The qualitative behavior of coupled predator-prey oscillations as deduced from simple circle maps

Abstract

Qualitative patterns of the non-equilibrium behavior of coupled consumer/resource systems, formulated as a standard circle map, are explored. Patterns of demographic locking, reflected as a "devil's staircase", are observed in simulations of the MacArthur consumer/resource equations. As coupling strength increases the steps in the staircase become broader, eventually overlapping to form a chaotic pattern. In addition to the standard form of chaos formation from overlapping Arnold tongues, a pattern of period doubling in the non-invertible standard circle map corresponds to the pattern observed in the classic consumer/resource equations.

Models of coupled population oscillators using 1-D maps

“…At a general ecological level we report behaviors similar to that reported for coupled consumer resource systems (Vandermeer, 1993;1994), with either entrainment or phase reversal, followed eventually by chaos, although the exact approach to chaos is substantially different in the maps than in differential equations. But the biological conclusions remain substantially congruent whether dealing with differential equations or 1D maps, the populations tend to be locked together at low levels of coupling, becoming chaotic as the coupling intensity increases.…”

“…Another example might be two consumer resource systems weakly connected through the common consumption of the consumers (e.g. Vandermeer, l993;1994). As evolution of generalization procedes (i.e.…”

“…In phenomena as diverse as simple coupled pendula and neural signals in slime molds (Sachsenmaier, Remy and Plattner-Schobel, 1972) complicated chaotic patterns can result from linking oscillators. While biological populations frequently illustrate oscillatory tendencies (Huffaker, 1958;May, 1981;Naeem, personal communication), only recently has the behavior of population models been examined from the point of view of coupled oscillators (Vandermeer, 1993(Vandermeer, , 1994(Vandermeer, , 1996, although other biological oscillations have been so examined, ranging from cardiac cells (Torre, 1976) to firefly flashes (Buck and Buck, 1976).…”

New forms of structure in ecosystems revealed with the Kuramoto model

“…Coupling oscillator sets has already gained traction in the ecological literature [9][10][11][12][13][14][15][16][17][18][19][20], mostly from a theoretical perspective, including the potential to generate chaotic behaviour [20,21]. However, a perhaps more basic question stimulated by the pioneering work of Arthur Winfree [22] is what will be the patterns of synchrony within the collection of coupled oscillators?…”

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

“…By analyzing the transformed parameter Θ, it is possible to study the qualitative behavior of the underlying system (Vandermeer, 1994;Vandermeer et al, 2001) much as is done in the study of physical oscillators (Bak, 1986;Bohr et al, 1984;Cvitanovic et al, 1990;Jensen et al, 1984). It is a simple matter to describe the general behavior of the overall system simply by knowing the value of Θ.…”