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Dynamic Analysis of Beddington-DeAngelis Predator-Prey System with Nonlinear Impulse Feedback Control.

1. Introduction

The predator-prey system plays an important role in the relationship of biological populations, so many predator-prey systems with different functional responses have been studied, such as the Monod type [1-5], the Holling type [6-13], and the Ivlev type [14-18]. Currently, the use of chemicals is more and more widespread in agriculture. Also, when the number of pests reaches a critical value, we can release natural enemies. Therefore, the feedback control of pulse state is proposed [17,19-22]. The Beddington-DeAngelis functional response was introduced by Beddington [23] and DeAngelis et al. [24]. The Beddington-DeAngelis functional response avoided some of the singular behaviors of the ratio-dependent model at low density [25]. Cantrell and Cosner discussed the following predator-prey system with Beddington-DeAngelis functional response [23,24,26]:

[mathematical expression not reproducible], (1)

where r, K, m, a, b, c, [epsilon], and [mu] are positive constants. u(t) and v(t) represent the population density of prey and predator at time t, K is the environmental carrying capacity of the prey, and r is the intrinsic growth rate of prey. Function mu/(a + bv + cu) indicates the Beddington-DeAngelis functional response, and bv stands for mutual interference between the predators. The constants [epsilon] and [mu] represent the rate of conversion and death rate of predators, respectively.

For simplicity, we determine dimensionless system (1) and scale it as follows:

t [right arrow] rt,

u [right arrow] u/K,

v [right arrow] bv/cK. (2)

Then, we get

[mathematical expression not reproducible], (3)

where

s = m/br,

[delta] = m[epsilon]/cr,

d = c[mu]/m[epsilon],

A = a/cK. (4)

In recent years, many pulse equations have been studied that simulate the ecological processes of populations, and most of these studies are pulse differential equations at the fixed time [1,27-32]. However, feedback control of time pulse has certain defects, which may reduce crop yield and possibly increase management costs. Therefore, we can choose to spray the pesticide when the quantity of pests reaches a certain threshold instead of spraying the pesticide at a fixed time. This measure avoids the possibility of the explosive growth of the number of pests and is more suitable for pest control. This paper studies the Beddington-DeAngelis system with pulse state feedback control strategies:

[mathematical expression not reproducible]. (5)

Capturing or using chemicals on predators and prey may impulsively reduce the density of the predators and prey. P(D) and Q(D) represent the survival rate of prey and predator populations when a given dose D of insecticides is applied and 0 [less than or equal to] P(D) < 1 and 0 [less than or equal to] Q(D) < 1. We assume that insecticides have different insecticidal rates for these two populations, where [mathematical expression not reproducible] and [mathematical expression not reproducible] is the constant number of natural enemies released [33,34].

Some published articles focus on the property of the successor function and Poincare map to discuss the existence of order-1 periodic solution and the local stability. Besides, if the proposed model has the first integrals, the existence of order-2 periodic solutions can be discussed [35-40]. However, due to the complexity of such model as the Beddington-DeAngelis system in this paper, the problem of global dynamics such as the global stability of the model and the existence of order-k(k [greater than or equal to] 2) periodic solution has not been well solved. Also, there a few researches on the property of Poincare map while it is applied. So the main arrangement of this paper is as follows. In Section 2, some preliminaries about the pulse semidynamic system and system (3) are given. In Section 3, we construct the Poincare map, deduce the expression of Poincare map function of system (5), and then give some of its properties such as the monotonicity, differentiability, fixed point, and asymptote. In Section 4, we prove the existence and stability of the boundary periodic solution and order-k(k [greater than or equal to] 1) periodic solution of system (5). In Section 5, we conducted a numerical simulation.

