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Unperturbed

Unperturbed

Experimental data

In-box

Experimental data

In-box

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FIGURE 5.3 I The first row of plots shows the effects on L-stage numbers of applying the in-box and out-box perturbation protocols described in the text to a chaotic attractor of the deterministic LPA model. Parameter values are those in Table 4.1 with Cpa — 0.35 and /ia = 0.96. Notice the suppressed amplitudes caused by the in-box perturbations applied from week 134 to week 212. The dashes above the plots during the perturbation period indicate those weeks at which three adults were added to the population. The out-box procedure, on the other hand, does not suppress the oscillatory amplitudes (although it does regularize the chaotic fluctuations by producing a periodic orbit). Notice the return to chaotic oscillations when both procedures are terminated. The second row of plots shows the result of applying the perturbation schemes when using the demographic stochastic LPA model (4.7). These stochastic orbits can be considered simulations of the experiment and indeed they are very similar to the data shown in the third row of plots. These data are from one replicate in an experiment in which the same perturbation schemes were applied to laboratory cultures of T. castaneum.

region or "hot spot" of state space. If an orbit is slightly perturbed in a certain way when it enters the region (specifically, if only three adults are added to the culture), then the amplitude of the chaotic oscillations in the L-stage are significantly dampened. See Fig. 5.3. This predicted reduction in oscillatory amplitude led to the following experiment.

We maintained nine populations of T. castaneum according to the experimental protocol for the chaos treatment described in Section 4.1 for 132 weeks, after which we chose six cultures at random and divided them into two treatment groups of three cultures each. We maintained the remaining three cultures as controls. Starting at week 134, we subjected the two treatment groups to perturbations as follows. The region in state space where L < 150 and A<3 contains the hot spot on the chaotic attractor. We call this region the "hot box." The three cultures in one of the treatment groups were perturbed according to an "in-box" procedure: if, at any time, the life stage vector {Lh Pt, At) from a culture was in the hot box, then three adults were added to that culture; otherwise no perturbation was made. According to the LPA model, the oscillations in these cultures should have significantly reduced amplitudes.

The three cultures in the other treatment group were subjected to exactly the opposite perturbation scheme: three adults were added when and only when the life stage vector (Lf, Pt, At) fell outside the hot box, i.e., when either Lt > 150 or At > 3. The LPA model prediction for this "outbox" protocol is that no amplitude reduction in the chaotic oscillations will occur. See Fig. 5.3. The out-box perturbation scheme serves to test whether any reduction in the oscillatory amplitude observed in the inbox treatment group is due simply to the addition of adults, rather than the chaotic property of sensitivity to initial conditions and where in state space the addition of adults is performed.

We sustained the two perturbation protocols for a total of 78 weeks. At that time we ceased the perturbations and maintained the cultures for another 54 weeks. During these final weeks, the LPA model prediction was, of course, for the oscillatory amplitudes of the cultures to return to the levels attained prior to the application of perturbations. See Fig. 5.3.

The in-box perturbations did have the predicted effect. Figure 5.3 shows the observed time series data for the L-stage numbers from one of the replicate cultures. Notice these cultures exhibit large-amplitude L-stage fluctuations prior to the in-box perturbations, but their oscillations dampened dramatically after the in-box perturbations were applied. When this perturbation scheme was discontinued, the large-amplitude oscillations quickly returned.

On the other hand, those cultures subjected to the out-box perturbation scheme show large chaotic fluctuations in L- stage numbers similar to those observed in the control cultures (i.e., in the chaos treatment of the route-to-chaos experiment). The large-amplitude fluctuations occurred even though the experimental protocol assigned out-box perturbations more often than in-box perturbations (see Fig. 5.3). This demonstrates that the dampening effect of the in-box treatment was due to the timing of the perturbations to coincide with the occurrence of life-stage numbers in a sensitive region of state space, near the hot spot on the chaotic attractor.

We emphasize that relatively small perturbations were used to obtain the large decrease in the amplitude of the fluctuations seen in the inbox populations. During the time when the treatment perturbations were applied (weeks 134 to 210), a total of 156 adults were added to the three experimental cultures in accordance with the in-box rule. During the same period of time we counted a total of 6545 adults in the three unperturbed control populations. Thus, the perturbations represent only 2.4% of the number of adults we could have expected if the perturbations were not applied. Nevertheless, these in-box perturbations resulted in a 82.7% decrease in the number of adults and a 48.3% decrease in the total number of insects (L-stage plus P-stage plus A-stage) counted during this period, relative to the unperturbed controls. By contrast, over the same period of time, we added a total of 234 adults to the out-box populations and obtained only a 4.3% decrease in the total number of adults and a 3.7% decrease in the total number of insects relative to the unperturbed controls.

