The thesis includes three parts. In the first part, we study a nonlocal evolution equation which describes the seed dispersal of single species and prove the existence, uniqueness and stability of positive steady state solution to this equation. In the second part, we study the principal eigenvalue problem and given a sufficient condition that ensures the existence of coexistence state to a nonlocal evolution system. Then we also consider a competition model which involves two similar species. The existence of coexistence states and their stability are investigated. In the third part, we establish the existence, uniqueness and continuous dependence on initial values for the solutions to a nonlocal phase field system. We also discuss the asymptotic behavior of the solution and prove the global boundedness of the solutions.
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