2. Preliminaries

2.1. Preliminaries of Pulse Semidynamic Systems. A pulsed semipowered system with state-dependent feedback control can be expressed as [41,42]

[mathematical expression not reproducible]. (6)

Here, (u, v) [member of] [R.sup.2], P, Q, [alpha], and [beta] are continuous functions from [R.sup.2] to R. Let M [subset] [R.sup.2] be the impulse set of system (6), and for any f(u, v) [member of] M, the impulse occurs; the map I is defined as

[f.sup.+] = I(f) = (u + [alpha](u, v), v + [beta](u, v)) = ([u.sup.+], [v.sup.+]) [member of] [R.sup.2], (7)

where [f.sup.+] is the impulse point for f. We define N = I(M) as the phase set of the system and N [intersection] M = [empty set], X = [R.sup.2] is the metric space, and [R.sub.+] is the set of all nonnegative reals; we call (X, [PI], R) as a semidynamic system. For any f [member of] X, [PI](f, 0) = f, [PI]([PI](f, t), s) = [PI](f, t + s), where t, s [member of] [R.sub.+] [43]. The set

[D.sup.+](f) = {[PI](f, t)| t [member of] [R.sub.+]}, (8)

is called the positive orbit of [phi]. Furthermore, for any set M [member of] X, let

[M.sup.+](f) = [D.sup.+](f) [intersection] M - {f}. (9)

Next, we give the definition of impulsive semidynamic system and order-k periodic solution.

Definition 1 (see [44,45]). The pulsed semidynamic system (X, [PI], M, I) consists of the nonempty closed subset M of X, the continuous semidynamic system (X, [PI]), and the continuous function I.

We denote the points of discontinuity of [[PI].sub.f] by {[f.sup.+.sub.n]} and call [f.sup.+.sub.n] an impulsive point of [f.sub.n]. We define a function [THETA] from X into the extended positive reals [R.sub.+] [union]{[infinity]} as follows: let f [member of] X; if [M.sup.+](f) = [empty set], we set [THETA](f) = [infinity]; otherwise [M.sup.+](f) [not equal to] [empty set], and we set [THETA](f) = s, where [PI](f, t) [not member of] M for 0 < t < s but [PI](f, t) [member of] M.

Definition 2 (see [46]). For trajectory [[PI].sub.f] in (X, [PI], M, I), if there are nonnegative integers m [greater than or equal to] 0 and k [greater than or equal to] 1, k is the minimum integer satisfying [f.sup.+.sub.m] = [f.sup.+.sub.m+k] and [T.sub.k] = [[summation].sup.m+k-1.sub.i=m] [THETA]([f.sub.i]) = [[summation].sup.m+k-1.sub.i=m] [s.sub.i]; then, the period of [[PI].sub.f] is T, and there is a period of order-k.

Definition 3 (see [42,47]). The T-periodic solution (u, v) = ([xi](t), [eta](t)) of thwe system

[mathematical expression not reproducible], (10)

is orbitally asymptotically stable and enjoys the property of asymptotic phase if [[mu].sub.1] satisfies [absolute value of [[mu].sub.1]] < 1, where

[mathematical expression not reproducible], (11)

with

[mathematical expression not reproducible], (12)

and [phi] is continuously differentiable with respect to u and v. (u, v) [not member of] M can also be denoted by [phi](u, v) [not equal to] 0. P, Q, [partial derivative][alpha]/[partial derivative]u, [partial derivative][alpha]/[partial derivative]v, [partial derivative][beta]/[partial derivative]u, [partial derivative][beta]/[partial derivative]v, [partial derivative][phi]/[partial derivative]u, and [partial derivative][phi]/[partial derivative]v are calculated at the point ([xi]([t.sub.k]), [eta]([t.sub.k])), [P.sub.+] = P([xi]([t.sup.+.sub.k]), [eta]([t.sup.+.sub.k])), and [Q.sub.+] = Q([xi]([t.sup.+.sub.k]), [eta]([t.sup.+.sub.k])).

2.2. Preliminaries of System (3). We know that the solution of system (3) is positive and bounded for all t. If d [greater than or equal to] [(1 + A).sup.-1], then the equilibrium point (1,0) is globally asymptotically stable [25].

