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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 $x^{'}_1(t)=-D_{11}(t,x_1(t))+D_{12}(t,x_2(t)) +c_1(t)x_1(t-\tau_1(t))e^{-\gamma_1(t)x_1(t-\tau_1(t))},$ $x^{'}_2(t)=-D_{22}(t,x_2(t))+D_{21}(t,x_1(t)) +c_2(t)x_2(t-\tau_2(t))e^{-\gamma_2(t)x_2(t-\tau_2(t))}.$

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D25 Population dynamics (general) 34K25 Asymptotic theory of functional-differential equations
##### Keywords:
Nicholson-type delay system; permanence
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##### 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), John Wiley and Sons NY · 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, 1, 436-445, (2011) · Zbl 1208.34120 [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 [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). · Zbl 1242.34145 [7] Smith, H.L., () [8] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
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