The studies contained in this book were motivated primarily by Robert May's suggestion that some of the complexity observed in the dynamics of biological populations might be attributable to simple laws. Our approach first required the investigation of a more fundamental question. Can "simple" (low-dimensional) mathematical models describe and predict the dynamics of real biological populations with quantitative accuracy sufficient to permit an inquiry into complex dynamics? In dealing with this question, we are confronted with issues concerning modeling methodology, including model construction, parameterization (calibration), validation, and stochasticity. Furthermore, our approach entails the documentation of model predictions by means of experimental observations.

Thus, at one level our studies deal with modeling methodology and address the question, "Can simple mathematical models 'work'?"

On another level our studies address May's hypothesis directiy and, in so doing, investigate many related nonlinear phenomena. Will a real biological population undergo model predicted changes (bifurcations) when manipulated in a prescribed way? Will the population follow a model predicted route-to-chaos? Can tiny system perturbations, through nonlin-earity, produce inordinately large effects on system behavior?

A third aspect of our studies involves arriving at new insights into the dynamics of biological populations. The studies in the previous chapters are not just about flour beeties. The beetles simply serve as a useful animal model for addressing the questions and issues just discussed—questions and issues that are important to the general study of population dynamics and ecology. In what follows we summarize some of the lessons learned.

Model assessment. It is important to distinguish between procedures to parameterize (calibrate) a model and those to validate a model. As J. D. Aber puts it [1], there should always be "some attempt to compare model predictions against independent data sets: data not used in any way in the derivation of the model's parameters ... there are very few aspects of ecology for which no validation data exist." One can set some of the available data aside from the parameter estimation procedure and use it as data to be predicted by the parameterized model. The goal is to obtain confidence not only in the model's ability to fit data, but also in the model's ability to predict data. Prediction is, of course, at the heart of the scientific method. In his book on methods of ecological research, E. D. Ford writes ([67], p. 390-391):

Model assessment against data provides a test of a model's effectiveness. Three levels of assessment can be made for complete models: fitting, predicting, and revealing different results. Fitting is not a strong assessment criteria—yet it can be difficult to achieve and when it is achieved there has to be understanding of how that was done____Prediction is often considered as more valuable than fitting and is widely used in both statistical modeling and systems simulation as validation____Revealing results of a different kind is the strongest assessment, i.e., that the model predicts something not expected and when searched for through more research, is found to occur.

At one level, we can view the studies in this book as accomplishing these feats for the LPA model and the flour beetle laboratory system. Having done this, we are in a position to use the model with some confidence and authority for studies in nonlinear population dynamics, including the phenomenon of chaos. As Ford goes on to say, "If chaos is found in ecological systems then it will be an example of the third level of model assessment, where a model explains an unexpected result." However, incorporating the chaotic behavior predicted by models into ecological theories "is likely only if fitting and/or predicting have already been attained."

Importance of "mechanistic" models. Part of the success of the deterministic LPA model in our projects can be attributed to the model's design around the dominant mechanism known to drive the dynamics of the species used in the experiments. The interactions of interstage cannibalism in flour beetles are reflected in the number and type of model state variables that appear in the model, as well as in the placement and type of nonlinearities. An advantage of mechanistic models (over phenomeno-logical and nonparametric fitting models) is that they in general have lower dimensionality and, when they are shown to be accurate, provide quantitative predictions.

Even though the laboratory beetle system might be considered a simple biological system (and it certainly is when compared to most natural ecosystems), it is considerably more complicated than the LPA model suggests. Left out of the model are many important biological factors that affect fertility, mortality, and growth, including gender, the egg stage, chronological age, and many other genetic, physiological and behavioral traits of the beetles. Almost certainly one could derive a more accurate model by incorporating more of these factors. However, to do so would come at the expense of higher dimensionality and an increase in parameters, costs that should be introduced only if warranted by failures of the LPA model and if the availability of data is sufficient to support (i.e., parameterize and validate) such a model.

