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Use streamline_plot:

streamline_plot([u*(1-u-0.2*v),v*(1-v-0.5*u)],[u,0,2],[v,0,2])

Use streamline_plot:

streamline_plot([u*(1-u-0.2*v),v*(1-v-0.5*u)],[u,0,2],[v,0,2])
streamline_plot([u*(1-u-0.2*v),v*(1-v-0.5*u)], [u,0,2], [v,0,2])