  We suggest a novel approach to treating symbiotic relations between
biological species or social entities. The main idea is the characterisation of
symbiotic relations of coexisting species through their mutual influence on
their respective carrying capacities, taking into account that this influence
can be quite strong and requires a nonlinear functional framework. We
distinguish three variants of mutual influence, representing the main types of
relations between species: (i) passive symbiosis, when the mutual carrying
capacities are influenced by other species without their direct interactions;
(ii) active symbiosis, when the carrying capacities are transformed by
interacting species; and (iii) mixed symbiosis, when the carrying capacity of
one species is influenced by direct interactions, while that of the other
species is not. The approach allows us to describe all kinds of symbiosis,
mutualism, commensalism, and parasitism within a unified scheme. The case of
two symbiotic species is analysed in detail, demonstrating several dynamical
regimes of coexistence, unbounded growth of both populations, growth of one and
elimination of the other population, convergence to evolutionary stable states,
and everlasting population oscillations. The change of the dynamical regimes
occurs by varying the system parameters characterising the destruction or
creation of the mutual carrying capacities. The regime changes are associated
with several dynamical system bifurcations.