Stochastic versions of models are necessary. Another aspect of the mechanistic modeling approach taken in this book is the inclusion of stochasticity. The connection of the deterministic model to real data is accomplished by modeling the expected deviations of data from model predictions in terms of probabilistic assumptions concerning environmental and demographic sources of noise. However, the stochastic versions of the LPA model are more than just the means to accomplish the statistical tasks of model parameterization and validation. They provide predictions of the population dynamics and how they are influenced by both deterministic and stochastic factors. Like deterministic models, stochastic models can serve as exploratory tools for studies of temporal and state-space patterns expected to be observed in data. They provide explanations of detailed patterns appearing in data that result from the interplay of deterministic forces and noise, patterns that are not explained by either deterministic or stochastic forces alone. They also provide, as do their deterministic counterparts, testable predictions that can be confronted with data.

Low-dimensional models can "work." One conclusion to be drawn from the studies in this book is that the deterministic LPA model provides a remarkably accurate description of the flour beede cultures under various laboratory conditions. It made accurate a priori predictions of the long-term dynamics of beetle populations when demographic parameters of the populations were changed by direct manipulation. Although deterministic model attractors, including chaotic attractors, were the focus of the experimental studies in Chapters 4 and 5, the LPA model also accounted for population dynamics when numbers were not near an at-tractor, either because they were not initially near the attractor at the start of the experiment or because stochastic disturbances occurred during the experiment that moved them away from the attractor. Census data taken from experimental treatments often showed state-space patterns and temporal episodes unlike those of the predicted attractor and yet the LPA model offered the means to explain these events in terms of, for example, "saddle flybys" — dynamic motion predicted to occur in the vicinity of unstable deterministic entities — and "lattice effects."

A particularly striking example comes from the chaos treatment where a complicated, subtle, unstable 11-cycle lying on the chaotic attractor is observed in data, as are 6-cycle patterns predicted by the model to occur on the lattice of census data values. That such subtle patterns are observed in real data gives testimony to the (often astonishing) degree to which the LPA model accurately describes and predicts the dynamics of the beetle populations.

The studies appearing in this book are not the only success stories for the LPA model. In a variety of other circumstances the model, or a modification of the model, not only has provided new explanations for patterns observed in data, but has made predictions — sometimes unexpected and nonintuitive predictions—that were borne out by experiments.

Saddle flybys, for example, have proved to be very common in flour beetle population data. The ability of the LPA model to account in this way for unusual episodes of quiescence in some data time series has provided a previously unavailable explanation for these puzzling temporal patterns [35,36]. The existence of multiple attractors has been used to account for certain kinds of patterns seen in data and to explain differences among replicate populations in terms of stochastic "bouncing" between attractors (or, more accurately, their basins of attractions) [91 ]. In another study, model-predicted multiple attractors were documented by means of laboratory experiments [90]. The startling feature of that experiment was not only the unexpected existence of two attractors but, from a biological point of view, the nonintuitive nature of one of them. We have seen in this book how the model also explains phase shifts in the oscillations of the beetle census numbers and hence another difference that can occur among replicate cultures; also see [89]. In yet another study, an unexpected increase in population numbers that occurred in an experiment involving a periodically fluctuating habitat [97] was neatly explained—as a kind of resonance—by a periodically forced version of the LPA model [88,90].

These additional studies, taken together with those in this book, support the (sometimes uncanny) accuracy of the LPA model as a mathematical description of the dynamics of flour beetle cultures; and they support this model's ability to generate accurate predictions that can be documented by experimental evidence. In short, these studies show that the LPA model "works" and does so across a wide range of circumstances.

Of course, the laboratory system we studied in this book—although close to the "natural" habitat of Tribolium—is certainly far less complex than communities of interacting species found in nature. Ecological systems in nature are undoubtedly high-dimensional dynamical systems, and a significant problem is to determine to what extent, if any, their dynamics can be described and predicted by low-dimensional models. Certainly there is no chance of doing this if mathematical models cannot accurately describe and predict the dynamics of low-dimensional biological systems. The flour beede system and the studies in this book provide a cornerstone example of a successful modeling effort that both describes and predicts a wide range of complex dynamics of a real biological population.

