  We investigate the classical two species ODE and PDE Lotka-Volterra
competition models, where one of the competitors could potentially go extinct
in finite time. We show that in this setting, classical theories and intuitions
do not hold, and various counter intuitive dynamics are possible. In
particular, the weaker competitor could avoid competitive exclusion, and the
slower diffuser may not win. Numerical simulations are performed to verify our
analytical findings.
