Bifurcation Behavior and Hybrid Controller Design of a 2D Lotka–Volterra Commensal Symbiosis System Accompanying Delay

: All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing ﬁxed point theorem, inequality tactics and an appropriate function, we gain the sufﬁcient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive deﬁnite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis sys-tem. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes.


Introduction
It is well known that the delayed dynamical model is a vital tool to solve many biological questions.In order to describe the interrelation and internal law of biological populations, many scholars pay great attention to the construction of various predator-prey models.By exploring the various dynamical behaviors of predator-prey models, we can effectively control the densities of predators and prey in the natural world.Recently, many works on predator-prey models have been published and a great deal of excellent works have been presented.For instance, Xiang and Wang [1] focused on the stabilization and boundedness of a prey-predator system involving disease in predator and prey-taxis.Peng and Yu [2] discussed the Turing pattern in a diffusive prey-predator system involving herd behavior and nonlocal delay.Khan et al. [3] investigated the bifurcations and chaos of a 2D discrete prey-predator system.Yan et al. [4] analyzed the bifurcation and stationary pattern in a Leslie-Gower prey-predator system involving prey-taxis.For more detailed studies, one can see [5][6][7][8][9][10].
In 2020, Zhu et al. [11] proposed the following Lotka-Volterra commensal symbiosis system: . ( where u 1 (t) represents the density of the first species and u 2 (t) represents the density of the second species, r 1 , r 2 stand for the intrinsic growth of u 1 , u 2 , respectively, l 1 , l 2 are the carrying capacities of u 1 , u 2 , a is the relationship coefficient between u 1 and u 2 , q 1 , q 2 are catchability parameters, E stands for a fishing business that is used for harvest, m i (i = 1, 2, 3, 4) is the proper real constant.All the parameters r 1 , r 2 , l 1 , l 2 , q 1 , q 2 , E, m 1 , m 2 , m 3 , m 4 are positive constants.For a more concrete meaning of model ( 1) one can see [11].Zhu et al. [11] explored the partial survival extinction and global stability of the equilibrium point of model (1).Here we would like to point out that in many cases, the development of species relies on not only the current time but also the history time, based on this viewpoint, it is necessary to introduce delay into the biological models.According to this idea, we assume that there exists a self-feedback time from the first species u 1 to the first species u 1 and a self-feedback time from the first species u 2 to the first species u 2 .Then we can lightly formulate the following delayed Lotka-Volterra commensal symbiosis system: where θ > 0 is a time delay that stands for self-feedback time.
From the viewpoint of mathematics, delay is a vital factor that affects the dynamical traits of various differential systems.In various cases, a delay will result in an alteration of stability, the emergence of bifurcation and the onset of chaos and so on [12].One can also see [13][14][15][16][17][18][19].In particular, delay-caused Hopf bifurcation is an important dynamical phenomenon.Biologically, delay-caused Hopf bifurcation can give a good description of the balanced relationship among the density of various biological populations.In order to reveal the interaction relationship of various biological populations, we argue that it is of great importance to explore the delay-caused Hopf bifurcation for many biological models.Inspired by this idea above, we are to focus on the delay-caused Hopf bifurcation and control aspect of bifurcation for system (2).To be specific, we are going to deal with the following key questions: (1) Analysis of the peculiarity of solution (e.g., non-negativeness, existence and uniqueness and boundedness) of solution to system (2).(2) Study the emergence of Hopf bifurcation phenomenon and stability issue of system (2).(3) Construct both distinct controllers to adjust the domain of stability and the time that Hopf bifurcation of system (2) generates.
The main contributions of this study are introduced as follows: (i) On the basis of the previous publications, a new delay-independent bifurcation and stability criterion for system (2) is set up. (ii) Making use of distinct controllers, the domain of stability and the time that Hopf bifurcation of system (2) generates are controlled with effect.(iii) The influence of delay on commanding Hopf bifurcation and stabilizing the densities of the first species and the density of the second species of system (2) are offered.(iv) By constructing a suitable positive definite function, we obtain the sufficient condition ensuring the global stability of system (2).
The structure of this article is stated as follows.The peculiarity of the solution (e.g., boundedness, non-negativeness, existence and uniqueness) of system (2) is discussed in Section 2. Section 3 deals with the bifurcation phenomenon and stability of system (2).Section 4 explores the global stability of system (2).Section 5 focuses on the control problem of the bifurcation phenomenon for system (2) by formulating a reasonable hybrid delayed feedback controller involving parameter perturbation accompanying delay and state feedback.Section 6 handles the the control problem of bifurcation phenomenon for system (2) via formulating a reasonable extended hybrid delayed feedback controller including parameter perturbation accompanying delay and state feedback.Section 7 displays Matlab software (latest veresion 2023b) simulation outcomes to test the validity of the acquired key results.A laconic conclusion is drawn to complete this work in Section 8. 1) is an ordinary differential system, model ( 2) is a delayed differential system that is more reasonable than model (1) and can better describe the objective reality in biology.Thus, model ( 2) is new.

