Eigenvalue problem with indefinite weight function and advection term and its applications

This paper is absorbed in the eigenvalue problem with indefinite weight function and advection term. We establish some results (one can see Theorem 1.2 for more details) to answer completely the questions concerning the positive eigenfunctions on the following eigenvalue problem with the flux boundary conditions: { − d ϕ x x + a ϕ x = λ m ( x ) ϕ , a m p ; 0 > x > L , − d ϕ x ( 0 ) + a ϕ ( 0 ) = − b u a ϕ ( 0 ) , a m p ; d ϕ x ( L ) − a ϕ ( L ) = − b d a ϕ ( L ) . {equation*} {cases} -dφ ₓₓ+aφ _x=λ m(x)φ ,\,&0>x>L,\\ -dφ _x(0)+aφ (0)=-b_uaφ (0),\;&dφ _x(L)-aφ (L)=-b_daφ (L). {cases} {equation*} Furthermore, we give two examples to show the applications of Theorem 1.2, i.e., by employing eigenvalue theory, the theory of monotone dynamical system and the method of upper-lower solution, we obtained the global dynamic behavior of a single population model (3.4) and two-species competitive model (3.6) when the boundary condition parameters vary in all different situations. It is worth noting that it is not trivial to derive these conclusions in this paper.