Trading space for time in population dynamics

Abstract

Classic models, such as the Lotka–Volterra or the Nicholson–Bailey, assume that we can understand the dynamics of populations without worrying about the fact that interactions between individuals are played out in space. These models were developed using the analogy to the ‘mass-action’ interactions between molecules in a volume of gas. An analogy that is without-a-doubt an attractive one, because the subsequent half-century of theoretical ecology stuck faithfully to that particular line of reasoning. The apparent take-home message appeared to be that population dynamics can be understood as a fairly low-dimensional system, and allowance need not be made for the spatial component to dynamics. The advent of modern computer power, however, has raised doubt about the validity of this assertion, because numerical simulations provide mounting evidence that space is indeed important. The direct implications are graver than we tend to acknowledge, because dynamical systems theory (see, for example, Taken's theorem for prediction in deterministic systems and the Cayley–Hamilton theorem for stochastic systems) says that the number of time lags involved in regulation should scale with the number of interacting populations. [Theoretically speaking, therefore, the regulation of a population embedded in a predator–prey metapopulation with ten subpopulations should involve studying ∼10–20 lags of regulation.] Acknowledging space would hence seem logically to imply abandoning any hope of having simple low-dimensional rules for population dynamics. For theorists dabbling in spatiotemporal dynamics, it is only human to have the left hand (‘space’) ignore what the right hand (‘time’) is doing.Keeling and co-workers 1xRe-interpreting space, time lags and functional responses in ecological models. Keeling, M.J et al. Science. 2000; 290: 1758–1761Crossref | PubMed | Scopus (74)See all References1 have now published what might promise to provide an elegant way for theoretical ecology to both have the cake and eat it. They study two different predator–prey metapopulation models where local dynamics follows Lotka–Volterra dynamics. The patches are linked through global dispersal of both the predator and the prey. The dynamics of this spatial system differ from its nonspatial analogue in being more stable. The stability arises because the limited movement (only a fraction of the individuals move) induces spatial association between the two species. Using moment techniques – a mathematical modeling technique for stochastic systems that involves writing down a sequence of equations for the mean, the variance, the skewness, etc.– the authors show that simple equations can still be used to understand the aggregate dynamics. The trick is to include a correction factor for the interspecific covariance in the equation for the mean.Similar results have been obtained from previous applications of moment equations 2xSpatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Bolker, B and Pacala, S.W. Am. Nat. 1999; 153: 572–602Crossref | Scopus (315)See all References2. Keeling et al., however, go one exciting step further by developing an explicit approximation for the covariance, as a delayed density-dependent function of the mean. This demonstrates that the complex spatiotemporal interactions in a parasitoid–host metapopulation can, if care is taken, be modeled using low-dimensional rules. In the example developed by Keeling et al., this results from trading in the slightly simplistic spatial dynamics (global dispersal within the metapopulation for all individuals that disperse) for one additional density-dependent lag in regulation. My personal enthusiasm for this work is, first, for the results themselves. I am also excited about the new theoretical road that is opening. I envisage a theory, incorporating localized dispersal, that trades off spatial interactions for temporal lags in a fashion orders-of-magnitude more economic than that of Taken, Cayley and Hamilton, and closes our much denied gap between spatial and temporal theories of population dynamics.