The stochastic LPA model (4.7) was remarkably effective at predicting the dynamic behavior of all cultures in this experiment, including the response of the experimental cultures to both the in-box and out-box perturbations (Fig. 5.3). Model simulations exhibited the same amplitude and general pattern of fluctuations as seen in the experimental data. The results are particularly significant given that the model parameter values used in the experiment were obtained from a previous study (Table 4.1) and the model was not "fit" to the data obtained from the experiment.

5.2 | TEMPORAL PATTERNS

In all systems of equations exhibiting chaotic behavior recurring patterns are observed over time.

Mathematical attractors reside in state space. One advantage of plotting attractors in state space is that doing so will often reveal distinctive geometric and topological structures. This is particularly true for chaotic attractors where temporal patterns are usually not so apparent from time series plots. For example, Fig. 5.1 shows the distribution of data points from the chaos treatment (c^ = 0.35) of the route-to-chaos experiment in relation to the geometric configuration of the chaotic attractor in state space. However, the temporal motion of the data is not indicated in this graph.

Since the temporal dynamics of the demographic triples (Lt, Pt, At) are chaotic, a time series plot of, say, its ¿-stage component Lt appears erratic (see Fig. 5.4). The erratic motion of chaotic dynamics does not, however, necessarily imply the total lack of temporal order. Discernible patterns are often identifiable within the complex oscillations of chaotic time series. The chaotic attractor shown in Fig. 5.2 does have some distinctive temporal patterns, as we will see. Do these patterns help explain the data from the route-to-chaos experiment? That is to say, are these patterns found in the data?

Power spectrum plots associated with chaotic orbits on the attractor indicate a dominant period of approximately (but less than) 3 time units, or 6 weeks (Fig. 5.4). A few other periods are indicated by these plots— the most significant of which is a period of approximately 4 time units, or 8 weeks — but these periods play less significant roles. The power spectra associated with the stochastic LPA model orbits are more "spread out" than those predicted by the chaotic attractor. However, the stochastic

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FIGURE 5.4 I (A) The l-stage time series plot is from a chaotic orbit of the deterministic LPA model, with the parameter values of the chaos treatment (cm = 0.35) treatment. The neighboring graph is its power spectrum. (B) These plots are from orbit of the demographic noise model (4.7). (C) These plots are for replicate 13 of the chaos treatment in the route-to-chaos experiment. (Reprinted from Chaos, Solitons, and Fractals, Vol. 12, No. 2, J. M. Cushing,

S. M. Henson, R. A. Desharnais, B. Dennis, R. F. Costantino, A. A. King, "A Chaotic Attractor in Ecology: Theory and Experimental Data," pp. 219-234,2001, with permission from Elsevier Science.)

orbits also indicate a strong tendency for oscillations with period 3. The similarity between the power spectra of the stochastic model orbits and the experimental data in Fig. 5.4 is notable.

In addition to chaotic orbits there typically exist infinitely many unstable periodic cycles on a chaotic attractor. It turns out, however, that the chaotic attractor for the chaos treatment Cpa — 0.35 does not contain a 3-cycle, as might be expected from the spectral plots in Fig. 5.4. This assertion has not been rigorously proved mathematically, although extensive computer studies indicate that it is true.

While carefully examining the data from the chaos treatment of the route-to-chaos experiment, one of the authors (R. F. Costantino) identified a distinctive temporal pattern that frequently appears in each of the three replicate cultures as the demographic triples {Lt, Pt, At) move in state space. This pattern consists of 11 consecutive triples and therefore spans a 22-week period. (Fig. 5.7 shows an example.) The "11-pattern" appears often enough in the data that one wonders if the chaotic attractor can in some way account for its presence. This question stimulated an investigation into the temporal patterns associated with the chaotic attractor.

An analysis of the bifurcations occurring in the LPA model suggested the existence, on the chaotic attractor, of an unstable (saddle) 11-cycle that has a strong influence on orbits lying on and near the attractor [37]. Although we have no rigorous mathematical proof of its existence, we can numerically find this unstable 11-cycle. A plot appears in Fig. 5.5. Note the similarity of the cycle's power spectrum in Fig. 5.5 to that of the chaotic attractor. This is one indication of the cycle's influence on the chaotic dynamics on and near this attractor.