If 0 < d < [(1 + A).sup.-1], system (3) has three equilibrium points, (0,0), (1,0), and ([u.sub.*], [v.sub.*]), respectively, where u, and [v.sub.*] are all positive and satisfy the following conditions:

[mathematical expression not reproducible]. (13)

For completeness, we summarize global results for system (3) in Lemma 1 [25,48,49].

Lemma 1

(i) If d > [(1 + A).sup.-1], (1,0) is globally asymptotically stable

(ii) If d < [(1 + A).sup.-1] and tr(J([u.sub.*], [v.sub.*])) [less than or equal to] 0, the ([u.sub.*], [v.sub.*]) is globally asymptotically stable

(iii) If d < [(1 + A).sup.-1] and tr(J([u.sub.*], [v.sub.*])) > 0, there is an exact limit cycle

3. Poincare Map

3.1. Domains of the Poincare Map. In the following parts, we only discuss the case of ([u.sub.*], [v.sub.*]) as the globally asymptotically stable point of system (5).

System (5) has two isoclinal lines, which are defined as [L.sub.1] and [L.sub.2]:

[L.sub.1] : v = (1 - A)u - [u.sup.2] + A/s + u - 1;

[L.sub.2] : v = (1 - d)u/d - A. (14)

Next, the lines associated with the phase set and impulse set are defined as [L.sub.3] and [L.sub.4]:

[mathematical expression not reproducible]. (15)

In this case, the value range of TH is 0 < TH < [u.sub.*] and lines [L.sub.3] and [L.sub.4] always intersect with line [L.sub.1]. The intersection point of [L.sub.1] and [L.sub.4] presents as G(TH, [v.sub.G]), and the intersection point of [L.sub.1] and [L.sub.3] presents as [mathematical expression not reproducible], where [v.sub.G] and [v.sub.H] are, respectively,

[mathematical expression not reproducible]. (16)

The open set defined in [R.sup.2.sub.+] is as follows:

[OMEGA] = {(u, v) | u > 0, v > 0, u < TH} [subset] [R.sup.2.sub.+]. (17)

The impulse set M is the part of line [L.sub.4] above the U-axis and below point G:

M = {(u, v) [member of] [R.sup.2]| u - TH, 0 [less than or equal to] v < (1 - A)TH - [TH.sup.2] + A/TH + s - 1}. (18)

The continuous function I is expressed as

[mathematical expression not reproducible], (19)

so the phase set N is

[mathematical expression not reproducible], (20)

where [mathematical expression not reproducible]. We assume that the initial point ([u.sup.+.sub.0], [v.sup.+.sub.0]) is always on the [L.sub.3] in the following sections.

3.2. Construction of Poincare Map. The Poincare map of system (5) can be defined in different ways. In this work, we choose the [L.sub.3] to define the Poincaree map.

Because ([u.sub.*], [v.sub.*]) is chosen as the globally asymptotically stable point in system (5), and the value range of TH is defined in the impulse set [L.sub.4] : u = TH as 0 < TH < [u.sub.*], any trajectory starting from the point [mathematical expression not reproducible] on the phase set must intersect with the impulse set [L.sub.4] : u = TH at the point [Z.sub.k+i] (TH, [v.sub.k+1]). From Cauchy-Lipschitz theorem, we know the value of [v.sup.+.sub.k] is only determined by [v.sub.k+1]; in order to discuss fluently, we make [v.sub.k+1] = g([v.sup.+.sub.k]). The point [Z.sub.k+1] on the impulse set [L.sub.4] is mapped to the point [mathematical expression not reproducible] on [L.sub.3] after experiencing an impulse. [mathematical expression not reproducible] is obtained by the impulse function [mathematical expression not reproducible] of system (5). [Z.sup.+.sub.k+1] is the initial point of the next impulse function on the phase set.

So we can express the Poincare map of system (5) as

[mathematical expression not reproducible]. (21)

We can determine the Poincaree map from the points on the phase set. Next, we infer the expression of Poincare map function [phi] and discuss its properties according to the expression of [phi].

