Mathematical Models And Data

For the things of this world cannot be made known without a knowledge of mathematics.

Argument is conclusive ... but... it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment.

10 In thinking about noise and stochasticity, one is forced to think about randomness and chance. It is an amusing philosophical aside to note that, in the quote reproduced in Section 1.1, Poincare defines chance in deterministic terms, by invoking what we now term sensitivity to initial conditions. From his point of view randomness (and hence noise) is deterministic chaos!

In this book we take an approach to May's hypothesis that differs from those based on the examination of individual historical time series of data for evidence of chaos. The complexity of natural systems, along with the inherent difficulties in confidently linking data from such systems with theory and models, points to the need for controlled laboratory experiments — experiments designed and analyzed with the specific intent of testing the predictions of nonlinear population theory.

May's hypothesis embodies a theoretical possibility or hypothesis. It is widely recognized in science that laboratory experiments are one of the best ways to test theory (although this is perhaps less recognized in ecology than it is in other scientific disciplines [132]). In the laboratory, one can carefully control environments and eliminate, or at least reduce, the effects of confounding elements, identify important and unimportant mechanisms, make accurate census counts and other measurements, replicate results, and manipulate parameters. Although laboratory microcosms are no substitute for field experiments, they are useful for testing or disproving basic ecological concepts, mechanisms, and hypotheses.

Another feature of our approach is that it does not focus on individual data sets and ask whether they possess one kind of dynamic characteristic or another, such as equilibrium dynamics, periodicity, or chaos. Instead we return to the bifurcation diagram, as a mathematical metaphor for May's hypothesis, with an intent to investigate the range of dynamic possibilities that a real biological population can display and whether these dynamics can include the dynamic bifurcations predicted by a simple mathematical model.

A bifurcation diagram is a summary of a mathematical theory's predictions as to how a population will respond to disturbances of a specific kind. Disturbances can generally be more easily invoked and studied in laboratory experiments. While such manipulated experiments might seem unduly remote from circumstances in the natural world, one need only reflect on the nonstatic natural world, in which populations and their physical and biological environments are continually subjected to perturbations, disturbances, and manipulations. Of special note, of course, are the deliberate and inadvertent disturbances caused by humans. What are the consequences of such disturbances for the dynamics of populations and their ecosystems? How do populations respond if survival and recruitment rates are changed? Such changes can be the result of any number of causes: changes in availability of food and habitat resources, changes in mortality rates, competition pressures, exposure to predation and diseases, genetic modifications, harvesting by humans, and so on. Nonlinear theory says that responses may be unexpected, nonintuitive, and complex— and, as ecologists in general recognize, the natural world is nonlinear [144],

The laboratory setup used in our studies is in fact less artificial than it might appear, in that it is rather similar to the natural environment of the biological organism we use. Beetles species of the genus Tribolium, or "flour beetles," as the insects in this genus are often called, have lived in containers of stored grain products produced by humans for literally thousands of years. Furthermore, the demographic manipulations we impress on the populations in our studies are not unlike those that would result from natural or human-induced causes — for example, a pest eradication program, a disease, or a genetic mutation.

An important role played by laboratory experiments is that of providing a detailed understanding of the dynamics of a particular species. As in other scientific disciplines, controlled experiments, in which one probes, manipulates, perturbs—even distorts — a system and observes how it responds, can lead to an understanding of how that system works and to insights into the mechanisms and causes that drive its dynamics. However, understanding the flour-beetle system is not the primary goal of the experimental projects described in this book. We use our laboratory system as a tool to assist in the development of modeling methodologies and mathematical and statistical techniques upon whose foundations we can study, document, and provide insights into the role of nonlinearityin population dynamics — particularly into the complexities that can result from nonlinearity.

There is a demanding prerequisite to testing a theoretical hypothesis about dynamic transitions and bifurcations, namely, the identification of adequate mathematical models. We require models that provide more than adequate statistical fits to data. We need biologically based models derived from mechanisms known to be important for the dynamics of the particular organism under consideration and in whose predictions we have a high degree of confidence. Unfortunately, models of this type are rare in ecology. The track record of relating mathematical models to ecological data is not good, particularly with respect to the formulation of testable hypotheses and predictions. In this regard, ecology differs from most other mature scientific disciplines in which verifiable quantitative predictions of models play a central role. This may be the single most important reason why a great many ecologists do not use models as serious tools in their work and are skeptical or disbelieve the insights of modeling exercises [1]. Besides the lack of prediction, shortcomings common to modeling endeavors include weaknesses in (or lack of full disclosure of) model structure, failure to incorporate biologically relevant mechanisms, poor model parameterization or calibration (including the use of overparameterized models), lack of model validation against independent data, the absence of a robustness or sensitivity analysis, and the failure to include stochasticity.

The first step in our investigation, then, will be to confront these difficulties and to build a convincing model of the biological system we use. Only with that preliminary goal accomplished will we be in a position to investigate, both analytically and experimentally, complex nonlinear phenomena predicted by the model (including a route-to-chaos).

A mathematical model is not sufficiently connected to population data unless it accounts for variability in the data. By containing a description of how data deviate from its predictions, a model can provide the means for parameter estimation, model evaluation, and generation of realistic predictions. One source of deviation is measurement error, which affects our estimation of the values of state variables. This kind of "noise" can be present even in a fully deterministic system. A different cause of variability in data is "process error." Process errors change the values of the state variables themselves because no deterministic model can account exactly for the dynamics of a biological population—many extrinsic and intrinsic processes and forces are inevitably left unaccounted for by any model. We describe these deviations probabilistically, and the resulting system is stochastic. Since measurement errors are negligible in our experimental studies, it is primarily process errors that account for "noise" in our data. Thus, in this book what we mean by "stochasticity" is deviation from deterministic model predictions due to process errors.