We can obtain further evidence of the 11-cycle's importance from lag metric calculations for a typical chaotic orbit with respect to the cycle. Recall that we calculate the lag metric at time t for a selected phase of the 11-cycle by computing and averaging the distances between each of the eleven orbit triples at times t - 10, t - 9, ..., t - 1, t and the corresponding points from the selected phase of the cycle.2 The result, after doing this for each time t, is a time series of distances, each measuring how close the orbit was to (the selected phase of) the 11 -cycle during the past 11 time units.

Notice that the lag metric is quite demanding as a measure of how close an orbit is to a cycle. To be close to a cycle by this metric means that the state space points on the orbit remain close to those on the cycle, in the correct temporal sequence, for the entire period of the cycle. In the case of the 11 - cycle, a small lag metric means a sequence of data triples remains

2 See the caption for Fig. 2.12.

Chaotic Attractor
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FIGURE 5.5 I The unstable 11-cycle on the chaotic attractor of the LPA model in the treatment Cpa = 0.35 has a power spectrum with frequencies similar to those of the chaotic orbit in Fig. 5.4. (Reprinted from Chaos, Solitons, and Fractals, Vol. 12, No. 2, J. M. Cushing,

S. M. Henson, R. A. Desharnais, B. Dennis, R. F. Costantino, A. A. King, "A Chaotic Attractor in Ecology: Theory and Experimental Data," pp. 219-234,2001, with permission from Elsevier Science.)

near all of the cycle state space triples, in sequence, for 11 time steps — that is to say, for a total of 22 weeks or a little more than five generations.

A periodic cycle has different phases depending on its initial point. An 11-cycle has, therefore, 11 phases. An orbit close to one phase of the 11 -cycle might not (and in general will not) be close to another phase. In looking for patterns near the 11 -cycle, we perform lag metric calculations for each of the 11 phases of the 11-cycle.

Figure 5.6 shows plots resulting from lag metric calculations for a chaotic obit using two different phases of the unstable 11-cycle lying on the chaotic attractor. One can see in these plots how, over the course of time, the chaotic orbit recurrentiy comes close to and moves away from these particular phases of the 11-cycle. These incidents are analogous to the saddle equilibrium (or "1-cycle") flybys we have seen in other beetie cultures, and therefore we refer to them as "11-cycle flybys." Using lag metric plots such as these one can identify episodes of 11-cycle flybys and hence time intervals during which the chaotic orbit traces a path in state space similar to the 11-cycle. Some examples are shown in Fig. 5.6.

Certainly, there are times when an orbit on the chaotic attractor will not resemble the 11-cycle. However, in what follows it is convenient to

FIGURE 5.6 I The top graph shows plots resulting from lag metric calculations for a chaotic orbit of the deterministic LPA model using two different phases of the unstable 11-cycle (shown in Fig. 5.5). The low points in these plots indicate times at which the orbit is near the corresponding phase of the 11-cycle for 11 consecutive time steps, i.e., times at which the orbit undergoes an " 11-cycle flyby." The plots show frequently occurring 11 -cycle flybys during 2000 time steps. Plots resulting from lag metric calculations using the other nine phases of the 11-cycle are similar and show many other 11-cycle flybys.. Because of the horizontal scale in the graph, the temporal durations of these flybys are not evident. For example, the flyby occurring shortly after f = 1000 has a lag metric less than 10 for 41 time steps (from t = 1028 to 1069). This means that during a period of 82 weeks the orbit was on average less than 10 individuals from the 11-cycle. The plots also indicate times when the orbit is not near the selected phases of the 11-cycle. This is the case, for example, during the time interval t = 400 to 900. (It turns out the lag metric plots for the other nine phases of the 11-cycle—which are not shown in the graph—are also large during this interval.) The similarity of the orbit to the 11-cycle during a flyby is evident in the accompanying L-stage time series plots and, perhaps more strikingly, the state-space plots.

use the 11-cycle as a kind of surrogate for (or a signature of) the chaotic attractor.

