  In this work, we present a reduced-order model for a nonlinear
cross-diffusion problem from population dynamics, for the
Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The
finite-difference discretization of the SKT equation in space results in a
system of linear--quadratic ordinary differential equations (ODEs). The reduced
order model (ROM) has the same linear-quadratic structure as the full order
model (FOM). Using the linear-quadratic structure of the ROM, the reduced-order
solutions are computed independent of the full solutions with the proper
orthogonal decomposition (POD). The computation of the reduced solutions is
further accelerated by applying tensorial POD. The formation of the patterns of
the SKT equation consists of a fast transient phase and a long steady-state
phase. Reduced order solutions are computed by separating the time, into
two-time intervals. In numerical experiments, we show for one-and
two-dimensional SKT equations with pattern formation, the reduced-order
solutions obtained in the time-windowed form, i.e., principal decomposition
framework (P-POD), are more accurate than the global POD solutions (G-POD)
obtained in the whole time interval. Furthermore, we show the decrease of the
entropy numerically by the reduced solutions, which is important for the global
existence of nonlinear cross-diffusion equations such as the SKT equation.
