We are concerned with the coexistence states of the diffusive Lotka-Volterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the one-dimensional problem, we employ the generalized Jacobi elliptic function method to find semi-exact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the above-mentioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semi-trivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included.
This paper considers the problem: if coexistence occurs in the long run when a third species w invades an ecosystem consisting of two species u and v on R, where u, v and w compete with one another. Under the assumption that the in uence of w on u and v is small and other suitable conditions, we show that the three species can coexist as a non-monotone travelling wave. Such type of non-monotone waves plays an important role in the study of three-species phenomena. However, fewer results are known for the existence of such waves in the literature. Our approach, based on the method of sub-and super-solutions and bifurcation theory, provides a new approach to construct non-monotone waves of this type. Moreover, we show that the waves we construct are stable. To the best of our knowledge, this is the rst rigorous result of stability for such type of waves.
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