Abstract
This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.
Highlights
We are concerned with the following nonlocal diffusive competition-cooperation systems of three species with delays:
Li et al in [13] introduced the nonlocal diffusion and time delays into the classical Lotka–Volterra reaction diffusion system and proved the existence, asymptotic behavior, and uniqueness of the traveling wave front of this system connecting the equilibria on the two axes
As we know, there is no conclusion about the existence, asymptotic behavior, and other properties of the traveling wave solutions of (1), inspired by [13], we mainly consider the asymptotic behavior, strict monotonicity, and uniqueness of the traveling wave fronts of (1), connecting e3 and e4
Summary
We are concerned with the following nonlocal diffusive competition-cooperation systems of three species with delays:. J1 ∗ u1(x, t) − d1u1(x, t) + r1u1(x, t)1 − u1 x, t − τ11 + a12u2 x, t − τ12 − a13u3 J2 ∗ u2(x, t) − d2u2(x, t) + r2u2(x, t)1 + a21u1 x, t − τ21 − u2 x, t − τ22 − a23u3 J3 ∗ u3(x, t) − d3u3(x, t) + r3u3(x, t)1 − a31u1 x, t − τ31 − a32u2 x, t − τ32 − u3 x, t − τ13, x, t − τ23, x, t − τ33,. Where (Ji ∗ ui)(x, t) RJi(x − y)ui(y, t)dy, i 1, 2, 3, Ji denote the diffusive kernel functions of three species, respectively, and ui(x, t) represent the density of three species. Ri > 0, 0 < aij < 1, i ≠ j, i, j 1, 2, 3 and τij ≥ 0, i, j 1, 2, 3
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