The Unit Circle in the Complex Plane y

FIGURE 1.2 I Complex numbers have the form k = a + bi, where a and b are real numbers and i satisfies i2 = -1 (and is sometimes written i = One can also write a complex number in the form k = reie and represent it geometrically as in this figure. The real number r is the "magnitude" of k (i.e., r = \k\ = yja2 + b2). The real number d is the "polar angle." Using this geometric representation, we see that the stability criterion for a linear system is that all eigenvalues lie inside the unit circle defined by r = 1. At a bifurcation point, the linearization has an eigenvalue on the unit circle and, therefore, of the form X = e'e.

Suppose an equilibrium loses stability because an eigenvalue k (actually, a complex conjugate pair of eigenvalues) crosses the unit circle at a point other than +1 or —1. In this event, what typically occurs is the creation in state space of an invariant set which has the form of a closed, one-dimensional loop [76,192]. Near the bifurcation point this "invariant loop" is nearly elliptic in shape, but further away it can become considerably distorted. If the loop is an attractor, then the final state of nearby orbits depends on the dynamics that take place on the invariant loop itself.

One possibility is the existence on the loop of an orbit whose limit set is the entire invariant loop. Such a "quasiperiodic" orbit moves around the loop, never quite repeating and in the process coming arbitrarily close to every point on the loop. In this case, the entire invariant loop is an attractor. The complicated oscillatory dynamic resulting from an "invariant loop" bifurcation7 is not chaotic, however. It does not possess sensitivity to initial conditions.

7 Sometime called a "discrete Hopf" bifurcation or a "Neimark-Sacker" bifurcation [55,133, 154],

1.2 | Bifurcations and Chaos We can use the two-dimensional system Jt+1 =bA,exp{-c1At)

At+X = (1 - ¿¿j)/texp(—c2/i) + (1 - (ia)At to illustrate invariant loop bifurcations and quasiperiodic oscillations. We can view this system of difference equations as a generalization of the Ricker model (1.2). In the Ricker model all individuals are treated as identical and the population is described by a single state variable, the total population abundance. The model (1.6), on the other hand, distinguishes two types of individuals belonging to two distinct life-cycle stages, an immature juvenile stage J and a reproductive adult stage A. In addition this model, unlike the Ricker model, allows for the overlapping of generations, since a fraction 1 - /¿a of the adults survive from one census time to the next. The unit of time is that of the maturation period so that no juvenile remains a juvenile longer than one unit of time. The exponential terms model the effects of population density on vital rates. The term bexp{-CiA) is the per unit time production of juveniles per adult (incorporating fecundity and survival) in the presence of A adults. The term (1 - fij) exp(-c2/) equals the fraction of juveniles that survive and mature to adulthood in a unit of time (in the presence of / juveniles).

Figure 1.3 shows a bifurcation diagram generated by the system of equations (1.6). Attractors are plotted against the parameter b, while the other parameters remain fixed at certain assigned values. As in the Ricker model, when b increases from 0, a transcritical bifurcation occurs with an accompanying exchange of stability between the extinction equilibrium (/, A) = (0,0) and a survival equilibrium with positive components for both juveniles J and adults A. This positive equilibrium destabilizes with further increase in b and an invariant loop bifurcation occurs (at approximately b = 7.6). The resulting oscillations, although nearly periodic, never exacdy repeat. In state space they trace out the loop shown in Fig. 1.3 (i.e., the orbit comes arbitrarily close to every point on the loop).

Another possibility for the dynamics on an invariant loop is the existence of a periodic cycle that attracts all orbits on the loop. This situation is called "period locking." If, in addition all nearby orbits approach the loop (and therefore they approach the cycle on the loop), then this cycle is an attractor. In state space, the attractor consists of finitely many points lying on the loop, which is now "invisible." Typically, parameter intervals on which such period locking occurs are interspersed with intervals of quasi-periodic dynamics, forming period-locking windows in the bifurcation diagram. See Fig. 1.3.

The stable loops that result from an invariant loop bifurcation can, upon further changes in the model parameter, lose their stability and

FIGURE 1.3 I The top plot shows a bifurcation diagram for the juvenile-adult model (1.6) with parameter values Ci = 0.01, c2 = 0.10, ¡ij = 0, /ia = 0.90.AsinFig. 1.1, points from the attractor are plotted above the value of b. An invariant loop bifurcation occurs at approximately b = 7.6. The middle graph shows time series plots of the attractors before and after the bifurcation. In the lower graph these attractors are shown in state space.

FIGURE 1.3 I The top plot shows a bifurcation diagram for the juvenile-adult model (1.6) with parameter values Ci = 0.01, c2 = 0.10, ¡ij = 0, /ia = 0.90.AsinFig. 1.1, points from the attractor are plotted above the value of b. An invariant loop bifurcation occurs at approximately b = 7.6. The middle graph shows time series plots of the attractors before and after the bifurcation. In the lower graph these attractors are shown in state space.

cause new bifurcations to occur. Unlike the one-dimensional case where there is a typical bifurcation sequence (period doubling), in higher dimensional systems there can be a variety of different types of bifurcation sequences and cascades that eventually result in chaos. Researchers have classified and studied several types of bifurcation sequences, but a complete catalog has yet to be made. One possibility is for an invariant loop attractor to bifurcate into a double loop, in a kind of period-doubling bifurcation. In other cases the loop can become twisted and convoluted, even break into separate pieces as the parameter changes. A period-locking window might "open" and the resulting cycles undergo a period-doubling sequence of bifurcations, or even invariant loop bifurcations, toward chaos. Chaotic attractors frequently occur abruptly upon the closing of a period-locking window, in what are termed "crises."

Thus, in progressing from one to just two dimensions we can encounter a considerable increase in dynamic complexity.

From a bifurcation theory point of view, chaos is often embedded within parameter regions that include a complicated variety of other dynamic possibilities (such as periodic and quasiperiodic cycles). Moreover, as attractors destabilize across a bifurcation diagram they often do not disappear, but may survive as unstable invariant sets or leave behind their influence in the form of transient dynamics (i.e., the temporal route that orbits take to the attractor). In this way, cycles, quasiperiodic orbits and even chaotic sets can leave their mark on the dynamics of a system even when they are unstable or only present for nearby parameter values. In such a regime it is difficult, and may make little sense, to relate a population's dynamic to a specific type of attractor. Parameter estimates come with confidence intervals that are likely to incorporate a range of different types of dynamic characteristics. Under these circumstances, an attempt to identify and label a particular time series of data as chaotic becomes problematic, even in a deterministic setting. Such an attempt is made even more difficult in the presence of stochasticity

We will need to deal with stochasticity in some detail in the following chapters. For now, we only point out that random disturbances, applied to an orbit during its journey toward an attractor, induce continual transient behavior and even allow for visits to regions of state space far from the attractor. As a result, a population's dynamics may not be dominated by a deterministic attractor. Instead the dynamics might involve a mix of characteristics — deriving from transients and even unstable invariant sets, in addition to attractors. This is particularly likely when the deterministic component of the dynamics is complicated, involving multiple, quasiperiodic, or chaotic attractors. From this point of view, even if deterministic chaos plays a role in a population's dynamics, it is unlikely to be the sole player.

Perhaps the most significant punch line resulting from nonlinearity is the potential for a complicated array of complex dynamics, in which perturbations in state variables and parameters can lead to unusual and perhaps unexpected dynamic consequences.

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