Noise-Induced Bistable States and Their Mean Switching Time in Foraging Colonies

Abstract

We investigate a type of bistability occurring in population systems where noise not only causes transitions between stable states, but also constructs the states themselves. We focus on the experimentally well-studied system of ants choosing between two food sources to illustrate the essential points, but the ideas are more general. The mean time for switching between the two bistable states of the system is calculated. This suggests a procedure for estimating, in a real system, the critical population size above which bistability ceases to occur.

Motif analysis for small-number effects in chemical reaction dynamics

“…26 As is discussed in Introduction, recent numerical studies support the validity of chemical Fokker-Planck equation up to smaller value of Ω than the range adopted for LNA. 7,31,32 In fact, we also demonstrated that the direct numerical simulations agree quite well with the predictions of chemical Fokker-Planck equation for all motifs marked by "+" or "++" in Fig. 1, where small-number effects are well reproduced by the chemical Fokker-Planck equation, rather than by the LNA.…”

“…[5][6][7] Note that the first two reactions are necessary to satisfy the ergodicity condition because the absence of either reaction gives rise to an absorbing state. Since the total number of molecules N is conserved throughout the above reactions, N is set to be identical to a unitless volume Ω via proper transformation.…”

“…1,2 Even in the case of eukaryotic cells, protein abundance in a fine intra-cellular structure (such as the dendritic spine or organelle) is expected to be quite small. This small molecule number leads to non-negligible fluctuations and a discrete nature of molecular concentrations, which may in turn alter the frequencies of each chemical reaction event; indeed, several salient phenomena induced by the smallness in molecule numbers have been studied, both theoretically [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and experimentally. [17][18][19] Moreover, how such microscopic molecular discreteness can contribute to cellular functions at a larger scale has gathered much interest, and the effect induced by the discreteness is expected to provide a novel concept to understand cellular behaviors and function.…”

“…Specifically, several recent theoretical studies have reported that, in certain chemical reaction systems, the chemical compositions and underlying dynamics can be drastically altered under a small-number condition [3][4][5][6][7][8][9][10][11][12][13][14][15][16]23,24 compared to those expected in the rate equation assuming with a large volume and a large number of molecules. These phenomena are induced by molecular discreteness and its associated stochasticity, designated as "small-number effects" or "discreteness-induced transitions," which are exemplified by the emergence of multi-modality, [3][4][5][6][7][10][11][12] reversal of reaction current, 16 and slow relaxation. 25 In spite of the several examples of small-number effects reported to date, there is no unified understanding of these phenomena.…”

Modes of competition and the fitness of evolved populations

“…For instance, new effective extrema may be formed through the effects of noise [36], as well as the effective location of extrema being moved. In the present case, the effect is more closely related to the situation found in some kinds of stochastic pattern formation [37,38], where the region of stability of patterns can be increased by the stochastic effects giving a new effective condition for stability which differs from that found through a purely deterministic analysis.…”