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
.