Ecologists distinguish different sources of stochasticity. Two fundamental types that have been delineated and widely discussed as important to biological populations are environmental stochasticity and demographic stochasticity [11,13,14,63,64,123,156,164]. These two sources of noise act in different ways to produce random variations in population numbers from census to census. Environmental noise involves the chance variation in population numbers arising from extrinsic sources that affect all (or at least many) members of the population. Demographic stochasticity, on the other hand, is the variability in population numbers caused by independent random contributions of births, deaths, and migrations of individual population members. In all populations both types of stochasticity are undoubtedly present.

Whatever their source, random variations in population numbers cause deviations from predictions of a deterministic model. To describe these variations requires the inclusion of a probabilistic term in the model. The mathematical descriptions of environmental stochasticity and of demographic stochasticity are different, however, as are the resulting stochastic versions of the model. We will have occasion, it turns out, to model both kinds of noise in the following chapters.

In addition to providing a quantitative connection between model and data, a validated stochastic model can provide stochastic predictions for the time evolution of population numbers and be used for simulation studies of a population's dynamics. As we will see, even in our controlled laboratory situation a full understanding of a population's dynamics requires a mixture of stochastic and deterministic elements. The addition of stochasticity to nonlinearity brings a new level of complexity to the dynamics. It provides random perturbations that continually stir the system, bringing into play far more than just the attractors of the deterministic "skeleton." Although the underlying deterministic attractors can exert their influence in the form of discernible temporal patterns, the effect of stochasticity is to prevent a system from remaining on an attractor. Stochasticity allows the system to visit locations in state space that are not on or near the attractor, including the locations on or near unstable invariant sets. This introduces transient dynamics that can produce observable patterns in data. For example, a random perturbation that places a population sufficiently near an unstable equilibrium can cause the population to linger near the equilibrium before it attempts a return to an attractor. Moreover, there can be regions in state space where orbits are actually attracted to an unstable equilibrium (the so-called "stable manifold" of the equilibrium). In this case, an unstable equilibrium is called a "saddle." A population randomly placed near the stable manifold is tugged toward the saddle equilibrium before it is repelled, resulting in a saddle "flyby." A similar phenomenon can occur with an unstable periodic cycle (a "saddle cycle"). In this way, the transient behavior due to stochasticity can produce distinctive temporal patterns in the data that are unrelated to attractors.

Thus, in time series data under the influence of nonlinear dynamics and stochasticity, one should expect to see a complicated dance of attractors, transients, and unstable entities. It is more fruitful, in attempting to explain patterns observed in data, to study the relative influences of these various components, rather than try to explain the data in terms of a specific type of deterministic attractor. Stochastic models provide the means by which to do this. By their formulation and application we can obtain useful predictions that combine deterministic and stochastic aspects.

With regard to complicated dynamics such as chaos, the blend of stochasticity with nonlinearity can create a particularly complex array of patterns that is challenging to sort out. Chaotic attractors generally exist in the presence of unstable invariant sets and sometimes in the presence of other attractors. Furthermore, within confidence intervals of parameter estimates there can be a variety of other types of attractors and unstable sets — even unstable chaotic sets—and hence a whole suite of different kinds of transient dynamics.

Throughout the following chapters we will see how a mix of stochastic and deterministic ingredients is needed to provide a complete explanation of data obtained from our experimental populations. Given that these issues arise in studies of "simple" biological systems, in controlled laboratory settings, it is no wonder that in a natural setting there are formidable difficulties in "finding chaos in nature." These difficulties include, but go beyond, the similarity of chaos to stochasticity, which May warned would make chaos difficult to observe in ecological data. Diagnostics calculated from time series data that are based on characteristics of attractors can be highly contaminated with the influences of other dynamic entities. Quantities averaged over deterministic attractors, such as Lyapunov exponents, are prime examples [51].11

It has become clear that questions such as "Is this biological system chaotic?" or "Is this data set chaotic?" are too narrow. Instead one must find ways to sort out the extent to which various deterministic forces can contribute to the dynamics of particular populations or ecosystems when embedded in specific types of stochasticity [45,57,194], The expectation is that many properties of nonlinear systems will influence the dynamics of an ecological system, including attractors, transients, unstable invariant sets, stable manifolds, bifurcations, and multiple attractors, as well as new patterns that emerge from a stochastic mix of these ingredients. From this broader point of view, one might discover that the role of chaos in ecology is more substantial than it would otherwise appear and, as a result, one might formulate different answers to questions such as "Is chaos found in natural populations?", "Is it common or rare?", and "Do populations evolve away from chaos?". However, it is a significant challenge to devise and apply methods that identify those ingredients that play significant roles in data patterns. Controlled laboratory and field experiments are ideally suited for such an endeavor.

The studies in the following chapters — besides addressing specific points such as May's hypothesis, dynamics bifurcations, and routes-to-chaos — serve to illustrate and document the issues just discussed. The first step in these endeavors is the construction of an adequate model, to which we turn our attention in the next chapter.

11 Researchers have recently made modifications to time series methods in an attempt to surmount these difficulties (although the short lengths of available data sets is a serious drawback) [144],

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