  We introduce a mathematical model of symbiosis between different species by
taking into account the influence of each species on the carrying capacities of
the others. The modeled entities can pertain to biological and ecological
societies or to social, economic and financial societies. Our model includes
three basic types: symbiosis with direct mutual interactions, symbiosis with
asymmetric interactions, and symbiosis without direct interactions. In all
cases, we provide a complete classification of all admissible dynamical
regimes. The proposed model of symbiosis turned out to be very rich, as it
exhibits four qualitatively different regimes: convergence to stationary
states, unbounded exponential growth, finite-time singularity, and finite-time
death or extinction of species.
