Complex Dynamics in a Singular Delayed Bioeconomic Model with and without Stochastic Fluctuation

A singular delayed biological economic predator-prey systemwith andwithout stochastic fluctuation is proposed.The conditions of singularity induced bifurcation are gained, and a state feedback controller is designed to eliminate such bifurcation. Furthermore, saddle-node bifurcation is also showed. Next, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of roots of the corresponding characteristic equation, and the hybrid control strategy is used to control the occurrence of Hopf bifurcation. In addition, some explicit formulas determining the spectral densities of the populations and harvest effort are given when the system is considered with stochastic fluctuation. Finally, numerical simulations are illustrated to verify the theoretical results.


Introduction
The dynamic relationship between predator and prey has long been and will still be one of the dominant themes in both biology and mathematical biology because of its universal existence and importance.In the description of dynamics interactions, a crucial element of all models is the classical definition of functional response.Lots of predator-prey models with Holling type [1], Leslie-Gower type [2], and Beddington-DeAngelis type [3,4], and so forth have been investigated extensively by scholars.However, some predatorprey models in which prey population exhibits herd behavior, such as plankton-phytoplankton model [5], may appear in realistic world.Since the use of the square root properly accounts for the assumption that the interactions occur along the boundary of the population, Ajraldi et al. [6] proposed the following predator-prey system in which interaction terms use the square root of prey population rather than simply prey population: where  and  denote the prey and predator, respectively.The prey population exhibits a highly socialized behavior and lives in herds as the form √()(); that is, the weaker individuals are being kept at the center of their herd for defensive purpose.Braza [7] also investigated the dynamics of system (1) and showed that the prey exhibits strong herd structure and the predator interacts with the prey along the outer corridor of the herd of prey.Recently, Yuan et al. [8] considered a predator-prey system as follows: where − 2 represents the quadratic mortality for predator population.Predator-prey systems with such functional response have attracted little attention (see [6][7][8][9]).It is well-known that time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, capturing time, or other reasons.Delay differential equations often show much more complicated dynamics than ordinary differential equations because a time delay can cause a stable equilibrium to become unstable and cause the population to fluctuate.Many authors have been devoted to investigating the time delay effect on the dynamics of system and obtained some results (see [10][11][12][13][14][15]). Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in system (2), Xu and Yuan [9] introduced a time delay into system (2) and obtained the local stability and the existence of Hopf bifurcation of this system.However, bifurcate oscillation is harmful in some engineering applications, which has enormous potential in many technological disciplines such as power networks protection.Bifurcation control, which refers to the aim of designing a controller to suppress or reduce some existing bifurcation dynamics of a given system, can be useful.Various disciplines are attracted to bifurcation control and various methods of bifurcation control can be found [16][17][18].
In addition, Gordon [19] proposed the economic theory of a common-property resource, which focuses on the effect of the harvest effort on the ecosystem from an economic perspective.If () and () represent the harvest effort and the density of harvested population, respectively, then the total revenue TR = ()() and the total cost TC = (), where  represents unit price of harvested population and  represents the cost of harvest effort.Thus, an algebraic equation, which considers the economic interest V of the harvest effort on the harvested population, is established as follows: Based on the Gordon [19] theory and theory of singular system, Zhang et al. [20] first proposed a class of singular biological economic systems.Some results on those systems, such as the stabilities, bifurcations, and chaos, can be found in [20][21][22][23][24].
Based on the previous models, we establish the following predator-prey system consisting of two differential equations and an algebraic equation as follos: where , , and  are the prey, predator, and the harvest effort at the time , respectively. denotes the time delay which means the growth rate of predator species depends on the number of the prey species  units of time earlier.It is important to make some simplifying assumptions to discern the basic dynamics and to make the analysis more tractable.In order to simplify system (4), we use the following dimensionless transformations:  = /,  = / √ ,  = /, and  = .Then system (4) can be rewritten as follows: where  =    √ ,  =  √  b/,  = c,  = √  d/,  = m/, and  = p.
In the papers [6][7][8][9], the author used the simplifying assumption that  = 0; that is, the average handling time is zero.In line with the work in [6][7][8][9], we also assume that  = 0. Thus, system (5) takes the form: The initial conditions of system (6) are In reality, the environmental fluctuation is one of the important components for ecological systems.Many natural phenomena do not follow the deterministic law and usually oscillate randomly around some average values.The deterministic approach has limitations on mathematical modeling, which makes the accurate prediction for the future dynamics of system very difficult.Stochastic differential equation models play a significant role in various dynamic analysis, because they can provide some additional degree of realism compared to their deterministic counterpart [25].Recently, some results of the stochastic modeling of ecological populations are presented [26][27][28][29][30].
In order to study the effects of the environmental fluctuations on a delayed singular prey-predator bioeconomic model, the following stochastic model corresponding to the delayed system (4) in the fluctuating environment is given: The perturbed terms ξ ,  = 1, 2, 3, are mutually independent Gaussian white noises characterized by ⟨ ξ ⟩ and ⟨ ξ () ξ ( 1 )⟩ =   ( −  1 ), ,  = 1, 2, 3.The symbol ⟨⋅⟩ is the ensemble average due to the effect of the fluctuating environment,   is the Kronecker delta and represent the spectral density of the white noise, and  is the Dirac delta function with  and  1 being the distinct times.By taking the same translations and same notations, we can obtain the following form: where  1 () = ξ1 ()/,  2 () = ξ2 ()/ √ , and  3 () = ξ3 ()/.In addition, the initial conditions of system (9) are also similar to that of system (6).
The rest of paper is organized as follows.In the next section, some conditions for the existence of the positive equilibrium, saddle-node bifurcation, singularity induced bifurcation, and Hopf bifurcation are obtained and bifurcation controls in deterministic model are also showed.Some explicit formulas determining the spectral densities of the populations and harvest effort in the singular bioeconomic system with the fluctuation environment are gotten in Section 3. To support our theoretical predictions, some numerical simulations are included in Section 4. A brief discussion is also given in the last section.

