Cascading Thresholds to Heteroclinicity in an Ecosystem Model

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CASCADING THRESHOLDS TO HETEROCLINICITY IN AN ECOSYSTEM MODEL

J. VANDERMEER Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI, USA.

ABSTRACT Using a model of three consumer/resource pairs coupled in an ecosystem model, previous work has demonstrated a particular dynamical behavior in which a switch from a chaotic attractor to a heteroclinic cycle is the underlying mechanism that causes extinction of some components of the system. The complex bifurcation sequence from chaos to heteroclinic cycle in this model is described using numerical methods. Evidence is presented suggesting a structure of nested manifolds that gradually reverse stabilities as the bifurcation parameter changes. Keywords: chaos, consumer/resource, heteroclinic cycles.

1 INTRODUCTION Forging generalizations about dynamic behavior of model ecosystems has become far more complicated in recent years as complex nonlinearities are routinely added to models in search of more realism. One feature that has recently emerged as a generalized pattern is the existence of complicated heteroclinic cycles when several consumer resource systems are coupled together. Since consumption is the basis of energy transfer in ecosystems, such behavior is likely to become more important as further levels of complexity are incorporated into ecosystem models. Of particular importance is the way in which some systems generate heteroclinic oscillations that approach the origin, or at least one of the axes of the system, which is to say, the limiting case is extinction of one or more elements in the system. This is precisely the behavior observed in several recent cases of consumer/resource-based ecosystem models [1, 2]. The existence of heteroclinic cycles in ecological models is well known [3–7] and, as noted by Hofbauer and Sigmund [3, 4], are likely to appear more frequently the larger the ecosystem model. In one particular case, in a six-dimensional model, a seemingly complex bifurcation sequence from a constrained chaotic attractor to a heteroclinic attractor has been described [2]. The analysis of that situation recognized the heteroclinic nature of the behavior and also pointed out some unusual additional structural features that were not necessarily evident. Closer examination of this behavior reveals a threshold cascade of chaotic attractors with increasingly lower boundaries. The purpose of the present work is to describe in detail the nature of this cascade. While this paper treats only this special case, in light of the growing complexity of ecosystem models and the consequent inevitability of heteroclinic cycles to emerge [4], I fully expect that this sort of structure will be seen in other systems also.

2 THE MODEL The situation under study is a system of ordinary differential equations meant to represent the classical case of two competitors subject to an invading competitor, where the competition formulation is based on resource competition such that the dual two-dimensional systems {P1, x1} and {P2, x2} (where P stands for predator or consumer and x stands for resource) are subject to the invasion of the system P{P3, x3}. The question then is one of what parameter values generate what dynamical behavior patterns. The basic setup is illustrated in Fig. 1. Previously it has been established that a complex set of behavioral repertoires are possible with this system [2]. Here I am concerned with one particular

P1 P 23P

X12X X 3

Figure 1: Diagrammatic representation of the system under study. The xs represent prey items, the Ps represent predators, the arrowhead indicates a positive effect, and the small circle indicates a negative effect. x1, x2, and x3 are meant to be competitors on a resource gradient, such that x2 is intermediate and thus receives competitive pressure from both sides on the gradient. parameter setting, in which the appearance of heteroclinic cycles is common and tied to the important ecological force of extinction. The model is based on the standard Lotka–Volterra framework, as follows:

dPi/dt =−mPi + aPixiI , (1a)

dx1/dt = x1[1 − x1 − 1.1x2 − ax1P11], (1b)

dx2/dt = x2[1 − x2 − 1.1(x1 + x3) − aP22], (1c)

dx3/dt = x3[1 − x3 − 1.1x2 − P33], (1d)

1 = 1/(1 + bxi), (1e) where Pi are predators and xi are their prey (i = 1, 2, 3), m is the death rate of the predators, a is the consumption rate of the predators, and b is the parameter of the functional response. The ecological rationale for this formulation is that the three competitors (xis) are arranged on a gradient in which x2 is intermediate between x1 and x3, and the three predators are specialist on each of the three species (Fig. 1). The constants 1.1 are in the position of the classic competition coefﬁcients of ecology and ensure that x2 will be driven to extinction in the absence of control from the predators. Here I examine the nature of the approach to the heteroclinic cycle for this particular system with m = 0.8 and b = 2.