Peculiarity of Solution
In this part, we are going to explore the non-negativeness, existence and uniqueness, and boundedness of the solution for system (2) by virtue of fixed point theorem, inequality skills and a reasonable function. where For arbitrary U, Ũ ∈ ∆, one gains where Then it follows from Equation (5) that Thus f (U) conforms to Lipschitz condition for U (see [17]).Using fixed point theorem, we can easily conclude that Theorem 1 is true.Theorem 2. All solutions of system (2) starting with R 2 + are non-negative.
Proof.Assume that U(0) = (u 1 (0), u 2 (0)) is the initial value of system (2).By the first equation of system (2), we obtain which leads to Hence, Then, By the second equation of system (2), we obtain which leads to Hence, Then, The proof of Theorem 2 ends.
, then all solutions of system (2) starting with R 2 + are uniformly bounded. Then, where By Equation (18), we obtain Therefore, all the solutions of the system (2) are uniformly bounded.

Bifurcation Research
Assume that system (2) has the equilibrium point: Let The linear system of system (2) around E(u 1 , u 2 ) takes the following expression: The characteristic of Equation ( 25) owns the following expression: where If θ = 0, then Equation (26) reads as: is fulfilled, then the two roots λ 1 , λ 2 of Equation (28) have negative real parts.Thus the equilibrium point E(u 1 , u 2 ) of the model ( 2) under θ = 0 holds a locally asymptotically stable state.Suppose that λ = iφ is the root of Equation ( 26).Then Equation ( 26) takes which generates It follows from Equation (31) that It follows from Equation ( 33) that where It follows from Equations ( 34) and ( 35) that By Equation (36), one obtains where In view of cos 2 φθ + sin 2 φθ = 1, we gain where ) 2 < 0 holds, noticing that lim φ→+∞ ∆ 1 (φ) = +∞ > 0, then we know that Equation (38) admits at least one positive real root.Thus Equation (26) owns at least one pair of purely roots.
In the sequel, the following condition is given: Proof.By Equation (26), one gains which results in dλ dθ where Hence Re dλ dθ In view of (G 3 ), one gains Re dλ dθ which completes the proof.
Based on the study above, the following results are easily acquired.

Global Asymptotically Stability of System (2)
In this section, we explore the global asymptotic stability of the system (2).Assume that Theorem 5.If (G 4 ) holds, then the equilibrium point E(u 1 , u 2 ) of the model ( 2) is global asymptotically stable. Proof.Define Then If (G 4 ) is fulfilled, then dV(t) dt ≤ 0, which implies that Theorem 5 is true.
By (G 7 ), one obtains Re dλ dθ which completes the proof.
Based on the study above, the following conclusion is easily acquired.
Remark 2. In model ( 50), we adjust the growth rate of the density of the second species via changing state feedback and parameter perturbation with delay.

Bifurcation Domination via Hybrid Controller II
In this part, we are to explore the Hopf bifurcation control issue of the system (2) by virtue of a suitable hybrid controller consisting of state feedback and parameter perturbation involving delay.According to the idea in [22], one can lightly formulate the following controlled 2D Lotka-Volterra commensal symbiosis system: where θ stands for control parameter.System (50) and system (2) own the same equilibrium points E(u 1 , u 2 ).The linear system of system (80) around E(u 1 , u 2 ) takes the following expression: (82) The characteristic equation of system (83) owns the following expression: which leads to where If θ = 0, then Equation (85) reads as: is fulfilled, then the two roots λ 1 , λ 2 of Equation (87) have negative real parts.Thus the equilibrium point E(u 1 , u 2 ) of the model (80) under θ = 0 keeps a locally asymptotically stable state.Suppose that λ = iξ is the root of Equation (85).Then Equation (85) takes It follows from (89) that which leads to It follows from (91) that where It follows from ( 92) and (93) that where In view of cos 2 ξθ + sin 2 ξθ = 1, we obtain where Suppose that then we know that Equation (97) admits at least one positive real root.Thus, Equation (85) owns at least one pair of purely roots.Without loss of generality, here we assume that Equation (97) has eight positive real roots (say where ν } and assume that when θ = θ * 0 , (85) has a pair of imaginary roots ±iθ * 0 .
Next, the following condition is provided: where Hence, Re dλ dθ By (G 10 ), one gains Re dλ dθ which completes the proof.
Based on the study above, the following conclusion is easily acquired.Remark 3. In model (80), we adjust the growth rates of the densities of the first species and the second species via changing state feedback and parameter perturbation with delay.
Remark 4. Zhu et al. [11] dealt with the global stability and partial survival extinction of the model (1).In this paper, we set up a more reasonable delayed predator-prey model and explore the bifurcation behavior and hybrid controller design of the formulated Lotka-Volterra commensal symbiosis model.Theoretically speaking, the research methods have enriched the bifurcation theory of delayed differential equations to some degree.Biologically speaking, the obtained results of this article play a vital role in controlling the densities of predator species and prey species.Based on this viewpoint, we think that this paper has some novelties.

