Propagation phenomena for a two-species Lotka–Volterra strong competition system with nonlocal dispersal
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This paper is concerned with the propagation phenomena for a two-species Lotka–Volterra strong competition system with nonlocal dispersal. We first establish the existence of bistable traveling waves by appealing to the theory of monotone semiflows. Then we use a dynamical systems approach to prove that such bistable traveling waves are asymptotically stable and unique modulo translation. Finally, we study the spreading properties of solutions for a class of initial conditions by the comparison arguments and the method of super- and subsolutions. It is shown that for initial conditions where both species u and v are initially absent from the right half-line \(x > 0\), and the species v dominates the species u around \(x=-\infty \) initially, if v spreads in absence of u slower than u in absence of v, then solutions of the initial value problem will approach a propagating terrace, which connects the unstable state (0, 0) to the stable state (1, 0), and then the stable state (1, 0) to the other stable state (0, 1).
Mathematics Subject Classification35K57 35C07 35B40 92D25
G.-B. Zhang was supported by NSF of China (11861056), NSF of Gansu Province (18JR3RA093) and the postdoctoral fellowship at the Memorial University of Newfoundland, and X.-Q. Zhao was partially supported by the NSERC of Canada. We are also grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.
- 8.Coville, J.: Travelling Waves in a Nonlocal Reaction Diffusion Equation with Ignition Nonlinearity. Ph.D. thesis, Universit Pierre et Marie Curie, Paris (2003)Google Scholar
- 10.Coville, J.: Travelling fronts in asymmetric nonlocal reaction diffusion equation: the bistable and ignition case. Prpublication du CMM, Hal-00696208 (2012)Google Scholar
- 16.Fife, P.: Some nonclassical trends in parabolic-like evolutions. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds.) Trends in Nonlinear Analysis, pp. 153–191. Springer, Berlin (2003) Google Scholar