  In this paper we establish a rigorous gradient flow structure for
one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani
optimal transport geometry. This is achieved by first conditioning the
underlying stochastic process to non-fixation in order to get rid of
singularities on the boundaries, and then studying the conditioned $Q$-process
from a more traditional and variational point of view. In doing so we complete
the work initiated in [Chalub et Al., Gradient flow formulations of discrete
and continuous evolutionary models: a unifying perspective. Acta App Math.,
171(1), 1-50], where the gradient flow was identified only formally. The
approach is based on the Energy Dissipation Inequality and Evolution
Variational Inequality notions of metric gradient flows. Building up on some
convexity of the driving entropy functional, we obtain new contraction
estimates and quantitative long-time convergence towards the stationary
distribution.