The importance of the unstable 11-cycle on the chaotic attractor suggests — if the beede population dynamics are indeed well described by the LPA model and this chaotic attractor—that we should find evidence of the this cycle in the experimental data, that is to say, we should see 11 -cycle flybys. Such evidence would appear as a temporal sequence

Chaotic Attractor

FIGURE 5.7 I The lag metric indicates episodes when replicate 13 from the route-to-chaos experiment resembles one phase of the 11 -cycle on the chaotic attractor. Using this evidence we highlight some sample 11-patterns with open circles in the plot of the /.-stage data. The sample state space plot shows one temporal episode that clearly resembles the 11 -cycle in Fig. 5.5. Similar 11 -patterns appear later in the time series and in other replicates.

FIGURE 5.7 I The lag metric indicates episodes when replicate 13 from the route-to-chaos experiment resembles one phase of the 11 -cycle on the chaotic attractor. Using this evidence we highlight some sample 11-patterns with open circles in the plot of the /.-stage data. The sample state space plot shows one temporal episode that clearly resembles the 11 -cycle in Fig. 5.5. Similar 11 -patterns appear later in the time series and in other replicates.

of 11 data triples near the successive 11 triples on (some phase of) the 11-cycle shown in Fig. 5.5. In fact, the 11-patterns observed by Costantino match the model 11-cycle in precisely this way. We can quantify these

11 -cycle flybys, and the "influence" of the 11-cycle on the data, by using the lag metric (which we calculate for a time series of experimental data in the same way we do for model orbits). Figure 5.7 shows examples using one replicate from the chaos treatment of the route-to-chaos experiment.

The presence in the data of the distinctive 11-pattern predicted by the chaotic attractor provides yet another example of the ability of the LPA model to describe and predict the dynamics of flour beetle cultures— even quite subtle and complex details of their dynamics. However, there are some observable temporal patterns in the data for which the LPA model does not offer a ready explanation. An example is a distinctive 6-pattern—a recurrent temporal pattern occurring over 6 units of time (or

12 weeks). A few samples are shown in Fig. 5.8. We cannot account for

FIGURE 5.8 I Some distinctive 6-patterns in replicate 13 of the route-to-chaos experiment. Similar patterns appear later in the time series and also in other replicates.

these patterns in the same way we accounted for the 11 -patterns, that is to say, by the existence of a 6-cycle on the chaotic attractor. This is because there is no 6-cycle on the chaotic attractor (or anywhere else in state space). We have no rigorous proof of this assertion, although extensive numerical searches using computers suggest the absence of any 6-cycle solution of the LPA model in the chaos treatment case. It is often true that a failed explanation by an otherwise successful model raises questions whose investigation leads to new insights. This is the case with these 6-patterns, as we will see in the next section.

5.3 | LATTICE EFFECTS

What the chaotic motion of the infinite does is to force us to consider whether, within a very finite time, our actions in the finite have any relationship to... theoretical ideas from the infinite.

The protocol in the route-to-chaos experiment involved manipulation of the model parameters ixa and c^. These manipulations were accomplished by adjusting the number of A-stage individuals at each census (Section 4.1). Adult beetles come, of course, in whole numbers. As a practical matter in performing the experiment, we rounded (to the nearest whole integer) the results of the numerical calculations used to determine the adjusted number of A-stage individuals at each time step. A modified LPA model that incorporates this step in the experimental protocol is

Pt exp

With the parameter estimates and initial conditions for the chaos treatment, this modified LPA model produces the orbit shown in Fig. 5.9A. Notice how, after a short initial transient time, the orbit setdes into a periodic 6-cycle—a 6-cycle whose temporal pattern is similar to those observed in the experimental data (Fig. 5.8)!

Individuals in all life-cycle stages, not just adults, come in whole numbers. The experimental data consist of integer-valued triples (Lt, Pt, At).

FIGURE 5.9 I (A, upper) The ¿-stage component of an orbit generated by the lattice LPA model (5.1) settles into a periodic 6-cycle starting at t = 17 (triangles). (B, upper) The same is true of the orbit generated by the lattice LPA model (5.3). We calculated both orbits using the experimental initial conditions (L0, P0, A>) = (250, 5,100) and the parameter estimates in Table 4.1 (for V = 1) with treatment values ¡ia = 0.96, cm = 0.35. In state space the resemblance of the 6-cycles to the 6-patterns observed in the data is evident (Fig. 5.8).

