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Stability Analysis and Control Optimization of a Prey-Predator Model with Linear Feedback Control.

1. Introduction

In the practical production, effective control of pests is a very important issue of the world, which catches attention of scholars for pest management method [1-7]. Integrated pest management (IPM), also known as integrated pest control (IPC), is an effective approach that integrates biological, chemical tactics, and physical methods for pests control [8-11]. Due to population dynamics and its related environment, IPM utilize effective methods and techniques comprehensively to reduce the level of economic harm caused by pests. The aim of IPM is to control the density of the insects under the economic threshold by integrated usage of less harmful pesticides and biological control methods for maximizing the protection of the ecosystem.

In mathematics, impulsive differential equations (IDES) is such a powerful tool to describe these phenomena that rapid changes in biological populations are caused by the variety of the pests control by artificial intervention [12-22]. In recent years, the theoretical studies on IDES have produced a lot of good research results [23-34]. Based on the theoretical research, some scholars have introduced impulsive differential equations in Lotka-Volterra system such as the regular release of predators [35-37]; the periodic release of infected pests [38-40]; the periodic release of predators together with regular spray of pesticides [41-43]; the periodic release of predators and infected pests together with regular spray of pesticides [39, 44]. In the practical application, the two control measures can be adopted at two different levels of pest density concerning this case. Nie et al. [45], Tian et al. [46], Zhao et al. [47], and Zhang et al. [48] studied the following predator-prey system and assumed that different control measures were adopted at different thresholds,

[mathematical expression not reproducible], (1)

where the intrinsic growth rate of prey is denoted by a, the environment carrying capacity is denoted by K, the predation rate by natural enemies is denoted by b, and the transformation rate and the death rate of predator are denoted by [lambda] and d, respectively. The [eta] is a positive parameter, and the effect of pesticide to predator and prey species is denoted by [??] and [delta], respectively. The releasing quantity of natural enemy v(t) are denoted by [gamma] and [tau], respectively.

It is of great practical significance to adopt biological and chemical control strategies based on the different pest thresholds. But an important issue in this process should be pointed out, in which the biological control is carried out when the density of pest denoted by u(t) reaches the threshold [h.sub.1], and when the density u(t) reaches the threshold [h.sub.2], the integrated control strategy is adopted. But no strategy adopted for the density of pest denoted by u(t) = h, where [h.sub.1] < h < [h.sub.2], which is obviously unreasonable. In addition, from an economic and practical point of view, the control taken at threshold [h.sub.1] seems to be early and the amount of releasing predators will also be huge, while the control taken at threshold [h.sub.2] seems to be late and the intensity of chemical control will also be high. Considering the above problems, we should choose a pest control method between [h.sub.1] and [h.sub.2].

An outline of this paper is as follows. In next section, a pest management Smith model is formulated. Then the existence, uniqueness, and the asymptotically orbit stability of order-one periodic solution (OOPS) of system (7) are proved in Section 3. In Section 4, an optimization problem is formulated and obtained the minimized total cost in pest control. The theoretical results are verified by numerical simulations in Section 5. Finally, a conclusion is drawn.

2. Model Formulation

In biological mathematics, Logistic model [10]

dx/dt = rx (1 - x/K) (2)

is a classical mathematical model, where the predator and prey densities at time t are denoted by y(t) and x(t). r denotes the intrinsic rate of growth and K denotes the maximum environment carrying capacity, while system (2) is based on the assumption that the relative growth rate dx/xdt of the population size is linear function 1-x/K. In 1963, F.E.-Smith found that the data about the population of Daphnia did not conform to the linear function [49]. Thus, Smith assumed that the relative growth rate of population density at time t is proportional to the amount of remaining food; i.e.,

[1/x] [dx/dt] = r (1 - H(t)/T), (3)

where H(t) is the rate of food demand of the population at time t; T is the rate of demand for food in a population saturated state. Smith assumed that the food required to keep the population is [c.sub.1]x(t) and the food required for the population to reproduce is [c.sub.2](dx/dt). That is to say,

