Stability and direction for a class of Schrödingerean difference equations with delay

*Correspondence: jianguosun81@sina.cn 1Department of Computer Science and Technology, Harbin Engineering University, Harbin, 150001, China Full list of author information is available at the end of the article 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.


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 [, ]). May and Odter (see []) 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 []) proposed a three-species food chain model with chaotic dynamical behavior in , and then the dynamic properties of the model were studied. 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 © The Author(s) 2017. 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.

R E T R
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  , B  represent the half-saturation constants; R  , A  represent the intrinsic growth rate and the carrying capacity of the environment of the fish, respectively; C  , C  are the conversion factors of prey-to-predator; and D  , D  represent the death rates of Y and Z, respectively. In this paper, two different Schrödingerean delays in () 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 = , ) are positive; τ  and τ  represent the period of prey transitioning to predator and that of predator transitioning to top predator, respectively.

Bifurcation analysis
In this section we first study the Schrödingerean Hastings-Powell food chain system with delay, which undergoes the Schrödingerean Hopf bifurcation when τ = τ   . Next we confirm the Schrödingerean Hopf bifurcation's stability, direction and the periodic solutions of delay differential equations. Now we consider system () by the transformatioṅ We get the following Schrödingerean differential equation ( respectively. By (), () and the Schrödingerean Riesz representation theorem (see []), there exists a function η(θ , μ) of bounded variation such that where δ(θ ) is the Dirac delta function.

R E T R
It is easy to see that A * () and A() are adjoint operators. From (), (), (), () and (), we obtain that ±iωτ k are the eigenvalues of A(). So they are the eigenvalues of A * ().
In order to satisfy q * (s), q(θ ) = , we need to evaluate D. By the definition of bilinear inner product, we know that Then we chooseD as follows: It is easy to see that q * (s), q(θ ) =  and q * (s),q(θ) = .

A R T I C L E
In the remainder of this section, we also use the same notations to compute the coordinates, which describe the center manifold C  at μ = . Define where u t and W are real functions. By the definition of center manifold C  , we know that from (), where z andz are local coordinates for the center manifold C  in the directions of q andq * . If u t is real, then we know that W is also real. We only consider real solutions.
From (), () and the definition of A, we know that Similarly, we know that If we solve these for E  and E  , we compute W  (θ) and W  (θ) from (), (), () and confirm the following values to investigate the qualities of the bifurcation periodic solution in the center manifold at the critical value τ k (see []).
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 τ = τ k , then the stability and the direction of periodic solutions of the Schrödingerean Hopf bifurcation of system () are determined by the parameters μ  , β  and T  .
(i) The direction of the Schrödingerean Hopf bifurcation is determined by the sign of μ  : if μ  >  (resp. μ  < ), then the Schrödingerean Hopf bifurcation is supercritical (resp. subcritical), and the bifurcation periodic solution exists for τ > τ  (resp. τ < τ  ). (ii) The stability of the Schrödingerean bifurcation periodic solution is determined by the sign of β  : if β  >  (resp. β  < ), then the Schrödingerean bifurcation periodic solution is stable (resp. unstable). (iii) The sign of T  determines the period of the Schrödingerean bifurcation periodic solution: if T  >  (resp. T  < ), then the period increases (resp. decreases).