We study the properties of an ODE description of schools of fish (B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys. 128(1/2) (2007), 535–568.) and how they change in the presence of a random acceleration. The model can be reduced to one ODE for the direction of the velocity of a generic fish and another ODE for its speed. These equations contain the mean speed and a Kuramoto order parameter for the phases of the fish velocities. In this paper, we give a complete qualitative analysis of the system for large number of particles. We show that the stationary solutions of the ODEs consist of an incoherent unstable solution with and a globally stable solution with and a constant . In the latter solution, all the fish move uniformly in the same direction with and the direction of motion determined by the initial configuration of the school. This is called the “migratory solution”. In the second part of the paper, the directional headings of the particles are perturbed, in two distinct ways, and the speeds accelerated in order to obtain two distinct classes of non-stationary, complex solutions. We show that the perturbed systems have similar behavior as the unperturbed one, and derive the resulting constant value of the average speed, verifying the numerical observations. Finally, we show that the system exhibits a similar bifurcation to that in Vicsek and Czirok (T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75(6) (Aug 1995), 1226–1229.) between phases of synchronization and disorder. Either increasing the variance of the Brownian angular noise, or decreasing the turning rate, or coupling between the particles, cause a similar phase transition. These perturbed models represent a more realistic view of schools of fish found in nature. We apply the theory to compute the order parameter for a simulation of the Chile-Peru anchovy fishery.
The article offers a new perspective on the economic and business history of the German Democratic Republic (GDR). It argues that the historiographical emphasis on the “failure” of the East German planned economy should be replaced by an analysis of those factors which allowed the GDR economy to exist as long as it did and the effects they had on the system as a whole. To do so, the article first provides an overview of the structural conditions that characterized the East German economy and looks at the position and role of the publically-owned companies within the “Budgetverwaltungswirtschaft” (budget administration economy). To determine in which ways the East German companies functionally differed from capitalist ones, the article investigates in a second step the characteristics of capitalist economies, specifically the relevance awarded to companies and entrepreneurs as constitutive elements of the market. By distinguishing analytically the distribution of property rights bundles awarded to the different actors in the economy, the article provides a basis for comparing the functions of companies in capitalist and non-capitalist systems without assuming an essential difference between the two types of companies a priori. Finally, the article demonstrates the methodological insight gained from the comparative analysis by drawing on examples from the company Simson.
matter (prepared in
minium oxide (1500 g). Elution was affected with
light petroleum, followed by mixture of light petro-
leum-benzene (70 :30; 50 :50 ; 30 :70), benzene and
benzene-methanol (9 7 :3 ). Fractions (50 ml each)
After crystallisation from methanol, it melted at
2 8 5 - 2 8 6 °C ; reported8 285 - 286 °C.
Calcd: C 78.94 H 10.52,
Found: C 78.76 H 10.53.
The m.p. of the acetate showed no depression upon
admixture with the authentic material.
Calcd: C 77.11 H 10.04,
Found: C 77.06 H 10.10.
, single t.1.c. UV absorbing spot (ch1oroform:methanol 9:l ,
silica gel), displayed molecular formulae (C32H50N203, C33H50N203 and
c ? ~ H ~ ~ N ~ ~ ~ , respectively) suggesting,analogously to sarains 1-3, their belon-
ging to a homologous series. This was confirmed by some common spectral data:
IR = 1650 cm-l; UV = 238 mp; MS, loss of the same fragment C14H22N03. The com-
plex NMR spectra were seldom reproduceable being strongly influenced by tra-
ces of salts or acids and, also, by the concentration of the sample. However,
working on selected hermetically