H(x) = [c.sub.1]x(t) + [c.sub.2] [dx/dt]. (4)

Then

dx/dt = rx (1 - T - [c.sub.1]x/T + r[c.sub.2]x). (5)

Considering the demand for food of population reproduction, the Smith model uses the hyperbolic function (T - [c.sub.1]x)/(T + r[c.sub.2]x) instead of the linear function in the Logistic model. Thus, the Smith model is a further improvement of Logistic model. With the absence of predators, the per capita growth rate [l.sub.grow] of the pest is assumed to be the Smith growth [49] model.

[l.sub.grow] = rx [K - x/K - (r/c)x]. (6)

By the control strategy, the following predator-prey Smith system is investigated in this paper:

[mathematical expression not reproducible], (7)

where the releasing amount of the predator is denoted by [sigma](x) and [sigma]([h.sub.1]) = [[sigma].sub.max], [sigma]([h.sub.2]) = [[sigma].sub.min], where 0 [less than or equal to] [[sigma].sub.min] < [[sigma].sub.max]. The strength of chemical control to the prey is [sigma](x) and that to the predator is [beta](x), where the parameters [sigma](x), [alpha](x), [beta](x) are continuous functions and satisfies [alpha]([h.sub.2]) = [[alpha].sub.max], [beta]([h.sub.2]) = [[beta].sub.max]. A Pest control level h is between [h.sub.1] and [h.sub.2]. [[bar.y].sub.h] denotes the level of the predator at a lower density. By calculation we obtain [mathematical expression not reproducible] are constants. When the density of predator is below [[bar.y].sub.h], the chemical control is taken. Clearly, the control strategy of system (7) changes into the biological control strategy of system (1) when parameters [alpha](x), [beta](x), and [sigma](x), x of system (7), are chosen 0, 0, [gamma], [h.sub.1], respectively. When parameters [alpha](x), [beta](x), and [sigma](x), x of system (7), are chosen S, [??], [tau] [h.sub.2], respectively, the control strategy of system (7) turns into the integrated control strategy of system (1). Therefore, system (7) is the further promotion of system (1).

In our paper, [sigma](x), [alpha](x), and [beta](x) are assumed to have the following linear form [10]

[mathematical expression not reproducible] (8)

3. Dynamical Analysis of System (7)

In this section, we dynamically analyze system (7) to study the existence, uniqueness and orbital asymptotical stability of the OOPS. For convenience, OOPS is used to represent the order-one periodic solution.

3.1. Qualitative Analysis of System (7). We first study the following continuous system of system (7); i.e.,

[mathematical expression not reproducible]. (9)

Let

mx(t) - rx[(t).sup.2]/K + dx(t) - qx(t) y(t) = 0, [mu]x(t) y(t) - ly(t) = 0. (10)

Then we get three equilibria O(0, 0), [E.sub.0](m/r, 0), and [E.sup.*] ([x.sup.*], [y.sup.*]), where

[x.sup.*] = 1/[mu], [y.sup.*] = [mu]m - lr/q([mu]K + dl). (11)

Let

(I): l/[mu] > [square root of [DELTA]] - rK/rd, (12)

where [DELTA] = [r.sup.2][K.sup.2] + mKrd. Thus, we get the following theorem.

Theorem 1. The positive equilibrium point [E.sup.*]([x.sup.*], [y.sup.*]) is locally asymptotically stable, if (I) holds.

Proof. At the point [E.sup.*]([x.sup.*], [y.sup.*]), the Jacobian matrix is

[mathematical expression not reproducible], (13)

then

[mathematical expression not reproducible]. (14)

When (I) holds, then tr(J([E.sup.*])) < 0. Thus the point [E.sup.*] is locally asymptotically stable.

Theorem 2. If [mu]x [less than or equal to] l holds, then the point [E.sup.*] is globally asymptotically stable.

