Decisions, patterns, and oscillations in nonlinear competitive systems with applications to Volterra-Lotka systems

Author(s): Grossberg, S. |

Year: 1978

Citation: Journal of Theoretical Biology, 73, 101-130

Abstract: This paper describes new properties of competitive systems which arise in population biology, ecology, psychophysiology, and developmental biology. These properties yield a global method for analyzing the geometric design and qualitative behavior, e.g. limits or oscillations, of competitive systems. The method explicates a main theme about competitive systems: who is winning the competition? The systems can undergo a complicated series of discrete decisions, or jumps, whose structure can, for example, yield global pattern formation or sustained oscillations, as in the voting paradox. The method illustrates how a parallel continuous system can beanalyzed in terms of discrete serial operations, but notes that the nextoperation can be predicted only from the parallel interactions. It isshown that binary approximations to sigmoid signals in nonlinear net-works are not valid in general, It is also shown how a temporal series ofnested dynamic boundaries can be induced by purely nonlinear interac-tive effects. These boundaries restrict the fluctuations of population sizesor activities to ever finer intervals. The method can be used whereLyapunov methods fail and often obviates the need for local stabilityanalysis. The paper also strengthens and corrects some previous resultson the voting paradox.

Topics: Mathematical Foundations of Neural Networks, Applications: Biological Classification, Models: Other,

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Cross References

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