According to the following formula of system (5),

[mathematical expression not reproducible]. (22)

We can rewrite system (5) as a scalar differential equation on the phase set:

[mathematical expression not reproducible]. (23)

For model (23), we only focus on the region

[[OMEGA].sub.1] = {(u, v); u > 0, v > 0, v < (1 - A)u - [u.sup.2] + A/s + u - 1}. (24)

The function [rho](x, y) is continuous and differentiable in the region [[OMEGA].sub.1]. Besides, let [mathematical expression not reproducible] and [v.sup.+.sub.0] = J, where J [member of] N, J < [v.sub.G], and ([u.sup.+.sub.0], [v.sup.+.sub.0]) [member of] [[OMEGA].sub.1], so

[mathematical expression not reproducible]. (25)

From (23), we get

[mathematical expression not reproducible]. (26)

According to (21) and (26), we can obtain the definition of the Poincaree map [phi]:

[mathematical expression not reproducible]. (27)

We simulate a numerical simulation of the Poincare map function of model (5) (see Figure 1). The two cases are [phi]([v.sub.H]) < [v.sub.H] (Figure 1(a)) and [phi]([v.sub.H]) > [v.sub.H] (Figure 1(b)).

3.3. The Main Properties of Poincare Map. Through our analysis of the expression and numerical model of the Poincare map [phi], and assuming [tau] > 0, the following properties of Poincaree map are given.

Theorem 1. The domain of [phi] is [0, +[infinity]), and the range of [phi] is [[tau], [phi]([v.sub.H])], where [mathematical expression not reproducible]. [phi] monotonically increases on [0, [v.sub.H]] and monotonically decreases on [[v.sub.H], +[infinity]), and as the value of [v.sup.+.sub.k] increases continuously, [phi] approaches the asymptote [phi] = [tau].

Proof. We first prove the domain of [phi] is [0, +[infinity]). Because ([u.sub.*], [v.sub.*]) in system (3) is globally asymptotically stable, and because TH < [u.sub.*], any initial point from the [L.sub.3] will reach the impulse set M; so the definition domain of [phi] is [0, +[infinity]).

Next, we divide L3 into two parts, which are [0, [v.sub.H]] and [[v.sub.H], +[infinity]). First of all, we can choose two points [mathematical expression not reproducible] and [mathematical expression not reproducible] and assume 0 < [v.sup.+.aub.a] < [v.sup.+.sub.b] < [v.sub.H], then the trajectory of these two points will intersect with [L.sub.4] at the two points [Z.sub.a+1] (TH, [v.sub.a+1]) and [Z.sub.b+1] (TH, [v.sub.b+1]). According to Cauchy-Lipschitz theorem, [v.sub.a+1] < [v.sub.b+1] is true for any [v.sup.+.sub.a] < [v.sup.+.sub.b], and the formula of [phi] is

[mathematical expression not reproducible]. (28)

So it is easy to find for any [v.sup.+.sub.a] and [v.sup.+.sub.b] on [0, [v.sub.H]] and v+a < v+b, [phi](v+a) < [phi](v+b) is always true; therefore, [phi] monotonically increases on [0, [v.sub.H]].

And then, we prove that the [phi] is monotonically decreasing on [[v.sub.H], +[infinity]). Similarly, we take two numbers [v.sup.+.sub.a] and [v.sup.+.sub.b] on [[v.sub.H], +[infinity]) and assume [v.sup.+.sub.a] < [v.sup.+.sub.b], then the trajectories of the two initial points [mathematical expression not reproducible] and [mathematical expression not reproducible] from [L.sub.3] will cross the isoclinal line [L.sub.2] and then intersect [L.sub.3] at [mathematical expression not reproducible] and [mathematical expression not reproducible], respectively, where [v.sub.a'] > [v.sub.b']. Then, these two curves will intersect [L.sub.4] at two points [mathematical expression not reproducible] and [mathematical expression not reproducible], respectively. According to Cauchy-Lipschitz theorem, [v.sub.a+1] > [v.sub.b+1] is always true, and the expression of [phi] is obtained in (27), so for any [v.sup.+.sub.a] < [v.sup.+.sub.b] on [[v.sub.H], +[infinity]), [phi]([v.sup.+.sub.a]) > [phi]([v.sup.+.sub.b]) is always true, and [phi] is monotonically decreasing on [[v.sub.H], +[infinity]).