Proof. Let B = 1/x, then we have

[mathematical expression not reproducible] (15)

when [mu]x [less than or equal to] l, then D < 0.

By the method in [48], the point [E.sup.*]([x.sup.*], [y.sup.*]) is globally asymptotically stable (see Figure 1.)

3.2. Existence and Uniqueness of the Periodic Orbit of System (7). For convenience, let H(x, y) = [H.sub.0] denote the first integral of system (7), where the implicit function H(x, y) = H([x.sub.0], [y.sub.0]) is divided into upper and lower branches by isoclinic line dx/dt = 0 denoted by [mathematical expression not reproducible], where the starting point is [P.sub.0]. The impulsive set of system (7) is denoted by [[summation].sub.M] = {(x, y) | x = h, 0 [less than or equal to] y [less than or equal to] [bar.[y.sub.h]] and the phase set is denoted by [[summation].sub.N] = {(x, y) | x = (1 - [alpha](h))h, y [greater than or equal to] 0}.

Let [L.sub.1] = {(x, y) | x = l/[mu], y [greater than or equal to] 0} and [L.sub.2] = {(x, y) | 0 [less than or equal to] x, y = (m-rx)/q(K + dx)} as the isoclinic line dy/dt = 0 and dx/dt = 0, respectively. For any point A([x.sub.A], [y.sub.A]), where [x.sub.A] and [y.sub.A] are denoted as the abscissa and ordinate of point A, respectively. By the location of the threshold h and positive equilibrium point [E.sup.*], we get the following theorem.

Theorem 3. If 0 < [h.sub.1] < h < min{[x.sup.*], [h.sub.2]} holds, then a uniqueness OOPS exists in system (7).

Proof.

Case I (0 < [h.sub.1] < h < min{[x.sup.*], [h.sub.2]}). In view of Theorems 1 and 2, for any point [Q.sub.0] [member of] [OMEGA] where [OMEGA] = {(x, y) | 0 [less than or equal to] x [less than or equal to] (1-[alpha](h))h, y [member of] R} the trajectory [y.sub.H]-(x, [Q.sub.0]) has an intersection point with phase set [[summation].sub.N]. Thus, we discuss the trajectory tendency of the initial point on the phase set [[summation].sub.N].

Assuming the intersection point of phase set [[summation].sub.N] and isoclinic line dx/dt = 0 is point A([x.sub.A], [y.sub.A]), where [x.sub.A] = (1-[alpha] [alpha](h))h and [y.sub.A] = (m-rh(1-[alpha](h)))/q[K + dh(1-[alpha])]. The trajectory [mathematical expression not reproducible] intersects with pulse set at point [mathematical expression not reproducible], then the impulsive function can transfer the point [mathematical expression not reproducible] into the point [mathematical expression not reproducible]. Thus, we have

[mathematical expression not reproducible]; (16)

define

[mathematical expression not reproducible]. (17)

By the magnitudes between [sigma](h) and [bar.[sigma]](h), one has

(i) [sigma](h) [less than or equal to] [bar.[sigma]].

If [sigma](h) = [bar.[sigma]], the subsequent function of point A([x.sub.A], [y.sub.A]) is g(A) = 0, then the trajectory [mathematical expression not reproducible] is an OOPS.

If [sigma](h) < [bar.[sigma]], the point [mathematical expression not reproducible] under the point A([x.sub.A], [y.sub.A]), thus the subsequent function of point A is

[mathematical expression not reproducible]. (18)

The phase set [[summation].sub.N] intersects with x-axis at point B([x.sub.B], 0), where [x.sub.B] = h(1-[alpha](h)). By the orbit tendency, [mathematical expression not reproducible] intersects with the impulsive set [[summation].sub.M] at the point [mathematical expression not reproducible] which jumps to the point [mathematical expression not reproducible]. Obviously, the point B(h(1-[alpha](h)), 0) is under the point [mathematical expression not reproducible]. The subsequent function of the point B(h(1-[alpha](h)), 0) is

[mathematical expression not reproducible]. (19)

According to the continuity of subsequent function, there must be a point C between point A and B, which makes

g(C) = 0. (20)

(See Figures 2(a) and 2(b).)

