In　this　paper,　we　consider　the　following　nonlinear　single　species　diffusive　system　(x)　over　dot(i)(t)　=　x(i)(t)　(b(i)(t)　-　Sigma(li)(k=1)　a(i)k(t)(x(i)(t))beta(ik))　+Sigma　D-n(j=1)ij(t)(x(j)(t)-x(i)(t)),　(i,j=1,2,...,n),　where　b(i)(t),a(ik)(t),D-ij(t),i,j　=　1,2,...,n;k　=　1,2,...,l(i)　are　all　continuous　omega-periodic　functions,　and　beta(ik),　i　=　1,　2,...,n;k　=　1,　2,...,l(i)　are　positive　constants.　Sufficient　conditions　which　guarantee　the　permanence,　extinction　and　existence　of　a　unique　globally　attractive　positive　omega-periodic　solution　are　obtained.　Examples　together　with　their　numeric　simulations　show　the　feasibility　of　the　main　results.