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Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms. (English) Zbl 1232.34109

Sufficient conditions are obtained for permanence of the following system

x 1 ' (t)=-D 11 (t,x 1 (t))+D 12 (t,x 2 (t))+c 1 (t)x 1 (t-τ 1 (t))e -γ 1 (t)x 1 (t-τ 1 (t)) ,
x 2 ' (t)=-D 22 (t,x 2 (t))+D 21 (t,x 1 (t))+c 2 (t)x 2 (t-τ 2 (t))e -γ 2 (t)x 2 (t-τ 2 (t)) ·

MSC:
34K60Qualitative investigation and simulation of models
92D25Population dynamics (general)
34K25Asymptotic theory of functional-differential equations
References:
[1]Nicholson, A.: An outline of the dynamics of animal populations, Australian journal of zoology 2, 9-65 (1954)
[2]Gurney, W.; Blythe, S.; Nisbet, R.: Nicholson’s blowflies revisited, Nature 287, 17-21 (1980)
[3]Nisbet, R.; Gurney, W.: Modelling fluctuating populations, (1982) · Zbl 0593.92013
[4]Berezansky, L.; Idels, L.; Troib, L.: Global dynamics of Nicholson-type delay systems with applications, Nonlinear analysis: real world applications 12, No. 1, 436-445 (2011) · Zbl 1208.34120 · doi:10.1016/j.nonrwa.2010.06.028
[5]Berezansky, L.; Braverman, E.; Idels, L.: Nicholson’s blowflies differential equations revisited: Main results and open problems, Applied mathematical modelling 34, 1405-1417 (2010) · Zbl 1193.34149 · doi:10.1016/j.apm.2009.08.027
[6]B. Liu, Permanence for a delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term, Annales Polonici Mathematici, 2011 (APM 2204, in press).
[7]Smith, H. L.: Monotone dynamical systems, Math. surveys monogr. (1995)
[8]Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993)