(ii) [sigma](h) > [bar.[sigma]]

If [sigma](h) > [bar.[sigma]], then the point [mathematical expression not reproducible] must be above the point A([x.sub.A], [y.sub.A]). Thus the subsequent function g(A) > 0. The orbit [mathematical expression not reproducible] will intersect with impulsive set [[summation].sub.M] at point [mathematical expression not reproducible], then hits phase set [[summation].sub.N] at point [mathematical expression not reproducible]. Clearly, the point [mathematical expression not reproducible] is under the point [mathematical expression not reproducible]. Thus the point [mathematical expression not reproducible] must be under the point [mathematical expression not reproducible]. The subsequent function of point [mathematical expression not reproducible] is

[mathematical expression not reproducible]. (21)

So there must be a point D [member of] [[summation].sub.N], such that g(D) = 0 (see Figure 2(c)).

Now, the uniqueness of OOPS of system (7) is to be discussed.

Assuming that [mathematical expression not reproducible], then orbit [mathematical expression not reproducible] are OOPS, where [mathematical expression not reproducible].

[mathematical expression not reproducible], (22)

Assume

[mathematical expression not reproducible], (23)

then

[mathematical expression not reproducible], (24)

where

[omega](y) = y/m - rx - qy (K + dx), (25)

and

w'(y) = y(r + dqy)/[[m - rx - qy (K + dx)].sup.2] > 0. (26)

So [mathematical expression not reproducible]; that is to say, [mathematical expression not reproducible] is a decreasing function. For x [member of] [(1-[alpha])h, h], then [mathematical expression not reproducible].

Thus

[mathematical expression not reproducible]. (27)

which is a contradiction.

When [sigma](h) > [bar.[sigma]], then there exists a point P [member of] [bar.[AA.sup.+]] such that [mathematical expression not reproducible]. Thus, for any P' [member of] [bar.[BA.sup.+]], the subsequent function of point P' is

[mathematical expression not reproducible]. (28)

According to the proof above, we have [mathematical expression not reproducible]. Thus, if [y.sub.P'] < [y.sub.P], then g(P') < 0.

If [y.sub.P'] > [y.sub.P], then g(P') > 0. Thus the uniqueness of OOPS of system (7) in case of [sigma](h) > [bar.[sigma]] is proved. The proof is completed.

Theorem 4. If max{[x.sup.*], [h.sub.1]} < h < [h.sub.2] holds, then we have two cases. If h > [h.sub.0] holds, then system (7) has no OOPS. If h < [h.sub.0] holds, then system (7) has a unique OOPS, where [mathematical expression not reproducible].

Proof.

Case II (max{[x.sup.*], [h.sub.1] < h [less than or equal to] [h.sub.2]}). Two cases are discussed according the magnitude of h and [h.sub.0].

(i) If h > [h.sub.0], the trajectory [mathematical expression not reproducible] tending to point [E.sup.*]([x.sup.*], [y.sup.*]) is based on the global asymptotical stability of [E.sup.*]([x.sup.*], [y.sup.*]). For any P(x, y) [member of] Q, the trajectory [mathematical expression not reproducible] tends to [E.sup.*]([x.sup.*], [y.sup.*]). (See Figure 3.)

(ii) If h < [h.sub.0], the trajectory [mathematical expression not reproducible] must intersect with the impulsive set [[summation].sub.M] at the point [mathematical expression not reproducible], which jumps to the point [mathematical expression not reproducible]. The subsequent function of point [mathematical expression not reproducible]. Similar to the proof of Case I, when [sigma] [less than or equal to] [bar[sigma]], an OOPS exists in system (7). If [sigma] > [bar.[sigma]], we also can prove that an OOPS exists in system (7) by same method of Case I. (See Figure 4.)