Matlab Simulations
Example 1.Consider the following Lotka-Volterra commensal symbiosis system accompanying delay: It is easy to acquire that system (107) admits a unique positive equilibrium point E(0.4678, 0.2431).One can easily verify that the conditions (G 1 )-(G 3 ) of Theorem 4 hold true.By applying Matlab software (latest version 2023b), one can obtain θ 0 ≈ 2.22.To Validate the correctness of the acquired assertions of Theorem 4, we choose both different delay values: θ = 2.17 and θ = 2.45.For θ = 2.17 < θ 0 ≈ 2.22, we obtain simulation diagrams which are presented in Figure 1.Based on Figure 1, we find that u 1 → 0.4678, u 2 → 0.2431 when t → +∞.In other words, the equilibrium point E(0.4678, 0.2431) of the model (107) holds a locally asymptotically stable state.Biologically speaking, the density of the first species and the density of the second species will tend to 0.4678, 0.2431, respectively.For θ = 2.45 > θ 0 ≈ 2.22, we obtain simulation diagrams which are presented in Figure 2. Based on Figure 2, we find that u 1 will keep the periodic vibrating level around the value 0.4678, u 2 will keep the periodic vibrating level around the value 0.2431.That is to say, a family of periodic solutions (namely, Hopf bifurcations) appear near the equilibrium point E(0.4678, 0.2431).Biologically speaking, the density of the first species and the density of the second species will keep periodic vibration around the values 0.4678, 0.2431, respectively.
nying delay: 3) It is easy to acquire that system (6.3)admits a unique positive equilibrium point E(0.4678, 0.2431).Let β 1 = 0.5, β 2 = 0. Example 2. Consider the following controlled Lotka-Volterra commensal symbiosis system accompanying delay:  It is easy to acquire that system (108) admits a unique positive equilibrium point E(0.4678, 0.2431).Let α 1 = 0.84, α 2 = 0.05.One can easily verify that the conditions (G 5 )-(G 7 ) of Theorem 6 hold true.By applying Matlab software, one can obtain θ * ≈ 2.5.To Validate the correctness of the acquired assertions of Theorem 6, we choose both different delay values: θ = 2.4 and θ = 2.9.For θ = 2.4 < θ * ≈ 2.5, we obtain simulation diagrams which are presented in Figure 3. Based on Figure 3, we find that u 1 → 0.4678, u 2 → 0.2431 when t → +∞.In other words, the equilibrium point E(0.4678, 0.2431) of the model (108) holds a locally asymptotically stable state.Biologically speaking, the density of the first species and the density of the second species will tend to be 0.4678, 0.2431, respectively.the domain of stability of system (107) and postpone the time of emergence of Hopf bifurcation of system (107) via the formulated two hybrid delayed feedback controllers.

Conclusions
It is well known that the delayed dynamical model is a vital tool for describing the interaction of different biological populations in the natural world.During the past decades, a great deal of work on predator-prey models has been carried out and rich fruits on this topic have been reported.In this paper, we propose a new delayed Lotka-Volterra commensal symbiosis model.The existence and uniqueness, non-negativeness and boundedness of the solution of the delayed Lotka-Volterra commensal symbiosis system are discussed.The Hopf bifurcation issue is discussed.Sufficient conditions on the stability and bifurcation of this model are obtained.The critical delay value θ 0 is acquired.In order to adjust the domain of stability and the time of appearance of the bifurcation phenomenon of this model, we have successfully designed two different hybrid delayed feedback controllers.Two critical delay values θ * , θ are acquired.In these two controllers, the role of delay is displayed.The exploration fruits have great theoretical value in controlling and balancing the densities of two species.By adjusting the delay value, we can delay or advance the time of cycle motion of the two species.In addition, the exploration ideas can be used to dominate the bifurcation phenomenon, stability and chaos in various fractional-order and integer-order dynamical systems in numerous fields.In 2020, Zhu et al. [11] investigated the partial survival, extinction and global attractivity of the positive equilibrium point of the model (1).In this work, we introduce a delay into model ( 1) and obtain model (2).We have dealt with the boundedness, existence and uniqueness of the solution, Hopf bifurcation and its control problem of the formulated model (2).The research method of this paper is different from that of Zhu et al. [11] and the gained results are entirely innovative.Based on this point, we think that our studies replenish the work of Zhu et al. [11] to a certain degree.From a biological point of view, we only consider the growth rates of the density of the first and the second species depending on the same feedback time.In the future, we will deal with the controlled models (48) and (50) involving two different delays.