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The existence of periodic solutions for some models with delay. (English) Zbl 1095.34549
Consider the system $$\align \frac{dv_1(t)}{dt}& = \tau\bigl(v_1^*+v_1(t)\bigr)\biggl[-a_1v_1(t)+a_2g\bigl(v_2 (t)\bigr)\biggr],\\ \frac{dv_2(t)}{dt} & = \tau \bigl(v_2^*+v_2(t)\bigr)\biggl[-a_1v_1(t)+ a_2g\bigl(v_2(t)\bigr)-a_3g\bigl(v_2 (t-1)\bigr)\biggr],\tag* \endalign$$ where $a_1,a_2,a_3,v_1^*,v_2^*$ are constants and $\tau$ is the bifurcation parameter. The authors derives conditions such that it holds: (i) There exists a sequence $\{\tau_n\}$ with $\tau_{n+1}> \tau_n$ such that (*) has a Hopf bifurcation at $\tau_n$, $n=0,1,1, \dots$ (ii) For $\tau>\tau_1$, system (*) has at least one nonconstant periodic solution. Finally, the results are applied to a predator-prey model with Michaelis-Menten type functional response.

34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
Full Text: DOI
[1] Edoardo Beretta, Yang Kuang, Global analysis in some delayed ratio-dependent predator--prey systems, Nonlinear Anal. 32 (1998) 381--408. · Zbl 0946.34061
[2] Erbe, L. H.; Krawcewicz, W.; Geba, K.; Wu, J.: S1-degree and global Hopf bifurcation theory of functional-differential equations. J. differential equations 89, 277-298 (1992) · Zbl 0765.34023
[3] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[4] Shigui Ruan, Junjie Wei, Periodic solutions of planar systems with delays, Proc. Royal Soc. Edinburgh, Ser A., in press. · Zbl 0946.34062
[5] Thieme, H. R.: Persistence under relaxed point-dissapativity (with application to an endemic model). SIAM J. Math. anal. 24, 407-435 (1993) · Zbl 0774.34030
[6] Wu, J.: Theory and applications of partial functional differential equations. (1996) · Zbl 0870.35116