4. The Orbital Asymptotical Stability of OOPS of System (7)

According to the discussion above, a unique OOPS exists in system (7), denoted by [mathematical expression not reproducible]. Then we get the following theorem.

Theorem 5. If [sigma] [less than or equal to] [bar.[sigma]], then the OOPS of system (7) is orbitally asymptotically stable and globally attractive to the point [E.sup.*].

Proof. We choose arbitrary point [A.sub.0] on the phase set [[summation].sub.N]. If [A.sub.0] [member of] N/[bar.AP], then after several pulse effects the trajectory will jump to the segment [bar.AP]. Thus we assume that [A.sub.0] [member of] [bar.AP]; the trajectory [mathematical expression not reproducible] will hit the impulsive set [[summation].sub.M] at point [A.sup.-.sub.1], which jumps to the point [A.sup.+.sub.1]. The trajectory [mathematical expression not reproducible] will intersect with impulsive set [[summation].sub.M] at point [A.sup.-.sub.2] and then jumps to the point [A.sup.+.sub.2]. Repeat the process above; we get a point sequences {[A.sup.+.sub.k]}, where k = 1, 2, 3, ... such that

[mathematical expression not reproducible]. (29)

The sequence [A.sup.+.sub.k][[parallel].sub.k=0,1,2,...] is a monotonic decreasing sequence with lower bound [y.sub.P]. According to the monotonic bounded theorem, there must exist a limit [y.sub.P'] such that [mathematical expression not reproducible], which means that

[mathematical expression not reproducible]. (30)

Since g(A) = 0, if and only if A = P, then P' = P. That is to say [mathematical expression not reproducible].

Similarly, we can use the above method to get an increasing point sequences [B.sup.+.sub.k][[parallel].sub.k=0,1,2,...] such that

[mathematical expression not reproducible]. (31)

There must exist a limit [y.sub.P'] such that [mathematical expression not reproducible], which means that

[mathematical expression not reproducible]. (32)

Since g(B) = 0, if and only if B = P, then P' = P. That is to say, [mathematical expression not reproducible]. By the arbitrariness of the point [A.sub.0] and [B.sub.0], one has

[mathematical expression not reproducible]. (33)

Thus the OOPS of system (7) is orbitally asymptotically stable and globally attractive (see Figure 5).

Theorem 6. If [sigma] > [bar.[sigma]] and [[gamma].sub.1][m-r[[eta].sub.0]-q[[gamma].sub.0] (K + d[[eta].sub.0])] (1-[beta]) (K + d[[eta].sub.1])/[[gamma].sub.0] (m-r[[eta].sub.1]-q[[gamma].sub.1]) (K + d[[eta].sub.0]) < 1, then the OOPS of system (7) is orbitally asymptotically stable.

Proof. Let x = [eta](t), y = [gamma](t) is a T-periodic orbit of system (7) and [[eta].sub.0] = [eta](0), [[gamma].sub.0] = [gamma](0), [[eta].sub.1] = [eta](T), [[gamma].sub.1] = [gamma](T), [[eta].sup.+.sub.1] = [eta]([T.sup.+]), [[gamma].sup.+.sub.1] = y([T.sup.+]), then

[[eta].sup.+] = [[eta].sub.0] = (1 - [alpha])h, [[gamma].sup.+.sub.1] = [[gamma].sub.0] = (1 - [beta]) [gamma].sub.1] + [sigma]. (34)

Let

[mathematical expression not reproducible]. (35)

Then

[mathematical expression not reproducible]. (36)

and

[mathematical expression not reproducible]. (37)

Thus, when [mathematical expression not reproducible]. Therefore, the OOPS is orbitally asymptotically stable.

