  Ecological and evolutionary processes show various population dynamics
depending on internal interactions and environmental changes. While crucial in
predicting biological processes, discovering general relations for such
nonlinear dynamics has remained a challenge. Here, we derive a universal
information-theoretical constraint on a broad class of nonlinear dynamical
systems represented as population dynamics. The constraint is interpreted as a
generalization of Fisher's fundamental theorem of natural selection.
Furthermore, the constraint indicates nontrivial bounds for the speed of
critical relaxation around bifurcation points, which we argue are universally
determined only by the type of bifurcation. Our theory is verified for an
evolutionary model and an epidemiological model, which exhibit the
transcritical bifurcation, as well as for an ecological model, which undergoes
limit-cycle oscillation. This work paves a way to predict biological dynamics
in light of information theory, by providing fundamental relations in
nonequilibrium statistical mechanics of nonlinear systems.