FIGURE 5.9 I (A, upper) The ¿-stage component of an orbit generated by the lattice LPA model (5.1) settles into a periodic 6-cycle starting at t = 17 (triangles). (B, upper) The same is true of the orbit generated by the lattice LPA model (5.3). We calculated both orbits using the experimental initial conditions (L0, P0, A>) = (250, 5,100) and the parameter estimates in Table 4.1 (for V = 1) with treatment values ¡ia = 0.96, cm = 0.35. In state space the resemblance of the 6-cycles to the 6-patterns observed in the data is evident (Fig. 5.8).

In other words, triples of real data are confined to a discrete "lattice" embedded within the continuum of three-dimensional Euclidean state space.3 However, neither the original deterministic LPA model (4.1) nor the modified LPA (5.1) model provide integer valued predictions for these data triples. One mathematical way to obtain realizable, whole-number predictions from these models is simply to round each entry in the triple (Lf+i, Pt+1, Af+i) to the nearest whole integer. The resulting formulas are

Lt+i — round Pt+1 = round[(1 - /x/)Lf] At+1 = round i A I Cel T Cea A

Ptexp( ) + a-ßa)At in the case of the general LPA model (4.1) and

P exp

in the case of the model (5.1) modified for the protocol of the route-to-chaos experiment. These "lattice" LPA models define dynamical systems on the lattice of realizable (integer valued) triples.

The two lattice models (5.2) and (5.3) also produce, for the experimental conditions, a 6-cycle attractor similar to the 6-cycle obtained from the model (5.1). For example, Fig. 5.9B shows the orbit and attracting 6-cycle generated by the lattice model (5.3). Given the clear resemblance of the 6-patterns observed in the data to the 6-cycles predicted by all of these "lattice" LPA models, it seems that the explanation for the presence of this pattern in the data is found in the fact that the data lie on a discrete lattice. However, as we will see, this assertion is only part of a full explanation of the data time series.

Each orbit of a deterministic (autonomous) dynamical system when confined to a finite discrete lattice must eventually be exactly periodic. With only a finite number of states available, an orbit must eventually visit some state twice, after which it will exactly repeat itself because the system is deterministic. As a result, attractors predicted by a deterministic

3 Even if measured in population densities, the observable data triples lie on a lattice, in this case a lattice of fractions.

model confined to a finite lattice in general will be "simpler" than the at-tractors in the underlying continuum state space. This is certainly true if the continuum state space attractor is not periodic—for example, if it is quasiperiodic or chaotic. Strictly speaking, dynamical systems on a finite lattice cannot have chaotic dynamics, at least not as they are defined in modern theories of dynamics. Thus, we have another reason (besides stochasticity) a biological population cannot be, strictly speaking, chaotic. Nonetheless the dynamics in the continuum state space can exert a strong influence on the lattice dynamics and, in particular, we can again argue that a population's dynamics can be influenced by a deterministic chaotic attractor. We mention three fundamental points in this regard.

The first point is a mathematical one: under certain conditions, the periodic attractors on the lattice converge to the continuum state space attractor as the lattice is "refined," that is to say, as the lattice spacing -— the "mesh size" — tends to zero. Therefore, on a very fine lattice the dynamics resemble the dynamics in continuum state space. This is one way the continuum state space attractor can be said to "exert its influence" on the lattice dynamics.

But in biological systems in which animals or plants come in whole numbers, how can the whole-numbered data lattice be refined? One way is to consider population densities and habitat size instead of population numbers. Before explaining this, we first take note of a relevant feature of the LPA model (4.1).

Orbits and attractors of the LPA model scale with the habitat size V. By this we mean the following: if (Lt, Pt, At) is a solution of the model equations with habitat volume V and if k > 0 is any positive number, then (kLt, kPt, kAt) is a solution of the model equations with habitat volume kV. This fact implies that if habitat size is multiplied by a factor k, then population numbers are multiplied by the same factor. It follows, in terms of population densities, that orbits and attractors remain unchanged if V is changed.4 In contrast to this, orbits and attractors on lattice versions of the LPA model do change when V is changed.

An increase in habitat size V corresponds to a refinement of the lattice for population densities, where in place of whole-number triples (L„ Pt, At) we consider density triples (Lt/ V, Pt/ V, At/ V). For example, the state space for L-stage densities consists of the numbers 0, 1/V, 2/ V, 3/ V,... and the lattice mesh size equals 1/ V. As the habitat volume

4 This scaling property is not peculiar to the LPA models and is enjoyed by many familiar ecological models. The Ricker model and the modified Nicholson-Bailey host-parasitoid model are other examples.

