Existence of Compromise States in the Competition of Alternative Opponents in the Presence of External Support

Models of dynamical conflict systems (DCS) are studied for a pair of alternative opponents competing for presence in different regions of the resource space. Our new result establishes the existence of compromise states (with simultaneous and constant presence of both opponents in at least one region) under the external support of each opponent. It is known that in the case of purely conflict interaction without any external support, these states are impossible because only one opponent wins in each region. We show that, in these states, the statuses of the opponents (the values of the probabilities of presence) are determined by the parameters of external support rather than by the initial distributions. We present some computer examples illustrating, in particular, the duration of competition prior to the attainment of a compromise. Our results are applicable, e.g., in the theory of formation of opinions.