Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition-diffusion system

作者：Du, Li-Jun; Li, Wan-Tong^*; Wang, Jia-Bing

来源：Journal of Differential Equations, 2018, 265(12): 6210-6250.

DOI：10.1016/j.jde.2018.07.024

• Periodic traveling front
• Asymptotic behavior
• Comparison principle
• Entire solution
• New spreading ways

This paper is concerned with a time periodic competition-diffusion system
{u(t) = u(xx) + u(r(1)(t) - a(1)(t)u - b(1)(t)v), t > 0, x is an element of R,
v(t) = dv(xx) + v(r(2)(t) - a(2)(t)u - b(2)(t)v), t > 0, x is an element of R,
where u(t, x) and v(t, x) denote the densities of two competing species, d > 0 is some constant, r(i) (t), a(i) (t) and b(i) (t) are T-periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T-periodic solutions (p(t), 0) and (0, q(t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment.

• 出版日期2018-12-15
• 单位中国地质大学（武汉）; 兰州大学