R E S E A R C H
Open Access
Stability and direction for a class of
Schrödingerean difference equations
with delay
Mingzhu Lai
1, Jianguo Sun
1*and Wei Li
2*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.
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 con-trol 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 atten-tion 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 inves-tigated 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. Berryman and Millstein (see []) studied the control of chaos of a three-species Hastings-Powell food chain model. The stability of biological feasible equilibrium points of the mod-ified food web model was also investigated. By introducing the disease in prey population, Shilnikov et al. (see []) modified the Schrödingerean Hastings-Powell model, and the sta-bility 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,xrepresents the prey,yandzrepresent two predators. Based on the Holling type II functional response, we know that the middle predatoryfeeds on the preyxand the top
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predatorzpreys upony. We write three-species food chain model as follows:
dX dT =RX
– X
K
–C
AXY
B+X
,
dY
dT = –DY+ AXY
B+X
– AYZ
B+Y
, ()
dZ
dT = –DZ+C AYZ
B+Y
,
whereX,Y,Zare the prey, predator and top-predator, respectively;B,Brepresent the
half-saturation constants;R,Arepresent the intrinsic growth rate and the carrying
ca-pacity of the environment of the fish, respectively;C, C are the conversion factors of
prey-to-predator; andD,D represent the death rates ofY andZ, 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:
dx
dt =x( –x) – ax
+bx
y(t–τ),
dy
dt= –dy+ ax
+bx
y– ax +bx
z(t–τ), ()
dz
dt= –dz+ ax
+bx
z,
wherex,yandzrepresent dimensionless population variables;trepresents dimensionless time variable and all of the parametersai,bi,di(i= , ) are positive;τandτrepresent 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 whenτ=τ. Next we con-firm the Schrödingerean Hopf bifurcation’s stability, direction and the periodic solutions of delay differential equations.
Now we consider system () by the transformation
˙
u(t) =x(t) –x∗,
˙
u(t) =y(t) –y∗, ()
˙
u(t) =z(t) –z∗,
wheret=τ+τ.
We get the following Schrödingerean differential equation (SDE) system (see []) inC=
C([–, ],R):
˙
u(t) =Lμ(ut) +f(μ,ut), ()
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whereu(t) = (u(t),u(t),u(t))T∈R,Lμ:C→Randf:R×C→Rare given by
By (), () and the Schrödingerean Riesz representation theorem (see []), there exists a functionη(θ,μ) of bounded variation such that
It follows from () that
η(,μ) = (τk+μ) It is easy to see that system () is equivalent to
˙
u(t) =A(μ)ut+R(μ)ut, () whereθ∈[–, ] andμt(θ) =μ(t+θ) is a real function.
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For anyψ∈C([–, ], (R)∗), we define operatorA∗ofAby
A∗(μ)ψ(s) =
⎧ ⎨ ⎩
–dψds(s), s∈(, ],
–τdη
T(t, )ψ(–t), s= ()
and
ψ(s),ϕ(θ)=ψT()ϕ() –
–
θ
ξ=
ψT(ξ–θ)dη(θ)ϕ(ξ)dξ, () whereη(θ) =η(θ, ).
It is easy to see thatA∗() andA() are adjoint operators. From (), (), (), () and (), we obtain that±iωτkare the eigenvalues ofA(). So they are the eigenvalues ofA∗().
Letq(θ) be an eigenvector ofA() corresponding toiωτkandq∗(θ) be an eigenvector of A∗() corresponding to –iωτk. Then we know that
A()q(θ) =iωτq(θ)
and
A∗()q∗(θ) = –iωτq∗(θ).
Suppose thatq(θ) = (,ρ,ρ)Teiωτkθ is an eigenvector ofA() corresponding toiωτk. It
follows from the definitions ofA(),Lμ(ϕ) andη(,μ) that
q(θ) = (,ρ,ρ)Teiωτkθ=q()eiωτkθ.
By the definition ofA∗(see [], p.), we know that
q∗(θ) =D(,γ,γ)Teiωτkθ=q∗()eiωτkθ.