Dynamics of the Deterministic Model
2.1.Existence of the Positive Equilibrium.From the viewpoint of biological interpretation, we only consider the positive equilibrium.There exists a positive equilibrium (, , ) of system (6), where the values , , and  satisfy the following equations: From (10b) and (10c), we can obtain  = (/)√ and  = /( − ), respectively.Substituting the above values into (10a), we can obtain that  satisfies the following equation: By simple computation, we obtain that Suppose that  sn > 0 and (1)  >  sn : there is no positive equilibrium; ( If the parameter  is taken as bifurcation parameter, the singularity induced bifurcation, which is first introduced by Venkatasubramanian et al. [31], may appear around the interior equilibrium.The singularity induced bifurcation does not occur in a normal ordinary differential equation system, which has been characterized by a singular system.Roughly speaking, a singularity induced bifurcation refers to a stability change of the singular system, which leads to an impulse phenomenon of the singular system and may even result in the collapse of the system (see [32,33]).Thus, we give some results on the singularity induced bifurcation of singular system.
Theorem 3. If (H1) holds, then system (13)  Thus, all conditions of Lemma 2 are satisfied.System (13) has a singularity induced bifurcation at the positive equilibrium  * ( * ,  * ,  * ) when the bifurcation parameter  = 0. Furthermore, the constants  and  can be computed: which follows that If parameter  increases through zero, then one eigenvalue of system (13) will move from  + to  − along the real axis by diverging into the infinity.Thus, the stability of the positive equilibrium  * ( * ,  * ,  * ) changes from unstable to stable.The Jacobian of system (13) evaluated at  * takes the form: According to the leading matrix Ξ in system (13) and   * , here Ξ = ( ), the characteristic equation of system (13) at  * is det(Ξ −   * ) = 0.That is, the characteristic equation can be expressed: It is obvious that the rest eigenvalue (denoted by  2 ) has negative real part.Furthermore, it follows from Theorem 1.1 [34] that there is only one eigenvalue diverging to infinity, but the rest eigenvalue is continuous, nonzero, and cannot jump from on half open complex plane to another one as  increases through 0. Table 1 shows the change in the signs of the real parts of eigenvalues ( 1 and  2 ) due to the variation of economic interest of harvest effort.According to Table 1, it can be concluded that system ( 13) is stable at  * as  < 0 and unstable as  > 0. Consequently, a stability switch occurs as  increases through 0.
Remark 4. According to [35], system (13) along the positive equilibrium locus yields index one matrix pencil (Ξ, ) for  ̸ = 0 and index two matrix pencil (Ξ, ) for  = 0.This shows that there exists one index jump at a bifurcation point  = 0.It leads to the rapid expansion of the population from the point of view of biology.If this phenomenon lasts for a long time, the population will be beyond the carrying capacity of the environment and the prey-predator system will be out of balance, which is disastrous.With the purpose of economic interest of harvest effort at an ideal level as well as maintaining the sustainable development of the biological resource, some related measures should be taken to eliminate the impulse phenomenon and stabilize system (13) when the economic interest is positive.Thus, a state feedback controller is designed to stabilize system (13) at the positive equilibrium  * .
By using Theorem 3-1.2 [36], a state feedback controller () = (() −  * ), where  is a feedback gain and  * is the component of the positive equilibrium  * , can be applied to stabilize system (13) at  * .Thus, a controlled system is as follows: where all variables and parameters have the same interpretations as that in system (13).The feedback gain can be given in the following result.
Theorem 5. Suppose the feedback gain  satisfies one of the following cases: the controlled system ( 24) is stable at the positive equilibrium  * .
Proof.The Jacobian of system (24) evaluated at the interior equilibrium  * has the form According to the leading matrix Ξ in system (13) and J * , the characteristic equation of the controlled system (24) at  * is By using the Routh-Hurwitz criteria [37], the sufficient and necessary condition for the stability of the controlled system (24) at  * is that the feedback gain  satisfies the following: if  − 2 > 0, then  < 2( −  − )/( − 2); otherwise,  > 2( −  − )/( − 2).This ends the proof.
Remark 6.After introducing the feedback controller into system (13), the controlled system (24) can be stabilized at the positive equilibrium.The elimination of the singularity induced bifurcation means the prey-predator system restores ecological balance.Both sustainable development of the preypredator system and the ideal economic interest of harvesting can be obtained by enhancing or reducing the harvest effort on the prey.

