Abstract
In this paper, we revisit a Nicholson's blowflies model with proportional delay, the stability and bifurcation of whose discrete version have not been studied. By using the method of semidiscretization, we dig deeply out the stability and Neimark–Sacker bifurcation of its discrete model. Especially, some results for the existence and stability of Neimark–Sacker bifurcation are derived by using the centre manifold theorem and bifurcation theory. Numerical simulations are also formulated to verify the existence of Neimark–Sacker bifurcation derived.
2010 Mathematics Subject Classification:
Acknowledgments
This work is partly supported by the National Natural Science Foundation of China (61473340), the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province, and the National Natural Science Foundation of Zhejiang University of Science and Technology (F701108G14).
Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Appendix
Definition 5.1
Let be a fixed piont of the system (Equation8(8) (8) ) with multipliers and .
If and , a fixed point is called sink, so a sink is locally asymptotically stable.
If and , a fixed point is called source, so a source is locally asymptotically unstable.
If and or and , a fixed point is called saddle.
If either or , a fixed point is called to be non-hyperbolic.
Lemma 5.2
Let , where B and C are two real constants. Suppose and are two roots of . Then the following statements hold.
If then
(i.1) and if and only if and C<1;
(i.2) and if and only if and ;
(i.3) and if and only if ;
(i.4) and if and only if and C>1;
(i.5) and are a pair of conjugate complex roots and, if and only if and C = 1;
(i.6) if and only if and B = 2.
If namely, 1 is one root of , then the other root λ satisfies if and only if
If then has one root lying in . Moreover,
(iii.1) the other root λ satisfies if and only if ;
(iii.2) the other root if and only if .