombeing a theoretical curiosity, can also be a real lifeexperience.REFERENCESBascompte, J. & Solé, R. (1998a). Spatiotemporal patterns in nature.Trends Ecol. Evol., 13, 173–174.Bascompte, J. & Solé, R. (1998b). Modeling Spatiotemporal Dynamics inEcology. Springer Verlag, Berlin.Bellows, T.S. (1981). The descriptive properties of some modelsfor density dependence. J. Anim. Ecol., 50, 139–156.Blasius, B., Huppert, A. & Stone, L. (1999). Complex dynamics andphase synchronization in spatially extended ecological systems.Nature, 399, 354–359.Cavalieri, L.F. & Koçak, H. (1995). Intermittent transition betweenorder and chaos in an insect pest population. J. Theor. 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Springer Verlag, Berlin,pp. 1–26.Ó2003 Blackwell Publishing Ltd/CNRS
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