Stability and Neimark–Sacker bifurcation for a discrete Nicholson's blowflies model with proportional delay

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.

Keywords:

• Discrete Nicholson's blowflies model with proportional delay
• semidiscretization method
• Neimark–Sacker bifurcation
• centre manifold theorem
• bifurcation theory

2010 Mathematics Subject Classification:

• 39A10

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 $E(x,y)$ be a fixed piont of the system (8(8) $\{\begin{array}{l}{x}_{n+1}=y_n,\\ y_{n+1}=(1-d)y_n+b{x}_n{e}^{-a{x}_n},\end{array}$(8) ) with multipliers ${\lambda }_1$ and ${\lambda }_2$.

1. If $|{\lambda }_1|<1$ and $|{\lambda }_2|<1$, a fixed point $E(x,y)$ is called sink, so a sink is locally asymptotically stable.

2. If $|{\lambda }_1|>1$ and $|{\lambda }_2|>1$, a fixed point $E(x,y)$ is called source, so a source is locally asymptotically unstable.

3. If $|{\lambda }_1|<1$ and $|{\lambda }_2|>1$ $($or $|{\lambda }_1|>1$ and $|{\lambda }_2|<1)$, a fixed point $E(x,y)$ is called saddle.

4. If either $|{\lambda }_1|=1$ or $|{\lambda }_2|=1$, a fixed point $E(x,y)$ is called to be non-hyperbolic.

Lemma 5.2

Let $F(\lambda )={\lambda }^2+B\lambda +C$, where B and C are two real constants. Suppose ${\lambda }_1$ and ${\lambda }_2$ are two roots of $F(\lambda )=0$. Then the following statements hold.

1. If $F(1)>0,$ then

   	(i.1)	$|{\lambda }_1|<1$ and $|{\lambda }_2|<1$ if and only if $F(-1)>0$ and C<1;
   	(i.2)	${\lambda }_1=-1$ and ${\lambda }_2\ne -1$ if and only if $F(-1)=0$ and $B\ne 2$;
   	(i.3)	$|{\lambda }_1|<1$ and $|{\lambda }_2|>1$ if and only if $F(-1)<0$;
   	(i.4)	$|{\lambda }_1|>1$ and $|{\lambda }_2|>1$ if and only if $F(-1)>0$ and C>1;
   	(i.5)	${\lambda }_1$ and ${\lambda }_2$ are a pair of conjugate complex roots and, $|{\lambda }_1|=|{\lambda }_2|=1$ if and only if $-2<B<2$ and C = 1;
   	(i.6)	${\lambda }_1={\lambda }_2=-1$ if and only if $F(-1)=0$ and B = 2.

2. If $F(1)=0,$ namely, 1 is one root of $F(\lambda )=0$, then the other root λ satisfies $|\lambda |=(⟨,⟩)1$ if and only if $|C|=(⟨,⟩)1.$

3. If $F(1)<0,$ then $F(\lambda )=0$ has one root lying in $(1,\mathrm{\infty })$. Moreover,

   	(iii.1)	the other root λ satisfies $\lambda <(=)-1$ if and only if $F(-1)<(=)0$;
   	(iii.2)	the other root $-1<\lambda <1$ if and only if $F(-1)>0$.

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China [grant number 61473340], the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province, and the National Natural Science Foundation of Zhejiang University of Science and Technology [grant number F701108G14].