Winnerless competition principle and prediction of the transient dynamics in a Lotka–Volterra model

Abstract

Predicting the evolution of multispecies ecological systems is an intriguing problem. A sufficiently complex model with the necessary predicting power requires solutions that are structurally stable. Small variations of the system parameters should not qualitatively perturb its solutions. When one is interested in just asymptotic results of evolution (as time goes to infinity), then the problem has a straightforward mathematical image involving simple attractors (fixed points or limit cycles) of a dynamical system. However, for an accurate prediction of evolution, the analysis of transient solutions is critical. In this paper, in the framework of the traditional Lotka-Volterra model (generalized in some sense), we show that the transient solution representing multispecies sequential competition can be reproducible and predictable with high probability.

Dopamine, Affordance and Active Inference

“…The actual form of the model is both mathematically quite simple and biologically plausible: we have just placed a slow pattern generator on top of a fast pattern generator and have then mapped to sensory consequences. Both pattern generators have the same universal form and show autonomous, metastable dynamics of the sort seen in the real brain [87].…”

“…In other words, the agent expects the world to change continuously, with an itinerant (wandering) trajectory, visiting different states in succession. These reflect prior beliefs about forthcoming sensory events, and are encoded by differential equations that embody metastable dynamics; e.g., winnerless competition; [87], [88]. The resulting attractors can be thought of as central pattern generators that can be nested hierarchically at different time scales [89] to produce exteroceptive and proprioceptive (i.e., sensorimotor) predictions.…”

“…This means that the orbit or trajectory in the four dimensional attractor space maps to a two-dimensional trajectory, which cycles through the four locations in a fixed order. We use this trajectory to generate forces that elicit pointing movements: See [87] and [23] for details.…”

Hierarchical Heteroclinics in Dynamical Model of Cognitive Processes: Chunking

“…Based on the results of [Rabinovich et al, 2014], in this paper we proved that in the phase space of the model describing chunking there exists a new mathematical object -heteroclinic of heteroclinic sequence (HoHS). Evidently, the HoHS is an adequate image of the chunking processing, and it plays a similar role in the hierarchical mental activities as the stable heteroclinic channel (SHC) for the nonhierarchical robust transient cognitive dynamics [Afraimovich et al, 2004b;Rabinovich et al, 2012].…”

“…These are exactly the equalities in [Afraimovich et al, 2004b] where they have been shown to guarantee the existence of heteroclinic sequences. We illustrate a heteroclinic connection between consecutive saddle points in Fig.…”