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Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system. (English) Zbl 1180.35536

The goal of this paper is to study the asymptotic speed of spread, the existence and nonexistence of travelling wave for the following nonlocal and time-delayed reaction-diffusion system

$\left\{\begin{array}{c}\frac{\partial u}{\partial t}=d{\Delta }u+d{\int }_{0}^{T}{\int }_{ℝ}F\left(s,y\right)u\left(t-s,x-y\right)dy\phantom{\rule{0.166667em}{0ex}}ds-\beta {u}^{2}\left(t,x\right)\hfill \\ \frac{\partial v}{\partial t}=D{\Delta }v-\gamma v+\alpha u-\alpha {\int }_{0}^{T}{\int }_{ℝ}F\left(s,y\right)u\left(t-s,x-y\right)dy\phantom{\rule{0.166667em}{0ex}}ds,\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $\tau \in \left(0,\infty \right],d,D,\alpha ,\beta ,\gamma$ are positive and $F:{ℝ}^{2}\to ℝ$ satisfies natural assumptions.

##### MSC:
 35R10 Partial functional-differential equations 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions of PDE 92D25 Population dynamics (general)
##### References:
 [1] Aiello, W. G.; Freedman, H. I.: A time-delay model of single species growth with stage structure, Math. biosci. 101, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U [2] Al-Omari, J.; Gourley, S. A.: Monotone travelling fronts in an age-structured reaction – diffusion model of a single species, J. math. Biol. 45, 294-312 (2002) · Zbl 1013.92032 · doi:10.1007/s002850200159 [3] Al-Omari, J.; Gourley, S. A.: Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. math. 70, 858-879 (2005) · Zbl 1086.92043 · doi:10.1093/imamat/hxh073 [4] Gourley, S. A.; Kuang, Y.: Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. lond. Ser. A 459, 1563-1579 (2003) · Zbl 1047.92037 · doi:10.1098/rspa.2002.1094 [5] Gourley, S. A.; Wu, J.: Delayed nonlocal diffusive systems in biological invasion and disease spread, Fields inst. Commun. 48, 137-200 (2006) · Zbl 1130.35127 [6] Hale, J.: Theory of functional differential equations, (1977) [7] Hutchinson, G. E.: Circular causal systems in ecology, Ann. New York acad. Sci. 50, 221-246 (1948) [8] Li, W.; Wang, Z.: Traveling fronts in diffusive and cooperative Lotka – Volterra system with nonlocal delays, Z. angew. Math. phys. 58, No. 4, 571-591 (2007) · Zbl 1130.35079 · doi:10.1007/s00033-006-5125-4 [9] Liang, X.; Zhao, X. -Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. pure appl. Math. 60, 1-40 (2007) · Zbl 1106.76008 · doi:10.1002/cpa.20154 [10] Martin, R. H.; Smith, H. L.: Abstract functional-differential equations and reaction – diffusion systems, Trans. amer. Math. soc. 321, 1-44 (1990) · Zbl 0722.35046 · doi:10.2307/2001590 [11] Ruan, S.; Xiao, D.: Stability of steady states and existence of traveling waves in a vector disease model, Proc. roy. Soc. Edinburgh sect. A 134, 991-1011 (2004) · Zbl 1065.35059 · doi:10.1017/S0308210500003590 [12] Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. surveys monogr. 41 (1995) · Zbl 0821.34003 [13] Smith, H. L.; Zhao, X. -Q.: Global asymptotic stability of traveling waves in delayed reaction – diffusion equations, SIAM J. Math. anal. 31, 514-534 (2000) · Zbl 0941.35125 · doi:10.1137/S0036141098346785 [14] So, J. W. -H.; Wu, J.; Zou, X.: A reaction – diffusion model for a single species with age structure. I. travelling wavefronts on unbounded domains, Proc. R. Soc. lond. Ser. A 457, 1841-1853 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789 [15] Thieme, H. R.: Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. reine angew. Math. 306, 94-121 (1979) · Zbl 0395.45010 · doi:10.1515/crll.1979.306.94 · doi:crelle:GDZPPN002195887 [16] Thieme, H. R.; Zhao, X. -Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction – diffusion models, J. differential equations 195, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X [17] Weinberger, H. F.: Long-time behavior of a class of biological models, SIAM J. Math. anal. 13, 353-396 (1982) · Zbl 0529.92010 · doi:10.1137/0513028 [18] Wu, J.; Zou, X.: Traveling wave fronts of reaction – diffusion systems with delay, J. dynam. Differential equations 13, 651-687 (2001) · Zbl 0996.34053 · doi:10.1023/A:1016690424892 [19] Zhao, X. -Q.: Dynamical systems in population biology, (2003) [20] Zhao, X. -Q.; Xiao, D.: The asymptotic speed of spread and traveling waves for a vector disease model, J. dynam. Differential equations 18, 1001-1019 (2006) · Zbl 1114.45001 · doi:10.1007/s10884-006-9044-z