Theoretical analysis and simulations of the generalized Lotka-Volterra model

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables $w_i,$ $i=1,\dots ,N.\mathrm{}$ The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments), and a time dependent saturation term (due to the finite size of the economy). The $w_i\mathrm{’}\mathrm{s}$ turn out to exhibit a power-law distribution of the form $P\left(w\right)\mathrm{}\sim w^{-1-\alpha }.$ It is shown analytically that the exponent $\alpha$ can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely, between the resources available to the poorest and those available to the richest in a given society. The value of $\alpha$ is found to be insensitive to variations in the saturation term, which represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with $1<\mathrm{}\alpha <2.\mathrm{}$