Nonlinearityand bifurcations. One of the features of nonlinearity is that responses are not (necessarily) proportional to disturbances. Indeed, responses might be very unexpected and nonintuitive. Not only can magnitudes of quantities change in unusual ways, but so can properties of their temporal dynamics.

For example, the dynamic changes predicted by the bifurcation diagrams appearing in Figs. 3.2 and 3.3 are not intuitive. That a population would change from an equilibrium dynamic to an oscillatory crash-and-boom dynamic and then return to an equilibrium dynamic is not an obvious or expected outcome of a steady increase in adult mortality. The complicated dynamics and route-to-chaos predicted by the bifurcation diagram appearing in Fig. 4.1 are even less intuitive. Yet, in both cases, experimental populations of flour beetles indeed follow these bifurcation scenarios when their demographic parameters are appropriately manipulated.

One can single out a treatment lying along the route-to-chaos, a treatment located in the bifurcation diagram where chaotic attractors reside, and study the replicates from that treatment as examples of chaotic populations. We did this in Chapter 5. However, a stronger message is contained in the whole bifurcation diagram. The route-to-chaos experiment clearly demonstrates that real biological populations can display the range of dynamics—the bifurcations and complexity—predicted by simple deterministic rules embodied in low-dimensional models. This is the heart of May's hypothesis.

Confidence intervals. Parameter estimates come with confidence intervals. Model dynamics should be investigated for parameter values ranging throughout these intervals. If the dynamics of a well-validated model are robust (i.e., are qualitatively unchanged) throughout the confidence intervals, then one has strong support for reaching the conclusion that the biological population also has these same dynamics. If, for example, the model possesses a stable equilibrium for all parameter values lying in the confidence intervals, then one concludes that the deterministic component of the population's dynamics is an equilibrium. The same can be said about other types of attractors, including chaotic attractors.

However, it is possible for bifurcations to occur within confidence intervals and as a result it is possible that several qualitatively different types of dynamics occur for parameter values ranging throughout these intervals. Indeed, when dealing with complex dynamics, such as chaos, it is likely that many bifurcations occur within confidence intervals. In such a situation it maybe inappropriate to assign a specific dynamic property to the modeled biological population according to the predicted dynamic at selected point estimates for parameters. Instead, one must in some way assess the various influences of the different attractors that are distributed throughout the confidence intervals. For example, one needs to decide what it means to assert that a population is "chaotic" if within the confidence intervals for the estimated parameters there is a complicated array of chaotic attractors and period locking windows containing stable cycles. Moreover, there can be additional complications in assigning a particular attractor type to a population—such as the complications that arise from transients, stochasticity, and lattice effects.

Lattice effects. An interesting insight, with potentially widespread implications for the study of populations dynamics in general, resulted from a study of the chaos treatment in Chapter 5. Observed patterns in the experimental data were found to be caused by the fact that census numbers are necessarily discrete integers. Animal numbers or densities come from a finite set of discrete numbers — a "lattice" — and being so constrained must ultimately display, in the absence of noise, equilibrium or periodic dynamics. It is another tribute to the LPA model that it provided, by its integerization, an accurate explanation of these originally unanticipated patterns.

In a technical sense population time series data cannot—because they lie on a lattice — display quasiperiodic or chaotic dynamics, even if they are purely deterministic. In Chapter 5 a study of some deterministic and stochastic lattice models revealed the relationship between the dynamics on a lattice and those on the underlying continuous state space.

First of all, transient lattice dynamics (i.e., the dynamics prior to becoming locked into a cycle) tend to resemble the continuous state-space dynamics, at least as best they can while being constrained to the lattice. In this way, the dynamics of a stochastic system on a lattice—being a system continually perturbed into transient behavior—can tend to resemble the underlying continuous state-space dynamics. This, then, is one way the continuous state-space attractor exerts its influence on the lattice system. Note how, in this case, stochasticity is an aid to discovering the characteristics of the underlying dynamics—rather than a hindrance.

