War: The dynamics of vicious civilizations.

The dynamics of ‘‘vicious,’’ continuously growing civilizations ~domains!, which engage in ‘‘war’’ whenever two domains meet, is investigated. In the war event, the smaller domain is annihilated, while the larger domain is reduced in size by a fraction e of the casualties of the loser. Here e quantifies the fairness of the war, with e51 corresponding to a fair war with equal casualties on both sides and e50 corresponding to a completely unfair war where the winner suffers no casualties. In the heterogeneous version of the model, evolution begins from a specified initial distribution of domains, while in the homogeneous system, there is a continuous and spatially uniform input of point domains, in addition to the growth and warfare. For the heterogeneous case, the rate equations are derived and solved and comparisons with numerical simulations are made. An exact solution is also derived for the case of equal-size domains in one dimension. The heterogeneous system is found to coarsen, with the typical cluster size growing linearly in time t and the number density of domains decreases as 1/t . For the homogeneous system, two different long-time behaviors arise as a function of e. When 1 2,e<1 ~relatively fair wars!, a steady state arises that is characterized by egalitarian competition between domains of comparable size. In the limiting case of e51, rate equations that simultaneously account for the distribution of domains and that of the intervening gaps are derived and solved. The steady state is characterized by domains whose age is typically much larger than their size. When 0<e, 1 2 ~unfair wars!, a few ‘‘superpowers’’ ultimately dominate. Simulations indicate that this coarsening process is characterized by power-law temporal behavior, with nonuniversal e-dependent exponents. Some of these features are captured by a deterministic self-similar model, for which the characteristic exponents can be computed easily. The transition point e5ec5 1 2 is characterized by slower than power-law coarsening. @S1063-651X~96!09508-6#