Entire solutions of Lotka-Volterra strong competition systems with nonlocal dispersal

This paper is devoted to considering entire solutions of the following two-species Lotka-Volterra strong competition system with nonlocal dispersals {u(t) = (R)integral k(x - y)u(y, t) dy - u + u (1 - u - av), x is an element of R, t is an element of R, v(t) = d (R)integral k(x - y)v(y, t) dy - dv + rv (1 - v - bu), x is an element of R, t is an element of R, (0.1) where a > 1, b > 1, d, and r are positive constants. By constructing appropriate super- and sub-solutions, and with the aid of corresponding comparison principle, we established the existence and related qualitative properties of entire solutions originating from three and four traveling wave solutions. Meanwhile, we considered the non-existence of entire solutions originating from more than seven traveling fronts by introducing the definition of non-extendable (terminated) sequence. Compared to the known works for Lotka-Volterra competition system with classical diffusions, we have to overcome many difficulties due to the appearance of nonlocal dispersal operators in current paper.