Secondly, on sufficiently fine lattices the dynamics will more closely resemble the continuous state-space dynamic. One way (but not the only way) in which lattices can be refined is by considering the population density in larger habitat sizes, at least for populations whose abundance scales approximately with habitat size. It follows that habitat size can be a significant factor in determining population dynamics. In particular, the potential for complex dynamics is lessened and obscured in smaller habitat sizes. Or, put another way, a study that determines a particular dynamic for a population in one habitat size might be invalid in a larger habitat size because the simplified dynamics on the coarser lattice of the smaller habitat hide the underlying tendency for complexity— a complexity that is revealed, perhaps unexpectedly, in a larger habitat. A kind of bifurcation occurs in the dynamics as a function of habitat size.

It should be noted that these lattice effects are not necessarily the result of extremely small population numbers. Nor are they manifested only on very coarse lattices. For example, the lattice in the chaos treatment in Chapter 5 contains more than a million points.

Finally, for populations dominated by demographic as opposed to environmental stochasticity, the coefficient of variation decreases to zero with increasing population abundance. For such populations, as habitat size increases the stationary density distribution of the stochastic lattice model converges to the deterministic attractor of the underlying continuous state-space dynamics. In particular, if the underlying continuous state-space dynamic of a population is chaotic, population density will more clearly exhibit the properties of this chaos in larger habitats.

Mixes of deterministic/stochastic, continuum/lattice, and attractor/ transient dynamics. A conclusion that emerges from our studies, particularly from those in Chapter 5, is that a full explanation of ecological time-series data is unlikely to be found by analyses that rely solely on deterministic model attractors. Instead, what one expects to see in data is a mix of several, if not many, patterns resembling different deterministic entities that occur—in whole or in part—during randomly appearing temporal episodes. These entities include attractors (perhaps more than one), transients, unstable invariant sets and their stable manifolds, and lattice cycles (perhaps more than one). Stochasticity, besides simply "blurring out" deterministic attractors, can repeatedly cause trajectories to visit regions in state space away from attractors and thereby bring populations perpetually under the influence of deterministic dynamics other than those of an attractor. This is what one finds, for example, in computer studies of stochastic toy models (Figs. 5.11 and 5.12) and the LPA model (Fig. 5.13), and this is what one observes in the experimental flour beetle data (Fig. 5.14). Furthermore, the complexity of the mix increases with the complexity of the deterministic component of the dynamics.

If an adequate explanation of observed dynamic patterns found in time-series data from a single population under highly controlled conditions in which stochasticity is minimized — as in our laboratory studies — cannot be obtained by means of deterministic model attractors alone, then certainly this fact is even more likely to be true for ecological data obtained from field observations or experiments. It is widely recognized that insufficient length of time-series data is a major problem in the study of chaos in ecological data [ 144]. In addition, if ecological data are typically a blend of deterministic properties and stochasticity as described earlier, then one has added difficulties. The problem is not only the length of the data set, but the length of time spent on or near the attractor (or any other deterministic entity of interest). The data may be "contaminated" not only by noise—which has long been recognized as a significant problem in the detection of chaos in data—but by other deterministic dynamics (transients and even unstable invariant sets) not due to the chaotic attractor at all.

Moreover, a converse holds. Nonchaotic dynamics may be mistakenly identified as chaotic for the same reason, viz., that stochasticity allows visits to regions of state space away—even far away—from attractors. A time series of data may often visit regions where orbits are separating, for example near an unstable equilibrium or cycle, producing a kind of sensitivity to initial conditions (as measured, for example, by the stochastic Lya-punov exponent SLE) that is not due to deterministic chaos. Desharnais etal. [51] provide an example, using the environmental stochastic Ricker model, in which a noisy equilibrium has a positive SLE (see [46] for more details). Because of stochasticity, diagnostic tests for chaos that require long-term residency on or near a chaotic attractor are not likely to be appropriate for ecological time series. The LPA model and the route-to-chaos experiment in Chapter 4 provide other examples, a particularly notable one ofwhich is the treatment cpa = 0.50. Fortius treatmentthede-terministic LPA model possesses a stable cycle throughout the confidence intervals of the estimated parameters, even though the SLE is positive throughout those intervals (see Table 4.3). In these cases the stochastic Lyapunov exponent is uninformative and misleading with regard to the detection of chaos, especially when one considers that a noisy stable equilibrium or cycle is a leading alternative hypothesis to chaotic dynamics.