Saddle-Node Bifurcation.
If  is taken as bifurcation parameter, the following result is true.

Hopf Bifurcation and Control.
From the discussion above, we know that if (H1) holds, there is only one positive equilibrium  * ( * ,  * ,  * ); if  <  sn , there are two positive equilibriums  + * ( where  = (, , )  .In order to analyze the local stability of the positive equilibrium of system (6), we first use the linear transformation   () =   (), where ) .
In order to investigate the distribution of roots of the transcendental equation ( 34), we introduce the following result proved by Ruan and Wei [10].
Lemma 9. Assume that (H1) and (H2) hold, then the two roots of (36) have always negative real parts; that is, system (6) with  = 0 is locally asymptotically stable.
Let  ( > 0) be the root of ( 34), and we have the following equations by substituting it into (34) and separating the real and imaginary parts: which gives where Let  2 = ; (38) takes the form It is obvious that (39) has no real roots when Δ =  2 1 −4 0 < 0. When  0 > 0, (39) has two negative roots; when Δ > 0 and  0 < 0, (39) has one positive root.Thus, we have the following results.

If condition (ii) of Lemma 10 holds, and we denote
then ± + are a pair of purely imaginary roots of ( 34) with  =  +  .Next, we will check whether the following transversality conditions are satisfied.
Next, we will check whether the following transversality condition is satisfied.Differentiating the two sides of (46) with respect to  and applying the implicit theorem, we get For simplify, we define ω+ as  and τ+  as , and we can obtain Since  2 11 − 2 10 > 0, then the transversality condition is satisfied.Summarizing the above results, we have the following theorem.
(2) If  2  10 −  2 00 ≤ 0, the positive equilibrium of controlled system (44) is locally asymptotically stable for  ∈ [0, τ0 ) and unstable when  > τ0 , and controlled system (44) undergoes a Hopf bifurcation at the equilibrium  * when  = τ0 .Remark 14.For the controlled system (44), it has been proven in Theorem 13 that we can delay the onset of Hopf bifurcation without changing the original equilibrium by choosing appropriate parameter.
(57) Thus, a new equation is given by where and the symbol  *  denotes the algebraic cofactor of   , ,  = 1, 2, 3.
If the function Υ() has zero mean value, then the fluctuation intensity of the components is the frequency band  and  + d is  Υ() d, where the spectral density  Υ() is formally defined by where the equation ⟨  ⟩ = 0 and ⟨  ()  ( 1 )⟩ =   ( −  1 ), ,  = 1, 2, 3, are used to obtain (62).Furthermore, from    () = 1,  = 1, 2, 3, the fluctuation intensities of (), (), and () are written as follows (64) It should be pointed out that when  > 0 the explicit values of the spectral densities of the populations and harvest efforts are difficult to obtain as the evaluation of the integrals is not easy at all.However, at a given value of time delay , the fluctuation intensities can be determined by numerical integration.

