Wada basins and qualitative unpredictability in ecological models: a graphical interpretation

Abstract

The capability of ecological models to make precise predictions was questioned with the discovery of chaos. Here it is shown that an alternative form of unpredictability is associated with some nonlinear models. The notion of a Wada basin, in which three or more basins share complexly interdigitated boundaries, represents this new form of unpredictability. It signifies that a single point seemingly balanced between two basins of attraction may in fact just as easily travel to a third, seemingly unconnected, basin. A circle map approximation to coupled predator prey pairs is used to demonstrate certain qualitative properties associated with the equilibria within the basins. Most important is the demonstration that the ultimate basin in which a trajectory comes to lie may not be confidently predicted from detailed knowledge of the point of initiation. Furthermore, when Wada basin boundaries become large, a small amount of stochastic forcing may create chaos-like behavior (though not formally chaotic) in a system that has, mathematically, only stable equilbria. The generation of Wada basins is discussed, using piecewise linear maps.

Dynamic Landscapes, Stability and Ecological Modeling

“…In the neighborhood of a fold bifurcation the forcing term (F(x,ẋ, t)) can produce system attractors with fractal basins of attraction, causing the system to manifest unpredictable behavior (see Thompson, 1989;Thompson and Soliman, 1990;Soliman and Thompson, 1992). Such complex basin-boundary interactions have been suggested as a source of unpredictability in ecosystems as well (Vandermeer and Yodzis, 1999;Vandermeer, 2004).…”

Ascertaining when a basin is Wada: the merging method

“…Therefore, in this context, Wada boundaries are usually referred to as those that separate three or more basins at a time, but the basins need not to be connected. Since the earliest references to the Wada property in dynamical systems, many authors claim that the boundaries have the Wada property for disconnected basins [7][8][9][10][11][12]. In this work, we adopt this latter definition: Wada boundaries are those that separate three or more basins, no matter whether the basins are connected or not.…”

“…Despite our primary intuition, Wada basins are a common feature appearing in many dynamical systems. Since its first report, Wada basins have been found in open Hamiltonian systems [10], ecological models [11], delayed differential equations [12], hydrodynamical systems [13], and many engineering problems [14][15][16]. This is possible because Wada bound-aries are related to iterative processes and fractal structures, which are a common feature in the basins of nonlinear dynamical systems [17].…”

Consequential classes of resources: Subtle global bifurcation with dramatic ecological consequences in a simple population model

“…Discussion: A considerable literature focuses on issues such as alternative basins of attraction (May, 1977;Vandermeer and Yodzis, 1999) or intermingled basin boundaries (Neubert, 1997;Huisman and Weissing, 2001;Vandermeer, 2004), scenarios that embody the notion that ecosystems may dramatically and Table 1, except r =0.12. The thick red line indicates the separatrix between the oscillatory attractor and the hyperbolic attractor.…”

“…Ecosystems are sometimes, it is argued, on the edge of a metaphorical cliff and the slightest perturbation might result in dramatic transformation. The theoretical formulations leading to this image are varied: the simple indeterminate competition model has a saddle point perched on a separatrix that divides the basins that dictate competitive exclusion of one or the other species (May, 1977); the boundaries of basins containing chaotic attractors may collide, creating abrupt and pronounced changes in the nature of combined attractors (Grebogi et al, 1982(Grebogi et al, , 1983Vandermeer and Yodzis, 1999); with three or more basins the basins may fractally intersect such that final outcomes are impossible to predict no matter how detailed the knowledge of initiation points is (Huisman and Weissing, 2001;Neubert, 1997;Vandermeer, 2004;Hastings, 1993). These and other examples provide sensible models of unexpected change, perhaps what we may be forced to admit are not so unexpected in complex ecosystems.…”