Another point concerning the detection of chaos in data comes out of the investigation of temporal patterns in Chapter 5. The appearance of regular, even near-periodic, patterns in time series data does not itself rule out the presence of chaos.

Efforts to find unequivocal evidence of chaos in ecosystems have generally focused on the analysis of available data sets. Although different approaches have been taken in undertaking these analyses, we have seen that there are many difficulties associated with this task. Another approach to the question of chaos in ecology (indeed, complex nonlinear dynamics in general) — and the one taken in this book—is to study the response of a system to perturbations. Depending on the nature of the perturbations, an ecosystem in or near a region of complex nonlinear dynamics such as chaos is likely to respond quite differently from one in a robust state of stable equilibrium (or period cycling) — and to do so often in unusual, surprising, and nonintuitive ways, even to small perturbations. In many ways, this is perhaps a more interesting and important question to ask anyway, rather than whether or not a specific system is, or is not, chaotic in its present state and configuration. Therefore, in our studies we have not focused on problems associated with the detection of chaos in available data sets. Nor have we addressed the often-debated and important questions concerning the role of chaos in "natural" populations, such as the frequency or rarity of its occurrence or whether it is selected for or against. Instead our studies relate to different questions: what roles do complex dynamics, such as chaos, play in the response of ecosystems to disturbances? Given that population interactions are nonlinear, what are the possible consequences of changes in environmental and demographic parameters? What unexpected and unintended consequences due to nonlinearity might occur? Given the continual disturbances that affect the physical and biological environments of many, if not most, biological species (especially when the activities of humans are brought to mind), these are also important questions to ask about natural ecosystems. Experiments using laboratory systems are a natural way to begin a rigorous study of these questions. However, rigorous study demands a close connection between the biological system and the mathematics of nonlinear theory—between data and models.

We have seen in this book that it is possible to describe and predict, by means of a low-dimensional mathematical model, the consequences that environmental disturbances have on the dynamics of a biological population. As P. Kareiva writes [98], "... simply documenting direct demographic effects of environmental change or environmental stress has little predictive power. Only with a solid understanding of underlying dynamics can ecologists ever hope to anticipate the consequences of environmental perturbations." Our studies in this book show how, as Kareiva continues to say, "the marriage of nonlinear models and experiments can help to accomplish this task."

Furthermore, as a result of our modeling and experimental efforts, we have seen that a real biological population can indeed follow a model-predicted route-to-chaos [72], the possibility of which had so intrigued

Robert May in his seminal papers. While documenting this fact, we gained new insights into the complex dynamic patterns that can arise from the mix of nonlinear, deterministic, and stochastic elements inevitably found in ecological systems [19]. However, despite the many complications that arise from this mix of elements, one should not lose sight of the fact that it was a simple, low-dimensional model whose predictions were fulfilled, and upon whose foundation were based accurate explanations for complex dynamic patterns observed in the data. The Tribolium project provides an example that indeed corroborates May's hypothesis that complex dynamics in ecological data can be the result of simple rules.

It is hoped not only that the studies in this book provide insights into nonlinear population dynamics, but that they also provide a modest step toward the "hardening" of ecological science — a step toward raising its explanatory and predictive power beyond purely theoretical speculation and a satisfaction with only qualitative accuracy, reasonable "guesstimates," and verbal metaphors. It is true that the laboratory system used in our studies is a relatively simple biological system, and that the low-dimensional LPA model is a simple mathematical model of that system. Nonetheless, we can find motivation and inspiration for the study of such systems from another quote taken from May's seminal 1976 paper [124]:

Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamic properties.

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