3 RESULTS The bifurcation pattern of the system as a function of the parameter a is shown in Fig. 2, at two magniﬁcations. When a = 7.4 there is a simple chaotic form to the solution, and when a = 6.6 the system decomposes through a stable heteroclinic cycle (stable in the sense that the cycle itself includes saddles at zero for each of the six elements in the system, such that the system is globally unstable and must disappear). This is precisely the pattern reported elsewhere as a bifurcation point that leads to extinction [1, 2], sometimes a rather sudden drop to extinction through the sudden appearance of the heteroclinic cycle focused on the origin. In trying to further elaborate the details of this bifurcation, a curious pattern is evident in Fig. 2. There is a stepwise behavior in which the chaotic attractor jumps to a different state in a noncontinuous fashion, precisely seven times. It appears that there are seven chaotic attractors located on seven

(a)

ln(X1)

(b)

ln(X1)

Figure 2: Bifurcation diagrams of the min[ln(x1)] versus the parameter a. (a) Long view illustrating the overall pattern of simple chaos at large a and heteroclinic cycles at low a. (b) Close view; the dotted box in (a) illustrates the discontinuous step nature of the bifurcation sequence. different manifolds, all but one of which are repelling. Reducing the parameter a causes a reversal in stability of one of the manifolds, thus causing threshold jumps to different manifolds seven times before the manifold with the heteroclinic cycle is encountered. Some insight into the structure of these multiple manifolds can be gained by constructing the ‘valley-to-valley’ map for one of the variables. In Fig. 3 two such maps are illustrated, one for a = 6.74, the other for a = 6.9. With reference to Fig. 2, what might be expected is that at a = 6.9, only three of the manifolds will be visible in the valley-to-valley map, while at a = 6.74, all seven manifolds will be visible. The two valley-to-valley maps are shown in Fig. 3, where it is evident that three manifolds are visible in the a = 6.9 version and seven in the a = 6.74 version. Furthermore,

(a) (t+1)] 1 ln[x

ln[x1(t)]

(b) (t+1)] 1 ln[x

ln[x1(t)]

Figure 3: Valley-to-valley maps of ln(x1): (a) with parameter a = 6.74; (b) with parameter a = 6.9. Opposite colored (horizontal) lines (black in (a) and red in (b)) indicate approximate projection to the next valley from local minima.

ln[x1 (t)]

Figure 4: Data from Fig. 3a superimposed on the upper part of data from Fig. 3b, illustrating the changing nature of the slope of the lower boundary of the valley-to-valley map (emphasized by eye-ﬁtted red (grey) and black lines connecting lower boundaries).

as indicated by small oppositely colored horizontal lines in each ﬁgure, the minimum of the valley- to-valley map for the case a = 6.9 almost precisely corresponds to the intersection of the next lower intersection of the presumed iterative function. Contrarily, in all of the local minima for the case a = 7.74, the horizontal line intersects the 45° line at a point lower than the next intersection of the function. Since the horizontal lines represent the potential iterations at the lowest local point in the valley-to-valley maps, if they intersect the 45◦ line below the point where the next functional intersection is located, they will project to a lower value. As can be seen (Fig. 3b) all but the lowest horizontal lines intersect below that critical intersection (the lowest local minimum is unclear since so few points actually descend to the lower manifold). If the two map traces (Fig. 3) are superimposed (Fig. 4), it is clear that the ﬁrst (color red) is angled more acutely than the second (black points), thus generating the basic pattern. The overall picture, then, is of a manifold structure that results in a return map with peaks and valleys around the 45◦ line. The entire map is oriented at an angle to the 45◦ line, and this angle is determined by the parameter a. As the angle becomes more acute (the red points versus the black points in Fig. 4), more of the map is incorporated into the chaotic attractor, until a threshold is reached at which point the manifold containing the heteroclinic cycle is exposed. The manifold structure can be thought of as a collection of seven cones nested together much like the famous Russian dolls. As the parameter a changes, the cones become more exposed and the chaotic attractor encompasses each new cone as it emerges from the stack. This hypothesized structure is illustrated in Fig. 5, for three nested cones.

(a)

(b)

(c)

Figure 5: Diagrammatic representation of the suspected nature of the nested manifolds, changing as the parameter a changes: (a) only inner cone is attractive; (b) the inner two cones are attractive; (c) all three cones are attractive.

4 DISCUSSION Complicated ecological models are likely to result in complicated dynamical behavior and it is the task of theoretical ecologists to understand this behavior. In the case at hand, three consumer/resource pairs coupled in a six-dimensional framework, extinction events are the consequence of a set of coupled consumer/resource oscillators that generate a heteroclinic cycle that is focused on the origin, meaning that the system will collapse eventually. This sort of collapse is quite different from the classic models in which an approach to zero (i.e. extinction) is hyperbolic, with one or more of the species asymptotically approaching zero. Here, there are oscillations that approach zero in the limit. At a qualitative level all that appears to be happening is that a chaotic attractor suddenly switches to a heteroclinic cycle as a parameter is changed (e.g. examine Fig. 2a with foggy lenses). However, closer examination reveals that complex dynamics are involved, with interlocking manifolds gradually becoming exposed.

The presentation here is of a single numerical study of a particular parameter space of a particular model. But the structure of nested manifolds emerges as a previously unreported element of consumer/resource theory. As further complex ecosystem models are studied, I speculate that this sort of structure will emerge repeatedly, and questions will arise as to how many manifolds are hidden in the collapsed region within the original chaotic attractor and to what extent knowing the answer to this question might provide insight into the qualitative behavior of the model system.

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