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
\[
x^{'}_1(t)=-D_{11}(t,x_1(t))+D_{12}(t,x_2(t)) +c_1(t)x_1(t-\tau_1(t))e^{-\gamma_1(t)x_1(t-\tau_1(t))},
\]
\[ x^{'}_2(t)=-D_{22}(t,x_2(t))+D_{21}(t,x_1(t)) +c_2(t)x_2(t-\tau_2(t))e^{-\gamma_2(t)x_2(t-\tau_2(t))}. \]
\[ x^{'}_2(t)=-D_{22}(t,x_2(t))+D_{21}(t,x_1(t)) +c_2(t)x_2(t-\tau_2(t))e^{-\gamma_2(t)x_2(t-\tau_2(t))}. \]
Reviewer: Leonid Berezanski (Beer-Sheva)
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 92D25 | Population dynamics (general) |
| 34K25 | Asymptotic 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), John Wiley and Sons: 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).; 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., (Monotone Dynamical Systems. Monotone Dynamical Systems, Math. Surveys Monogr. (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 0821.34003 |
| [8] | Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002 |
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