Existence and asymptotic of traveling wave fronts for the delayed Volterra-type cooperative system with spatial diffusion

Abstract

In the paper, we are concerned with the existence and the exponential asymptotic behavior of traveling waves for the delayed Volterra-type cooperative system with nonquasimonotone condition \t\t\t{∂u1(x,t)∂t=D1∂2u1(x,t)∂x2+r1u1(x,t)[1−a1u1(x,t)−b1u1(x,t−τ1)+c1u2(x,t−τ2)],∂u2(x,t)∂t=D2∂2u2(x,t)∂x2+r2u2(x,t)[1−a2u2(x,t)−b2u2(x,t−τ3)+c2u1(x,t−τ4)],\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\textstyle\\begin{cases} \\frac{\\partial{u_{1}}(x,t)}{\\partial t}={D_{1}}\\frac{\\partial ^{2}{u_{1}}(x,t)}{\\partial x^{2}} + r_{1}u_{1}(x,t)[ 1 - a_{1}u_{1}(x,t) - b_{1}u_{1}(x,t-\\tau_{1}) + c_{1}u_{2}(x,t-\\tau_{2})],\\\\ \\frac{\\partial{u_{2}}(x,t)}{\\partial t}={D_{2}}\\frac{\\partial ^{2}{u_{2}}(x,t)}{\\partial x^{2}} + {r_{2}}{u_{2}}(x,t)[1 - {a_{2}u_{2}}(x,t) - {b_{2}}{u_{2}}(x,t-\\tau_{3}) + {c_{2}u_{1}}(x,t-\\tau_{4})], \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} modeling the variation of the populations. Here, the major contribution to population model is the introduction of population self-regulation depending on not only the populations at time t, but also on the earlier population time t-tau_{i} (i=1,3). By constructing a pair of suitable upper and lower solutions we obtain the existence of traveling wave fronts connecting the trivial equilibrium and the positive equilibrium, which indicates that there is a transition zone moving the steady state with no species to the steady state with the coexistence of two species. Furthermore, with the help of Ikehara’s theorem, the exponential asymptotic behavior of traveling wave front is exactly derived for this system without quasi-monotone conditions. The results are not only an extension of existing results for the known logistic or cooperative system, but also can extend another type of delayed logistic equation with spatial diffusion \t\t\t∂u(x,t)∂t=D∂2u(x,t)∂x2+ru(x,t)[1−au(x,t)−bu(x,t−τ)].\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\frac{\\partial u(x,t)}{\\partial t}=D\\frac{\\partial^{2} u(x,t)}{\\partial x^{2}}+ru(x,t) \\bigl[1-au(x,t)-bu(x,t-\\tau) \\bigr]. \\end{aligned}$$ \\end{document}