5. Numerical Simulations and Optimization of Pest Control Level

5.1. Numerical Simulations. In this section, the feasibility of our conclusions is verified by an example. Let m = 1, r = 0.3, K = 2, d = 0.5, q = 0.6, [mu] = 0.5, and l = 0.4. By calculation, the equilibrium point [E.sup.*] of system (7) is [E.sup.*] (0.8, 0.528). Parameter values are taken into system (7), then

[mathematical expression not reproducible] (38)

Let h = 0.7 satisfy the condition 0 < [h.sub.1] < h < min{[x.sup.*], [h.sub.2]} and the initial value be (0.4, 0.4). Let [[sigma].sub.max] = 1.5, [[delta].sub.min] = 0.1, [[alpha].sub.max] = 1.75, [[beta].sub.max] = 2, [h.sub.1] = 0, and [h.sub.2] = 2. A directed calculation yields that [[alpha].sub.0.7] = 0.15, [[beta].sub.0.7] = 0.5, and [bar.[[sigma].sub.0.7]] = 0.2975. Let [sigma] = 0.26, [sigma] = [bar.[sigma].sub.0.7]] = 0.2975, and [sigma] = 0.3. Figures 6(a), 6(b), and 6(c) show that a unique and asymptotically stable OOPS exists in system (7).

Let h = 0.9 satisfy the condition 0 < max{[x.sup.*], [h.sub.1]} < h < [h.sub.0] < [h.sub.2] and the initial value be (0.5, 1). A directed calculation yields that [[alpha].sub.0.9] = 0.5, [[beta].sub.0.9] = 0.3, and [[sigma].sub.0.9] = 0.269. Let [sigma] = 0.24, [sigma] = [bar.[[sigma].sub.0.9]] = 0.269, and [sigma] = 0.3. Figures 7(a), 7(b), and 7(c) show that system (7) has a unique and asymptotically stable OOPS.

For the case of 0< max{[x.sup.*], [h.sub.1]} < [h.sub.0] < h [less than or equal to] [h.sub.2], for example, h = 1.5 and the orbit of system (7) starts from (0.5, 1), we get [[alpha].sub.1.5] = 0.5, [[beta].sub.0.9] = 0.3 and [[sigma].sub.1.5] = 0.24 by calculation. Figures 8(a), 8(b), and 8(c) show that system (7) has no OOPS.

5.2. Determination and Optimization of Pest Control Level. The goal to investigate the existence of OOPS of system (7) lies in that it can obtain the possibility of determining the frequency of releasing predators and spraying pesticides, which makes the density of pest below the damage level. Although the density of prey is inaccurate or biased, the system will eventually undergo periodic changes under the effective control. The following problems are considered to determine the optimal frequency for releasing predators and chemical controls.

Assuming that unit cost of releasing predator is denoted by [[iota].sub.1] and the unit cost of spraying pesticides is denoted by [[iota].sub.2], which include the price of chemical agent and the price of the damage to environment. Our goal is to reduce the unit cost in this process. In one period, the total cost is denoted by [F.sub.[iota]], which is a function about [alpha](h) (i.e., chemical control strength) and [sigma](h) (i.e., yield of releases of predator).

Then [F.sub.[iota]](h) = [[iota].sub.1][sigma](h) + [[iota].sub.2][alpha](h). So the optimization model is formulated as

max [F.sub.[iota]](h)/T(h) s.t. [h.sub.1] [less than or equal to] h [less than or equal to] [h.sub.2] (39)

The optimization problem is solved to yield the optimal pest level [h.sup.*], which the optimal release rate of predator is [mathematical expression not reproducible], the optimal strength of chemical control is [mathematical expression not reproducible], and the optimal impulse period of chemical control is [T.sup.*] = T([[sigma].sup.*], [[alpha].sup.*]). However, the optimum pest control level [h.sup.*] is dependent on the ratio of [omega] [??] [[iota].sub.2]/[[iota].sub.1]. The impulse period T varies with the threshold h, as shown in Figure 9(a). And Figure 9(b) shows the variation of cost per unit time F/T and the period T with the pest control level h, where [[iota].sub.1] = 1000, [[iota].sub.2] = 1000, i.e., [omega] = 1. The optimal pest level is [h.sup.*] = 0.9, the optimal strength of chemical control is [mathematical expression not reproducible], and the optimal release rate of predator is [mathematical expression not reproducible]. It is important to note that the optimum economic threshold h is dependent on [omega], as is illustrated in Figure 10.

