- Research
- Open Access
Stability and direction for a class of Schrödingerean difference equations with delay
- Mingzhu Lai^{1},
- Jianguo Sun^{1}Email author and
- Wei Li^{2}
https://doi.org/10.1186/s13662-017-1132-3
© The Author(s) 2017
- Received: 2 December 2016
- Accepted: 7 March 2017
- Published: 22 March 2017
Abstract
Exploring some results of Wang et al. (Adv. Differ. Equ. 2016:33, 2016) from another point of view, we first investigate the stability and direction for a class of Schrödingerean difference equations with Schrödingerean Hopf bifurcation. Next we obtain the stable conditions for these equations and prove that Schrödingerean Hopf bifurcation shall occur when the delay passes through the critical value.
Keywords
- local stability
- Schrödingerean difference equations
- delay
1 Introduction
A biological system is a nonlinear system, so it is still a public problem upon how to control the biological system balance. The predecessors have done a lot of research. Especially the research on the predator-prey system’s dynamic behaviors has received much attention from the scholars. There is also a large number of research works on the stability of a predator-prey system with time delays. The time delays have a very complex impact on the dynamic behaviors of the nonlinear dynamic system (see [2, 3]). May and Odter (see [4]) introduced a general example of such a generalized model, that was to say, they investigated a three-species model, and the results show that the positive equilibrium is always locally stable when the system has two same time Schrödingerean delays.
Hassard and Kazarinoff (see [5]) proposed a three-species food chain model with chaotic dynamical behavior in 1991, and then the dynamic properties of the model were studied. Berryman and Millstein (see [6]) studied the control of chaos of a three-species Hastings-Powell food chain model. The stability of biological feasible equilibrium points of the modified food web model was also investigated. By introducing the disease in prey population, Shilnikov et al. (see [3]) modified the Schrödingerean Hastings-Powell model, and the stability of biological feasible equilibria was also obtained.
2 Bifurcation analysis
In this section we first study the Schrödingerean Hastings-Powell food chain system with delay, which undergoes the Schrödingerean Hopf bifurcation when \(\tau=\tau_{0}^{0} \). Next we confirm the Schrödingerean Hopf bifurcation’s stability, direction and the periodic solutions of delay differential equations.
It is easy to see that \(A^{\ast}( 0 )\) and \(A ( 0 )\) are adjoint operators. From (6), (7), (8), (9) and (10), we obtain that \(\pm i\omega\tau_{k}\) are the eigenvalues of \(A ( 0 )\). So they are the eigenvalues of \(A^{\ast}( 0 )\).
In the remainder of this section, we also use the same notations to compute the coordinates, which describe the center manifold \(C_{0} \) at \(\mu=0\).
If we solve these for \(E_{1} \) and \(E_{2} \), we compute \(W_{20} ( \theta )\) and \(W_{11} ( \theta )\) from (8), (9), (10) and confirm the following values to investigate the qualities of the bifurcation periodic solution in the center manifold at the critical value \(\tau_{k} \) (see [9]).
From the above analysis, we obtain the following theorem.
Theorem
- (i)
The direction of the Schrödingerean Hopf bifurcation is determined by the sign of \(\mu _{2} \): if \(\mu_{2} >0\) (resp. \(\mu_{2} <0\)), then the Schrödingerean Hopf bifurcation is supercritical (resp. subcritical), and the bifurcation periodic solution exists for \(\tau>\tau_{0}\) (resp. \(\tau<\tau_{0}\)).
- (ii)
The stability of the Schrödingerean bifurcation periodic solution is determined by the sign of \(\beta_{2} \): if \(\beta_{2} >0\) (resp. \(\beta_{2} <0\)), then the Schrödingerean bifurcation periodic solution is stable (resp. unstable).
- (iii)
The sign of \(T_{2}\) determines the period of the Schrödingerean bifurcation periodic solution: if \(T_{2} >0\) (resp. \(T_{2} <0\)), then the period increases (resp. decreases).
3 Conclusions
In this paper, we provide a differential model to describe the dynamic behavior of the Hasting-Powell food chain system. And two different Schrödingerean delays are incorporated into the model. The stabilities of equilibrium point and Schrödingerean Hopf bifurcation are studied. We also get the system’s stable conditions, and there are four cases in this paper, which are discussed to illustrate the existence of Schrödingerean Hopf bifurcation. Based on the center manifold theorem and the normal form theorem, we control the direction and the stability of Schrödingerean Hopf bifurcation. Finally, we give numerical examples to verify theorems and results.
Declarations
Acknowledgements
The authors thank the anonymous referees for their valuable suggestions and comments, by which the paper was revised. The Schrödingerean manifold theorem in this paper was proved while the third author was at the Norwegian University of Science and Technology as a visiting scholar.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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