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# RETRACTED ARTICLE: Stability and direction for a class of Schrödingerean difference equations with delay

*Advances in Difference Equations*
**volume 2017**, Article number: 87 (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.

## 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.

In this paper, we provide a Schrödingerean difference equation to describe the dynamic of Schrödingerean Hastings-Powell food chain model. In the three-species food chain model, *x* represents the prey, *y* and *z* represent two predators. Based on the Holling type II functional response, we know that the middle predator *y* feeds on the prey *x* and the top predator *z* preys upon *y*. We write three-species food chain model as follows:

where *X*, *Y*, *Z* are the prey, predator and top-predator, respectively; \(B_{1} \), \(B_{2} \) represent the half-saturation constants; \(R_{0} \), \(A_{1} \) represent the intrinsic growth rate and the carrying capacity of the environment of the fish, respectively; \(C_{1} \), \(C_{2} \) are the conversion factors of prey-to-predator; and \(D_{1} \), \(D_{2} \) represent the death rates of *Y* and *Z*, respectively. In this paper, two different Schrödingerean delays in (1) are incorporated into Schrödingerean Tritrophic Hastings-Powell (STHP) model which will be given in the following.

We next introduce the following dimensionless version of delayed STHP model:

where *x*, *y* and *z* represent dimensionless population variables; *t* represents dimensionless time variable and all of the parameters \(a_{i} \), \(b_{i} \), \(d_{i}\) (\({i=1,2} \)) are positive; \(\tau_{1} \) and \(\tau_{2} \) represent the period of prey transitioning to predator and that of predator transitioning to top predator, respectively.

## 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.

Now we consider system (2) by the transformation

where \(t=\tau_{1}+\tau_{2}\).

We get the following Schrödingerean differential equation (SDE) system (see [7]) in \(C=C ( { [ {-1,1} ],R^{3}} )\):

where \(u ( t )= ( {u_{1} ( t ),u_{2} ( t ),u_{3} ( t )} )^{T}\in R^{3}\), \(L_{\mu}:C\to R^{2}\) and \(f:R\times C\to R^{3}\) are given by

and

respectively.

By (3), (4) and the Schrödingerean Riesz representation theorem (see [3]), there exists a function \(\eta ( {\theta,\mu} )\) of bounded variation such that

for any \(\theta\in C\), where \(\theta\in [ {-\tau,0} ]\).

It follows from (5) that

where \(\delta ( \theta )\) is the Dirac delta function.

For any \(\varphi ( \theta )\in C ( { [{-1,1} ],R^{3}} )\), we define the operator \(A ( \mu )\) as follows (see [1]):

and

It is easy to see that system (2) is equivalent to

where \(\theta\in [ {-1,1} ]\) and \(\mu_{t} ( \theta )=\mu ( {t+\theta} )\) is a real function.

For any \(\psi\in{C}' ( { [{-1,1} ], ({R}^{2} )^{\ast}} )\), we define operator \(A^{\ast}\) of *A* by

and

where \(\eta ( \theta )=\eta ( {\theta,0} )\).

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 \(A^{\ast}( 0 )\).

Let \(q (\theta )\) be an eigenvector of \(A ( 0 )\) corresponding to \(i\omega\tau_{k}\) and \(q^{\ast}( \theta )\) be an eigenvector of \(A^{\ast}( 0 )\) corresponding to \(-i\omega\tau_{k}\). Then we know that

and

Suppose that \(q ( \theta )= ( {1,\rho_{1} ,\rho_{2} } )^{T}e^{i\omega\tau_{k} \theta }\) is an eigenvector of \(A ( 0 )\) corresponding to \(i\omega\tau_{k}\). It follows from the definitions of \(A ( 0 )\), \(L_{\mu}( \varphi )\) and \(\eta ( {0,\mu} )\) that

By the definition of \(A^{\ast}\) (see [8], p.109), we know that

In order to satisfy \(\langle{q^{\ast}( s ),q ( \theta )} \rangle=1\), we need to evaluate *D*. By the definition of bilinear inner product, we know that

Then we choose *D̄* as follows:

It is easy to see that \(\langle{q^{\ast}( s ),q ( \theta )} \rangle=1\) and \(\langle{q^{\ast}( s ),\bar {q} ( \theta )} \rangle=0\).

In the remainder of this section, we also use the same notations to compute the coordinates, which describe the center manifold \(C_{0} \) at \(\mu=0\).

Define

where \(u_{t} \) and *W* are real functions.

By the definition of center manifold \(C_{0} \), we know that

from (11), where *z* and *z̄* are local coordinates for the center manifold \(C_{0} \) in the directions of *q* and \(\bar{q}^{\ast}\). If \(u_{t} \) is real, then we know that *W* is also real. We only consider real solutions. Since \(\mu=0\), we know that

from (11) for the solution \(u_{t} \in C_{0} \), where

By using (4), we know that \(x_{t} ( \theta )=W ( {z,\bar {z},\theta} )+2\operatorname{Re} \{ {z ( t )q ( \theta )} \}\), where

It follows from (12), (13) and (14) that

By comparing the coefficients with (9), we get \(g_{20}\), \(g_{11}\), \(g_{02}\) and \(g_{21}\). And we need to compute \(W_{20} ( \theta )\) and \(W_{11}\). By (7) and (13), we know that

where

On the other hand, by taking the derivative with respect to *t* in (4), we know that

from (13), (14), (15) and (16), which together with (4) and (5) gives that

By using (9) for \(\theta\in [ {-1,1} ]\), we know that

Comparing the coefficients with (4), we obtain that

and

From (5), (7) and the definition of *A*, we know that

Similarly, we know that

from (18) and (19), where \(E_{1} = ( {E_{1} ^{ ( 1 )},E_{1} ^{ ( 2 )}} )\in R^{2}\) and \(E_{2} = ( {E_{2} ^{ ( 1 )},E_{2} ^{ ( 2 )}} )\in R\) are constant vectors.

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]).

To this end, we express each \({g}'_{ij} \) in terms of parameters and delay. Then we obtain the following values:

From the above analysis, we obtain the following theorem.

### Theorem

*If*
\(\tau=\tau_{k} \), *then the stability and the direction of periodic solutions of the Schrödingerean Hopf bifurcation of system* (22) *are determined by the parameters*
\(\mu_{2} \), \(\beta_{2} \)*and*
\(T_{2} \).

- (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.

## Change history

### 28 April 2020

The Editors-in-Chief have retracted this article [1] because it overlaps with articles by other authors that were simultaneously under consideration at different journals [2,3].

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## 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.

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The authors declare that they have no competing interests.

### Authors’ contributions

WL drafted the manuscript. ML helped to draft the manuscript and revised written English. JS helped to draft the manuscript and revised it according to the referee reports. All authors read and approved the final manuscript.

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The Editors-in-Chief have retracted this article because it overlaps with two articles by other authors that were simultaneously under consideration at different journals. The article also shows evidence of authorship manipulation. Additionally, contrary to the statement in the Acknowledgements section that "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.", the Norwegian University of Science and Technology confirmed that he has not been affiliated with their institution.

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Lai, M., Sun, J. & Li, W. RETRACTED ARTICLE: Stability and direction for a class of Schrödingerean difference equations with delay.
*Adv Differ Equ* **2017**, 87 (2017). https://doi.org/10.1186/s13662-017-1132-3

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DOI: https://doi.org/10.1186/s13662-017-1132-3