6. Conclusion

A Smith prey-predator system with linear feedback control for integrated pest management is investigated in this paper. Integrated control strategy is more practical which can maximize the protection of the ecological environment and reduce the cost of pest management. First, the method of subsequent function and differential equation geometry theory are used to prove the existence, uniqueness, and stability of the OOPS of system (7). Second, a specific example is given to verify the conclusion of the impulsive strategy. Last, an optimized problem is formulated and the minimized total cost in pest control is obtained. However, the optimized results have some deviations which need to be further improved.

https://doi.org/10.1155/2018/4945728

Data Availability

We agree to share the data underlying the findings of the manuscript. Data sharing allows researchers to verify the results of an article, replicate the analysis, and conduct secondary analyses.

Conflicts of Interest

The authors declare that they have no conflicts of interest. Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371230 and 11501331), the SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001 and BS2015SF002), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, and SDUST Innovation Fund for Graduate Students (no. SDKDYC170225).

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Yaning Li, (1) Yan Li (iD), (2) Yu Liu, (3) and Huidong Cheng (iD) (1)

(1) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

(2) College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

(3) College of Foreign Languages, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Huidong Cheng; chd900517@sdust.edu.cn

Received 13 July 2018; Revised 8 November 2018; Accepted 26 November 2018; Published 5 December 2018

Guest Editor: Abdul Qadeer Khan

Caption: Figure 1: The phase diagram of system (7) with m = 1, r = 0.3, K = 2, d = 0.5, q = 0.6, [mu] = 0.5, and I = 0.4.

Caption: Figure 2: The existence of the OOPS of system (7). (a) Discussion in Case (i) if [sigma](h) = [bar.[sigma]]. (b) Discussion in Case (i) if [sigma](h) < [bar.[sigma]]. (c) Discussion in Case (ii).

Caption: Figure 3: The existence of the OOPS of system (2) if h > [h.sub.0] in Case (II).

Caption: Figure 4: The existence of the OOPS of system (7) if h < [h.sub.0] in Case (II).

Caption: Figure 5: The orbitally asymptotically stability of the OOPS of system (7).

Caption: Figure 6: Numerical simulations in case 0 < [h.sub.1] < h < min{[x.sup.*], [h.sub.2]}. (a) Phase portrait of x(t) and y(t) on h = 0.7. (b) Time series of x(t). (c) Time series of y(t).

Caption: Figure 7: Numerical simulations in case 0 < max{[x.sup.*], [h.sub.1]} < h < [h.sub.0] < [h.sub.2]. (a) Phase portrait of x(t) and y(t) on h = 0.9. (b) Time series of x(t). (c) Time series of y(t).

Caption: Figure 8: Numerical simulations in case < max{[x.sup.*], [h.sub.1]} < [h.sub.0] < h [less than or equal to] [h.sub.2]. (a) Phase portrait of x(t) and y(t) on h = 1.5. (b) Time series of x(t). (c) Time series of y(t).

Caption: Figure 9: The variety in the period T and the profit per unit time F/T on the threshold h. (a) The variety in the period T on the threshold h. (b) The profits per unit time F/T on the threshold h.

Caption: Figure 10: The change in the cost per unit time F/T on the control level h for [[iota].sub.2]/[[iota].sub.i] = 2.5, 1, 5, 10.
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Title Annotation:Research Article
Author:Li, Yaning; Li, Yan; Liu, Yu; Cheng, Huidong
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
Words:6120
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