×

The existence of periodic solutions for some models with delay. (English) Zbl 1095.34549

Consider the system
\[ \begin{aligned} \frac{dv_1(t)}{dt}& = \tau\bigl(v_1^*+v_1(t)\bigr)\biggl[-a_1v_1(t)+a_2g\bigl(v_2 (t)\bigr)\biggr],\\ \frac{dv_2(t)}{dt} & = \tau \bigl(v_2^*+v_2(t)\bigr)\biggl[-a_1v_1(t)+ a_2g\bigl(v_2(t)\bigr)-a_3g\bigl(v_2 (t-1)\bigr)\biggr],\tag{*} \end{aligned} \]
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.

MSC:

34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Edoardo Beretta, Yang Kuang, Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal. 32 (1998) 381-408.; 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., \(S^1\)-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), Springer: Springer New York · Zbl 0787.34002
[4] Shigui Ruan, Junjie Wei, Periodic solutions of planar systems with delays, Proc. Royal Soc. Edinburgh, Ser A., in press.; 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), Springer: Springer New York · Zbl 0870.35116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.