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Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. (English) Zbl 1243.34116
The author considers a Nicholson’s blowflies model on a patch environment allowing different maturation delay in different patches. The main concern is the global dynamics of the model system. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. The author also further studies a diffusive Nicholson-type model with patch structure, and establishes a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics (general)
35C07 Traveling wave solutions
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[1] Berezansky, L.; Idels, L.; Troib, L., Global dynamics of Nicholson-type delay systems with applications, Nonlinear anal. RWA, 12, 436-445, (2011) · Zbl 1208.34120
[2] Gourley, S.A.; Kuang, Y., A stage structured predator-prey model and its dependence on maturation delay and death rate, J. math. biol., 49, 18-200, (2004) · Zbl 1055.92043
[3] Liu, B., Global stability of a class of delay differential systems, J. comput. appl. math., 233, 217-223, (2009) · Zbl 1189.34145
[4] Takeuchi, Y.; Wang, W.; Saito, Y., Global stability of population models with patch structure, Nonlinear anal. RWA, 7, 235-247, (2006) · Zbl 1085.92053
[5] Faria, T.; Trofimchuk, S., Positive travelling fronts for reaction – diffusion systems with distributed delay, Nonlinearity, 23, 2457-2481, (2010) · Zbl 1206.34086
[6] Gourley, S.A., Traveling fronts in the diffusive nicholson’s blowflies equation with distributed delays, Math. comput. modelling, 32, 843-853, (2000) · Zbl 0969.35133
[7] So, J.W.-H; Wu, J.; Zou, X., A reaction diffusion model for a single species with age structure \(I\). travelling wave fronts on unbounded domains, Proc. R. soc. lond. A, 457, 1841-1853, (2001) · Zbl 0999.92029
[8] So, J.W.-H.; Zou, X., Traveling waves for the diffusive nicholson’s blowflies equation, Appl. math. comput., 122, 385-392, (2001) · Zbl 1027.35051
[9] Faria, T.; Trofimchuk, S., Nonmonotone travelling waves in a single species reaction – diffusion equations with delay, J. differential equations, 228, 357-376, (2006) · Zbl 1217.35102
[10] Gourley, S.A.; Kuang, Y., Wavefronts and global stability in a time delayed population model with stage structure, Proc. roy. soc. lond. A, 459, 1567-1579, (2003) · Zbl 1047.92037
[11] Xu, R.; Chaplain, M.A.J.; Davidson, F.A., Travelling wave and convergence in stage-structured reaction – diffusion competitive models with nonlocal delays, Chaos solitons fractals, 30, 974-992, (2006) · Zbl 1142.35477
[12] Faria, T.; Oliveira, J.J., Local and global stability for lotka – volterra systems with distributed delays and instantaneous feedbacks, J. differential equations, 244, 1049-1079, (2008) · Zbl 1146.34053
[13] Fiedler, M., Special matrices and their applications in numerical mathematics, (1986), Martinus Nijhoff Publ. (Kluwer) Dordrecht
[14] Oliveira, J.J., Global asymptotic stability for neural network models with distributed delays, Math. comput. modelling, 50, 81-91, (2009) · Zbl 1185.34107
[15] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[16] Liu, B., Global stability of a class of nicholson’s blowflies model with patch structure and multiple time-varying delays, Nonlinear anal. RWA, 11, 2557-2562, (2010) · Zbl 1197.34165
[17] Zhao, X.-Q.; Jing, Z.-J., Global asymptotic behavior in some cooperative systems of functional differential equations, Can. appl. math. Q., 4, 421-444, (1996) · Zbl 0888.34038
[18] Smith, H.L., ()
[19] Smith, H.L.; Waltman, P., The theory of the chemostat, (1995), University Press Cambridge · Zbl 0860.92031
[20] Huang, W., Monotonicity of heteroclinic orbits and spectral properties of variational equations for delay differential equations, J. differential equations, 162, 91-139, (2000) · Zbl 0954.34071
[21] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press Inc. New York · Zbl 0484.15016
[22] Liz, E.; Pinto, M.; Tkachenko, V.; Trofimchuk, S., A global stability criterion for a family of delayed population models, Quart. appl. math., 63, 56-70, (2005) · Zbl 1093.34038
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