|This dissertation is devoted to the study of the asymptotic dynamics, in particular, coexistence states and convergence of nonnegative solutions to the competition systems with immigration and time periodic dependence. This includes
two species competition systems of ordinary differential equations with positive sources and periodic dependence,
two species competition systems of nonlocal evolution equations with inhomogeneous boundary conditions and
cancer models with radiation treatment. The main results of the dissertation consist of the following parts.
Firstly, we look at the Volterra-Lotka competition systems of ordinary differential equations with positive sources. We show that (i) if the competition is weak between two species, a unique stationary solution can be obtained in the time independent case. (ii) As long as the positive sources are large enough, a unique positive stationary solution exists no matter the competition is weak or not in the time independent case. (iii) If the system is time periodic, uniqueness can also be achieved under weak competition.
Secondly, we obtain the existence and uniqueness of continuous coexistence states of competition systems with nonlocal dispersal. It is shown that inhomogeneous Neumann condition or/and Dirichlet condition guarantee not only the persistence of the two species, but also the continuous coexistence when the competition is weak between two species. Once again, it can also be shown that some large enough inhomogeneous boundary conditions allow the continuous coexistence even if the competition between two species is strong. A sufficient condition is also obtained for such continuous coexistence to be unique. In particular, this condition always holds true when the coefficients that account for the competition between the species are both small.
Thirdly, we investigate the cancer model with periodic radiation treatment. Normal cell, tumor cell, radiated normal cell and radiated tumor cell are being considered in the model. We have found that (i) in the absence of cancer cells, if the trivial solution is a stable solution, then it is globally stable. If the trivial solution is an unstable solution, then a unique periodic positive solution exists and it is globally asymptotically stable. (ii) Any solution of the four-species cancer model system converges to a time periodic nonnegative solution. Moreover, if the competition coefficients between unaffected normal and cancer cells, as well as the recombining rates for radiated normal and tumor cells are sufficiently small, the uniqueness of such periodic solution can be obtained.