V increases without bound, the mesh size tends to zero and the lattice "approaches" the continuum state space.

In a general mathematical setting it is known, for a continuum state space orbit attracted to a stable cycle, that the cycle attractors of nearby lattice orbits will converge to that stable cycle as the mesh size decreases to zero. The relationship between a chaotic continuum state-space at-tractor and the cycle attractors on lattices of decreasing mesh is not in general so clear.5 Computer simulations using the lattice LPA model for the case of the chaos treatment suggest, however, that the lattice attractors do converge, or at least are very close, to the chaotic attractor for decreasing mesh size (Fig. 5.10). Similar "convergence" properties are illustrated in Figs. 5.11 and 5.12 for the Ricker and the modified Nicholson-Bailey host-parasite models. Note the simplification of attractors resulting from confining the models to a lattice cannot be attributed to a small number of available lattice points—that is to say, it is not the result of an extremely small habitat size nor of very low population numbers. Complex continuum state attractors can collapse to low period cycles on very fine lattices. That is, as lattice cycle attractors converge to a complex continuum state attractor, their complexity does not necessarily "monotonically" increase.

A second point regarding the relationship between the lattice dynamics and the continuum state space dynamics concerns stochasticity. Deterministic model orbits confined to a lattice eventually "lock" onto a periodic cycle. The transient time of a lattice orbit—the initial time before it becomes periodic—typically resembles the continuum statespace dynamics [92], It is not difficult to see why this is true for the rounded LPA model (and the Ricker and host-parasitoid examples). In producing the next census triple from the current census triple, the lattice model first calculates the triple predicted by the continuum state-space model and then rounds the result to a nearby lattice point. Repeating this process causes an accumulation of errors, of course, but if the mesh is not too coarse and the number of steps is not too many, the lattice orbit and the continuum state space orbit will remain close. Thus, when the continuum state-space attractor is chaotic, a lattice orbit will likely have transients that tend to resemble the chaotic attractor.

5 The convergence of lattice (cycle) attractors as the mesh decreases to zero is a problem related to the so-called "shadowing" of orbits in dynamical systems theory. A fundamental shadowing theorem asserts, for small mesh size, that near a lattice orbit there is (for all time) a continuum state-space orbit [9, 21]. However, this theorem concerns a restricted class of mathematical models, namely, those defined by smooth invertible maps (diffeomorphisms) with uniformly hyperbolic attractors. Chaotic attractors are not typically uniformly hyperbolic [80]. The map defined by the LPA model also is not invertible (nor are the two examples in Figs. 5.11 and 5.12).

FIGURE 5.10 I The first two rows are plots showing the population density attractors of the deterministic lattice LPA model (5.3) for several habitat volumes. As the habitat volume increases the attractors come to resemble the chaotic attractor in the continuum state space shown in the bottom plot (also see Figs. 4.3 and 5.2). We calculated the lattice attractors using the experimental initial conditions (Lt, Pt, A,) = (250,5,100), the parameter estimates from Table 4.1, and Cpa = 0.35, na = 0.96 corresponding to the chaotic treatment in the route-to-chaos experiment. Population densities (Lt/ V, Pt/ V, A,/ V) on the lattice for a habitat size V result from the population numbers (Lt, Pt, At) obtained from the model (5.3). Each lattice attractor is a periodic cycle. The plots at lower volumes use larger circles for visibility.

FIGURE 5.10 I The first two rows are plots showing the population density attractors of the deterministic lattice LPA model (5.3) for several habitat volumes. As the habitat volume increases the attractors come to resemble the chaotic attractor in the continuum state space shown in the bottom plot (also see Figs. 4.3 and 5.2). We calculated the lattice attractors using the experimental initial conditions (Lt, Pt, A,) = (250,5,100), the parameter estimates from Table 4.1, and Cpa = 0.35, na = 0.96 corresponding to the chaotic treatment in the route-to-chaos experiment. Population densities (Lt/ V, Pt/ V, A,/ V) on the lattice for a habitat size V result from the population numbers (Lt, Pt, At) obtained from the model (5.3). Each lattice attractor is a periodic cycle. The plots at lower volumes use larger circles for visibility.

