New applications of Schrödinger type inequalities to the existence and uniqueness of Schrödingerean equilibrium

As new applications of Schrödinger type inequalities appearing in Jiang (J. Inequal. Appl. 2016:247, 2016), we first investigate the existence and uniqueness of a Schrödingerean equilibrium. Next we propose a tritrophic Hastings-Powell model with two different Schrödingerean time delays. Finally, the stability and direction of the Schrödingerean Hopf bifurcation are also investigated by using the center manifold theorem and normal form theorem.


Introduction
A biological system is a nonlinear system, so it is still a public problem how to control the biological system balance. Previously a lot of research was done. Especially, the research on the predator-prey system's dynamic behaviors has obtained much attention from the scholars. There is also much research on the stability of 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 is 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 equal time 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 modified food web model was also investigated. By introducing disease in the prey population, Shilnikov et al. (see []) modified the Hastings-Powell model and the stability of biological feasible equilibria was also obtained.
In this paper, we provide a differential model to describe the Schrödinger dynamic of a Schrödinger Hastings-Powell food chain model. In a three species food chain model x represents the prey, y and z represent two predators, respectively. 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 the 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  and K  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 Schrödinger delays 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 a 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 transitioned to predator and predator transitioned to top predator, respectively.

Equilibrium and local stability analysis
Letẋ = ,ẏ =  andż = . We introduce five non-negative Schrödinger equilibrium points of the system as follows: The Jacobian matrix for the Schrödinger system () at E * = (x * , y * , z * ) is as follows: Then we have where If τ  = τ  = , then the corresponding characteristic () is rewritten as follows: Lemma . Suppose that the following conditions hold (see []): Then the positive Schrödinger equilibrium E * of the Schrödinger system () is locally asymptotically stable for τ  and τ  .

Existence of Schrödingerean Hopf bifurcation
The characteristic () reduces to where Let λ = iω (ω > ) be a root of (). And then we have By separating the real and imaginary parts we know that From () we obtain which show that Let z = ω  . Then we have It is easy to see that ±iω is a pair of purely imaginary roots of (). It follows from () where k = , , ,  and j = , , , . . . . Proof Taking the derivative of λ with respect to τ in (), we have and For simplicity we define ω k = ω and τ k = τ . From (), (), (), and () we have This completes the proof of Lemma ..
By applying Lemmas . and ., we have the following result.
Theorem . For the Schrödinger system (), the following results hold.
Lemma . Suppose that P R Q R + P I Q I = . Then we have Proof By taking the derivative of λ with respect to τ  in (), we have (see []) Since P R Q R + P I Q I = , we obtain So we complete the proof of Lemma ..
By applying Lemmas . and ., we prove the existence of the Schrödinger Hopf bifurcation.
Theorem . For the Schrödinger system (), the following results hold.

Lemma . Suppose that z
Proof This proof is similar to the proof of Lemma ., so we omit it here.
By applying Lemmas . and . to () we have the following result.
Theorem . For the Schrödinger system (), the following results hold.
Case IV: τ  = τ  = . We consider () with τ  in the stability range. Regarding τ  as a parameter, and without loss of generality, we only consider the Schrödinger system () under the case I.