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On the distribution of dominance in populations of social organisms. (English) Zbl 0759.92011
The idea of dominance is of major importance in studies of animal behaviour. The origin of dominance considerations stems from the existence of a “peck right” in interactions between hens. The paper discusses the dynamics of dominance in a population of anonymous organisms like bumble bees in simple situations. It is assumed that three basic rules govern the development of dominance: (i) Individuals continually encounter other individuals. Each pairwise encounter results in a winner and a loser. (ii) A dominant individual is more likely to win. (iii) After an encounter, the dominance of the winner is incremented, while that of the loser is decremented.
In the first part of the paper a stochastic model is described that is based on a random variable \(X(t)\) that takes a value from the dominance space [0,1]. Using the Markov property a basic nonlinear Boltzmann-like integrodifferential equation for the probability distribution of dominance is developed. Two approaches were used to obtain information concerning the solutions of this equation. Hypothesizing that each interaction can slightly change the dominance distribution one solution results in a still nonlinear approximate Fokker-Planck equation with effective convection and diffusion coefficients depending on the overall probability distribution. Furthermore certain qualitative conclusions concerning the behaviour of the solutions are given.
In the second approach a discrete model was constructed. The resulting system of ordinary differential equations was integrated numerically in two cases. One case paralleled the analytic results, showing that dominance cannot be used as an automatic organizing variable for the population. The second numerical example showed in accordance with the analytical part a splitting property caused by the pregiven rule (ii): The population will be splitted into two groups, one with high dominance and one with low. The same result P. Hogeweg and B. Hesper (1983) got in a simulation study under different presumptions.

92D50 Animal behavior
45K05 Integro-partial differential equations
60J99 Markov processes
65R20 Numerical methods for integral equations
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