In a stochastic version of a lattice model, orbits cannot settie into the lattice attractor (which, as we have noted, is always a periodic cycle). Although a stochastic lattice orbit might occasionally resemble the periodic attractor on the lattice, there are continual random perturbations that produce transients. As a result, the stochastic lattice orbit can also episodically resemble the underlying continuous chaotic attractor. Stochastic lattice orbits can exhibit intermittent and recurrent temporal patterns, some of which resemble the lattice attractor and some of which resemble the underlying continuum state-space attractor. Here stochas-ticity is an aid rather than a hindrance, as it is often considered to be, in the sense that it helps "reveal" an underlying tendency toward chaos; it reveals the presence of an underlying deterministic dynamic that is prevented

FIGURE5.11 I (a) The chaotic time series produced by the Ricker model x,,i = bxte~kx,/v with parameters b = 17 and k = 1 appears in the right graph. The left graph shows a plot of the chaotic attractor. The graphs in (b)-(e) show plots of the periodic cycle attractors obtained from the integerized Ricker model xM = round(17x(e~JI,/l') for several different habitat sizes V and the initial condition x0 = 5 V. (Other lattice attractors might arise from other initial conditions.) In (b) there are 196 displayed lattice points and the attractor is a 2-cycle. In (c) there are 1225 displayed lattice points and the attractor is an equilibrium. In (d) there are 4900 displayed lattice points and the attractor is a 13-cycle. In (e) there are 4.41 x 106 lattice points (the lattice appears to fill the plane) and the attractor is a 117-cycle that somewhat resembles the chaotic attractor in (a). In (f) one sees the result of adding environmental noise to the case (c). We generated the orbit using the stochastic equation x,+i = round(17xfe~-t</ lV"l3£') where Et is from a standard normal probability distribution. The plot on the left resembles that of the chaotic attractor in (a). The time series on the right contains episodes that resemble the chaotic time series in (a) interspersed with episodes that resemble the lattice equilibrium in (c). (Reprinted with permission from S. M. Henson, R. F. Costantino, I. M. Cushing, R. A. Desharnais, B. Dennis, and Aaron A. King, Lattice effects observed in chaotic dynamics of experimental populations, Science 294 (2001), 602-605. Copyright 2001 American Association for the Advancement of Science.)

FIGURE 5.12 I As in Fig. 5.11 we obtain a sequence of lattice attractors for the modified Nicholson-Bailey host-parasite model [15]. The stochastic, integerized versions of the equations are

Ht+1 = round((Vflrer(1-'i</i:l/)-aP'/v' + <7i£ii)2)

in which the area of discovery is inversely proportional to the habitat size V. (a) For the deterministic, continuum state space (no rounding and cti = a2 = 0) the parameter values r = 3, K = 100,a = 0.01 ands = 4.4yieldachaoticattractor.In(b)-(d)appearphaseplotsandtime series of the periodic cycle attractors obtained from the deterministic lattice model (rounding andcri = a2 = 0) with the initial condition H(0) = 100 V, P(0) = 100F and three habitat sizes V. In (b) are 200 displayed lattice points and the attractor is a high-amplitude 4-cycle. In (c) there are 5000 displayed lattice points and the attractor is a low-amplitude 4-cycle. In (d) there are 8 x 108 displayed lattice points and the attractor is a 181-cycle that somewhat resembles the chaotic attractor in (a). In (e) we see the result of adding demographic noise (<T] = <r2 = 0.01)tothecase(c).Thephaseportraitontheleftresemblesthechaoticattractor in (a). The time series on the right contains episodes that resemble the chaotic time series in (a) interspersed with episodes that resemble the low-amplitude, 4-cycle lattice attractor in (c). (Reprinted with permission from S. M. Henson, R. F. Costantino, J. M. Cushing, R. A. Desharnais, B. Dennis, and Aaron A. King, Lattice effects observed in chaotic dynamics of experimental populations, Science 294 (2001), 602-605. Copyright 2001 American Association for the Advancement of Science.)

from manifesting itself because of the lattice constraint. The particular mix of lattice and continuous dynamics that occurs is a function of several balancing factors, including the lattice spacing, the magnitude of the stochastic perturbations, the characteristics of the attractors, the "strength" of their stability, and the influences of unstable sets (such as saddle equilibria and cycles).

These points are illustrated in Figs. 5.11 and 5.12 for the lattice versions of the Ricker and the modified Nicholson-Bailey host-parasitoid models.

Noise added to lattice LPA models helps to "reveal" the underlying deterministic attractors in the route-to-chaos experiment. Figure 5.13 illustrates this for the attractor in chaos treatment Cpa = 0.35. The plots in

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