In order to satisfyq∗(s),q(θ)= , we need to evaluateD. By the definition of bilinear inner product, we know that
q∗(θ),q(θ)=D¯(,γ¯,γ¯)(,ρ,ρ)T
–
–τ
θ
ξ=
¯
D(,γ¯,γ¯)eiωτk(ξ–θ)dη(θ)(,ρ,ρ)Teiωτkξdξ
=D¯
+ργ¯+ργ¯–
–τ
(,γ¯,γ¯)θeiωτkθdη(θ)(,ρ,ρ)T
=D¯ +ργ¯+ργ¯+τkeiωτk(A+Bργ¯)
. Then we chooseD¯ as follows:
¯
D= +ργ¯+ργ¯+τkeiωτk(A+Bργ¯)
–
. It is easy to see thatq∗(s),q(θ)= andq∗(s),q¯(θ)= .
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In the remainder of this section, we also use the same notations to compute the coordi-nates, which describe the center manifoldCatμ= .
Define
z(t) =q∗,ut, W(t,θ) =ut(θ) –zq–z¯q¯=ut(θ) – Rez(t)q(θ), () whereutandWare real functions.
By the definition of center manifoldC, we know that
W(t,θ) =Wz(t),z¯(t),θ=W(θ)
z
+W(θ)z¯z+W(θ)
z
+· · · ()
from (), wherezand¯zare local coordinates for the center manifoldCin the directions
By comparing the coefficients with (), we getg,g,gandg. And we need to
com-puteW(θ) andW. By () and (), we know that
˙
W=ut˙ –zq˙ –z˙¯q¯
=
⎧ ⎨ ⎩
AW– Re(q¯∗(θ)fq(θ)), θ∈[–, ],
AW– Re(q¯∗(θ)fq(θ)) +f(z,z¯), θ=
=AW+H(z,¯z,θ), () where
H(z,¯z,θ) =H(θ)
z
+H(θ)zz¯+H(θ)
¯
z
+· · ·. ()
On the other hand, by taking the derivative with respect totin (), we know that
˙
W=Wzz˙+WZ¯z˙¯ ()
from (), (), () and (), which together with () and () gives that (A– iωτ)W(θ) = –H(θ),
AW(θ) = –H(θ),
(A+ iωτ)W(θ) = –H(θ).
By using () forθ∈[–, ], we know that
H(z,¯z,θ) = –Req¯∗(θ)f(z,z¯)q(θ)
= –g(z,z¯)q(θ) –g¯(z,z¯)q¯(θ).
Comparing the coefficients with (), we obtain that
H(θ) = –gq(θ) –g¯q¯(θ) ()
and
H(θ) = –gq(θ) –g¯q¯(θ). ()
From (), () and the definition ofA, we know that
W(θ) =
ig
τω
q()eiωτθ+ig
iωq¯()e
–iωτθ+E
eωθ. ()
Similarly, we know that
W(θ) =
ig
τω
q()eiωτθ+ig¯
iωq¯()e
–iωτθ+
E ()
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from () and (), where E= (E() ,E ()
)∈R andE= (E() ,E ()
)∈Rare constant
vec-tors.
If we solve these forEandE, we computeW(θ) andW(θ) from (), (), () and
confirm the following values to investigate the qualities of the bifurcation periodic solution in the center manifold at the critical valueτk(see []).
To this end, we express eachgijin terms of parameters and delay. Then we obtain the following values:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
C() =ωτi (gg– |g|
–|g|
) +
g
,
μ= –ReRe{{Cλ(()τ)}}, β= Re{C()},
T= –Im{C()}+μIm{λ
(τ)}
ω .
() From the above analysis, we obtain the following theorem.
Theorem Ifτ=τk,then the stability and the direction of periodic solutions of the Schrödin-gerean Hopf bifurcation of system()are determined by the parametersμ,βand T.
(i) The direction of the Schrödingerean Hopf bifurcation is determined by the sign ofμ:
ifμ> (resp.μ< ),then the Schrödingerean Hopf bifurcation is supercritical
(resp.subcritical),and the bifurcation periodic solution exists forτ>τ(resp.τ<τ).
(ii) The stability of the Schrödingerean bifurcation periodic solution is determined by the sign ofβ:ifβ> (resp.β< ),then the Schrödingerean bifurcation periodic
solution is stable(resp.unstable).
(iii) The sign ofTdetermines the period of the Schrödingerean bifurcation periodic
solution:ifT> (resp.T< ),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 incorpo-rated into the model. The stabilities of equilibrium point and Schrödingerean Hopf bifur-cation 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 bifurca-tion. 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.
Competing interests
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.
Author details
1Department of Computer Science and Technology, Harbin Engineering University, Harbin, 150001, China.2Academic Affairs Department, Guizhou City Vocational College, Guiyang, 550025, China.
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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Received: 2 December 2016 Accepted: 7 March 2017
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