This symposium was presented at the 1996 Annual Meeting of the Ecological Society of America, held in Providence, Rhode Island.
DENSITY DEPENDENCE FOR THE 90's: NONLINEARITY IN MODELS AND DATA
Organized by: Hal Caswell, Biology Department, Woods Hole Oceanographic Institution, Woods Hole, MA, 02543. (508) 289-2751. firstname.lastname@example.org
SUNDAY 11 AUGUST, 1996, 1:00 p.m.-5:00 p.m.
1:00 Caswell Opening Remarks
1:15 Hastings Chaotic transients and their role in understanding natural systems.
1:45 Neubert and Caswell Riddled basins of attraction: new extremes in indeterminacy.
2:15 Dennis et al. Nonlinear population models, statistical methods, and biological experiments.2:45 Briggs et al. Nonlinearity and stage-structure in an experimental system: the Indian meal moth and its granulosis virus.
3:30 Ellner Combining mechanistic and statistical models to study population dynamics.
4:00 Little and Pascual Detecting nonlinear dynamics in spatio-temporal systems: examples from ecological models.
4:30 Sugihara and Milicich Nonlinear tools for ecologists.
1. Hastings, Alan. University of California, Davis, CA 95616, USA. CHAOTIC TRANSIENTS AND THEIR ROLE IN UNDERSTANDING NATURAL SYSTEMS.
Density dependence and nonlinearity can lead to transient dynamics as well as to chaotic dynamics. Within the context of one and two species models, how the addition of the spatial component makes likely the presence of transient dynamics will be discussed. Transient dynamics which may change suddenly with no external change in the system will be one focus. The other major points will be the importance of the role played by transient dynamics for our understanding of natural systems, such as interactions between predator and prey, and how recognizing the importance of transients will change our approaches for analyzing models.
The chaotic dynamics exhibited by many nonlinear population models suggests that quantitative predictability of species interactions over long time scales may be limited by sensitive dependence on initial conditions. Still, model trajectories asymptotically approach some attractor and one can usually predict the qualitative population dynamics. If there are multiple attractors, this ability to predict the ultimate qualitative dynamics is only slightly impaired -- unless one or more of the basins of attraction are riddled. In that case, every initial condition in the basin of one attractor has initial conditions arbitrarily near by which converge to a different attractor. Thus, even predicting basic qualitative outcomes of species interactions in simple models (such as coexistence versus extinction) can be practically impossible. We will numerically demonstrate the existence of a riddled basin in at least one model and discuss the implications of such extreme indeterminacy for ecology.
Our interdisciplinary project group has been conducting joint theoretical and experimental studies to test the hypothesis that population fluctuations can be explained and predicted by nonlinear dynamic models. Specifically, we constructed a nonlinear, stage-structured model describing the life history of the flour beetle Tribolium. We developed statistical methods for connecting the model to time series data. We designed and conducted laboratory experiments in which flour beetle demographic parameters were experimentally manipulated. Experimental results to date have strongly confirmed the dynamic behaviors predicted by the nonlinear models, including: saddle nodes, stable equilibria, stable two-cycles, stable-three cycles, invariant loops, and chaos. This rigorous verification of predicted shifts in dynamic behavior provides a convincing example of the effectiveness of nonlinear mathematical modeling in population ecology.
In single-species long-term laboratory experiments, populations of the stored product moth, Plodia interpunctella, exhibited dramatic cycles in density with a period slightly longer than the moth's generation time. Moth populations in the presence of a granulosis virus also cycled, but with a slightly longer period. We used stage-structured models first to investigate the mechanisms by which intraspecific competition leads to generation cycles in the single-species populations, and second to investigate three hypothsized mechanisms for the increase in the cycle period in the presence of the virus: (1) the dynamical interation with the lethal effect of the virus, (2) sub-lethal effects of the virus, and (3) evolution of moth life history traits in the virus-infected populations.
Models of population dynamics are often used for three distinct purposes: prediction and management, characterizing the dynamics (e.g. noise vs. chaos), and identifying underlying mechanisms. In most cases the models are either purely phenomenological time-series models (e.g. nonlinear autoregression), or mechanistic models with a minimal set of parameters having specific biological interpretations (e.g. the LPA model for Tribolium). However our true state of knowledge is usually somewhere between these extremes. I propose that a reasonable modeling strategy then is to develop a "semi-mechanistic" model, which incorporates reliably known mechanisms and processes, but retains statistical flexibility on less well known aspects. Comparing phenomenological, mechanistic, and semi-mechanistic models for epidemic dynamics data, a semi-mechanistic model is shown to provide a better description of the dynamics as measured by forecasting accuracy. The fitted model can then be used to characterize the dynamics, showing in this case (a) a mix of "noise" and "nonlinearity" with both components too large to ignore, and (b) oscillations between local (in state space) chaos and stability. This case study also illustrates how a suite of models can be compared objectively to identify the amount and kind of mechanistic information that should be hard-wired into a model to maximize accuracy and reliability.
In ecology, techniques for nonlinear data analysis have been primarily tested with temporal models. The analysis of spatio-temporal data introduces new challenges, the possible high-dimesionality of attractors and the interaction of spatial sampling with dynamics. We investigate this interaction with two spatial ecological models that exhibit chaos and with sampling regimes motivated by, but not limited to, aquatic environments. The sampling regimes caricature two common limitations of time series in spatial systems. First, most samples are spatial averages of some sort. Second, long-term sampling is most easily accomplished in a fixed frame of reference. For externally advected populations, this effectively means that temporally successive samples are not taken from the same location with respect to spatial population patterns. We examine the effects of these two sampling limitations on the performance of several recently developed methods for nonlinear data analysis. Results indicate that the ability to identify underlying nonlinear dynamics quickly degrades when the sampling location changes in time relative to population patterns. On the other hand, the techniques are more robust in the face of spatial averaging.
We examine issues concerning detection of nonlinearities and possible chaos, particularly with regard to stochastic chaos. We show how recent attempsts to measure meaningful Lyapunov exponents for ecological data are fundamentally flawed, and that when observational noise is convolved with process noise, computing Lyapunove exponents for the real system will be difficult. Although identifying chaos in real data is a controversial subject, there is much to be gained in ecology by adopting the nonlinear perspective. As a specific example, we shall discuss a study involving an apparently noisy spawner-recruitment relationship in coral reef fish (pomacentrids). For this system, where linear methods are shown to yield very little, nonlinear time series tools (S-maps, residual-delay maps, and multivariate embeddings) can reveal the hidden mechanisms involved. In particular, we shall show how in a multivariate context, forcing by physical factors can interact with the biology in a highly nonlinear manner to account for most of the population variation observed in the larval phase. The model is predictive, and the finding that the larval phase can be highly nonlinear is significant in that it helps to explain a classical density-dependence problem in fisheries: how some spawner-recruitment relationships may be obscured, even with perfect measurements.
Alan R. Johnson
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