Then, we prove that as the value of [v.sup.+.sub.k] increases, [phi] tends to be stable and approaches the asymptote [phi] = [tau]. We define the closure of [[OMEGA].sub.1] as

[[OMEGA].sub.1] = {(u, v); u > 0, v > 0, v < (1 - A)u - [u.sup.2] + A/s + u - 1}. (29)

Since [phi] increases monotonically on [0, [v.sub.H]] and decreases monotonically on [[v.sub.H], +[infinity]), [[OMEGA].sub.1] is the invariant set of system (5). Let

if

[[(P(u, v), Q(u, v)) x ([u.sup.2] + 2(s - 1)u + (A - 1)s + 1/[(s + u - 1).sup.2])].sub.L=0] [less than or equal to] 0. (31)

Here x is the scalar product of two vectors, and then, the vector field will eventually reach the boundary [[OMEGA].sub.1], so [[OMEGA].sub.1] is an invariant set, and by calculation, one obtains

[mathematical expression not reproducible]. (32)

Since [phi] is monotonically increases on [0, [v.sub.H]] and monotonically decreases on [[v.sub.H], +E), [phi]([v.sup.+.sub.0]) is bounded for any [v.sup.+.sub.0] [member of] [0, [v.sub.H]] and [phi]([[v.sub.H], +[infinity])) [subset] [phi]([0, [v.sub.H]]). From the Cauchy-Lipschitz Theorem, [v.sub.k+1] is only determined by [v.sup.+.sub.k] and can be expressed by [v.sub.k+1] = [sigma]([v.sup.+.sub.k]). And any point on the phase set N is always going to be (dv/dt) < 0; hence, [mathematical expression not reproducible]. Then, we have

[mathematical expression not reproducible]. (33)

So as the value of [v.sup.+.sub.k] increases, [phi] tends to be stable and approaches the asymptote [phi] = [tau].

Since [phi] increases monotonically on [0, [v.sub.H]] and decreases monotonically on [[v.sub.H], +E) [subset] [phi]([0, [v.sub.H]]), [phi]([[v.sub.H], +E)); thus, [phi] takes the maximum value at [v.sub.H] and takes the minimum value at 0, where [mathematical expression not reproducible] and [phi](0) = [tau], so the range of value is [[tau], [phi]([v.sub.H])].

Theorem 2. [phi] is continuously differentiable.

Proof. Here, we can use the initial conditions of continuity and differentiability theorem which is to take parameters of Cauchy theorem and Lipschitz theorem to determine the continuity and differentiability of [phi]; by system (5), we can get that both P(u, v) and Q(u, v) functions are continuously differentiable in the first quadrant; so by Cauchy theorem and Lipschitz theorem with parameters, we can get that the [phi] is a continuously differentiable function.

Theorem 3. [phi] always has at least one fixed point if [tau] > 0.

Proof. From Theorem 1, we know that [phi] is monotonically increasing on [0, [v.sub.H]] and monotonically decreasing on [[v.sub.H], +[infinity]). Then, we divide it into two cases to discuss the existence of the fixed point for [phi].

Case I: when [phi]([v.sub.H]) < [v.sub.H], on the one hand, [phi](0) = [tau] > 0, so [phi] has at least one number [v.sub.c] on [0, [v.sub.H]) such that [phi]([v.sub.c]) = [v.sub.c]; on the other hand, since [phi] is monotonically decreasing on [[v.sub.H], +E) and [phi]([v.sub.H]) < [v.sub.H], [phi] has no fixed point on [[v.sub.H], +E). In conclusion, when [phi]([v.sub.H]) < [v.sub.H], the [phi] has at least one fixed point.