Numerical Simulations
With the help of MATLAB, some numerical results of system (6) are provided to substantiate the analytic results in this section.

Singularity Induced Bifurcation and State Feedback Control.
By selecting harvest effort  as the parameter, singularity induced bifurcation at the interior equilibrium  * is obtained.By using Theorem 3, it can be shown that system (13) has a singularity induced bifurcation at the interior equilibrium  * , and a stability switch occurs as  increases through 0. Further, a state feedback controller () can be applied to stabilize system (13) at  * .According to Theorem 5, if the feedback gain  satisfies  > −17.24, then the controlled system ( 24) is stable at  * and the singularity induced bifurcation of system ( 13) is also eliminated.The dynamical responses of system (13) and the controlled system (24) can be shown in Figures 1 and 2.

Hopf Bifurcation and Hybrid Control.
Since the assumption (H2) is true, the positive equilibrium  * of system (6) without any time delay is locally asymptotically stable (see Figure 3).
Based on the given parameter values, the critical value of time delay  0 = 2.955 can be obtained by solving the corresponding expression.From Theorem 12, the corresponding waveforms are shown in Figures 4 and 5.That is,  when  = 2.950 <  0 = 2.955, the positive equilibrium  * of the system is locally asymptotically stable (see Figure 4), and Hopf bifurcations occur once  = 2.960 >  0 = 2.955 (see Figure 5).Now we choose appropriate value of parameter  to control singular biological economic system (6).By choosing  = 0.85, which has the same equilibrium point as that of the original system, we find that the periodic solutions of the controlled system (44) are eliminated (Figure 6).Of course,   7-8, it is clear to see that the environmental fluctuation plays a crucial role in determining the magnitude of oscillation (as the magnitude of delay parameters is the same in both cases).Figure 8 has shown that the intensity of fluctuation for the population and harvest effort increase gradually from their steady state values as the delay parameter  increases.

Conclusion and Discussion
After Ajraldi et al. [6] proposed a predator-prey system (1) with square root functional response, Xu and Yuan [9] introduced a time delay and studied the effect of time delay on system (2).In addition, Zhang et al. [30] studied the dynamics of a predator-prey system with stochastic fluctuation.Based on the above work, we propose a singular delayed biological economic system with herd behavior and within the deterministic environment or fluctuating environment, which yields system (4) and system (8).
For the deterministic model, we obtain the conditions for the existence of the positive equilibrium.Singularity induced bifurcation which only occurs in the singular system is also obtained.Thus, a state feedback controller is designed to eliminate the bifurcation phenomenon.The condition of saddle-node bifurcation is also given.In order to study time delay on the effect of system (6), we introduce the new normal form of differential-algebraic systems due to the work of literatures [38,39] and analyze the local stability of the positive equilibrium and the existence of Hopf bifurcation by taking time delay  as bifurcation parameter.From simulations, we found that the critical value of time delay  is less than that in the literature [9].From the biological point of view, the predator species has to shorten its time interval to survive when the prey species is predated by natural or manmade factors.Since bifurcation oscillation is harmful in some field, the hybrid control strategy is introduced into system (6).Thus, the Hopf bifurcation may advance, delay, and even eliminate by selecting the proper value of parameter.In fact, Hopf bifurcation of the controlled system (44) with hybrid control strategy can be eliminated when á = 0.85.In addition, for the stochastic model, we find that the populations and harvest effort oscillate when the system is effected by the stochastic fluctuation (see Figures 7 and 8).

Figure 1 :
Figure 1: The dynamical responses of system (13) without feedback control.

Figure 2 :
Figure 2: The dynamical responses of harvest effort of system (24) with feedback control.

Table 1 :
Signs of real parts of eigenvalues of system (13) at  * .