Case II: when [phi]([v.sub.H]) [greater than or equal to] [v.sub.H], because [phi] monotonically decreases on [[v.sub.H], +E) and as the value of [v.sup.+.sub.k] increases continuously, [phi] approaches asymptote [phi] = [tau], so [phi] has only one point [v.sub.c] on [[v.sub.H], +E) such that [phi]([v.sub.c]) = [v.sub.c]. And the number of fixed points on [0, [v.sub.H]) is unknown. So when [phi]([v.sub.H]) [greater than or equal to] [v.sub.H], the [phi] has at least one fixed point.

In conclusion, [phi] always has at least one fixed point.

4. Study on the Periodic Solutions of System (5)

4.1. Boundary Periodic Solutions of System (5). For system (5), if the predator population becomes extinct and the predator also terminates its release, then system (5) has a boundary periodic solution, which produces the following system:

[mathematical expression not reproducible]. (34)

Solving (34) with initial value [mathematical expression not reproducible],

[mathematical expression not reproducible]. (35)

The trajectory from the initial point will eventually intersect with the straight line of the impulse set over time:

[mathematical expression not reproducible]. (36)

Solving the equation of T and D, we get

[mathematical expression not reproducible], (37)

where T is the period of the boundary periodic solution and D is the insecticide dose required to control the number of pests below the TH. Then, the boundary periodic solution of system (5) with a period of T is

[mathematical expression not reproducible]. (38)

Theorem 4. The boundary periodic solution ([U.sup.T](t), 0) of system (5) is asymptotically stable if

[mathematical expression not reproducible]. (39)

Proof. By Definition 3, we obtain

[mathematical expression not reproducible]. (40)

From the above formula, we can get

[mathematical expression not reproducible], (41)

so

[mathematical expression not reproducible]. (42)

In addition,

[mathematical expression not reproducible]. (43)

The expression of [[mu].sub.1] is

[mathematical expression not reproducible]. (44)

If condition (39) is true, then [absolute value of [[mu].sub.1]] < 1. It means the periodic solution ([u.sup.T](t), 0) of the boundary is asymptotically stable.

4.2. Existence and Stability of Periodic Solutions of Order-k(k [greater than or equal to] 1) at [tau] > 0. From Theorem 3, the Poincare map function [phi] of system (5) has at least one fixed point; that is to say, system (5) must have at least one order-1 periodic solution.

Theorem 5. The order-1 periodic solution ([xi](t), [eta](t)) is orbitally asymptotically stable if and only if

[mathematical expression not reproducible], (45)

where

[mathematical expression not reproducible]. (46)

Proof. We use R(TH, [[eta].sub.0]) and [mathematical expression not reproducible] to represent the start point and the endpoint of the order-1 periodic solution, respectively. From Theorem 4, we know that the Floquet multiplier

[mathematical expression not reproducible]. (47)

If (45) is true, then [absolute value of [[mu].sub.1]] < 1, so the order-1 periodic solution is always orbitally asymptotically stable.

Theorem 6. If [phi]([v.sub.H]) < [v.sub.H], there is at least one locally asymptotically stable order-1 periodic solution in system (5).

Furthermore, if there is only one fixed point on [0, [v.sub.H]], then there is a globally asymptotically stable order-1 periodic solution of system (5).

Proof. If [phi]([v.sub.H]) < [v.sub.H], Poincare map [phi] has at least one fixed point. This also proves that there is at least one order-1 periodic solution in system (5). From Theorem 5, we know that the periodic solution is always asymptotic stable if [absolute value of [[mu].sub.1]] < 1. So if [phi]([v.sub.H]) < [v.sub.H], then there is at least one locally asymptotically stable order-1 periodic solution in system (5).

If [phi]([v.sub.H]) < [v.sub.H], there is only one fixed point on [0, [v.sub.H]]. This proves that there is a unique order-1 periodic solution in system (5). According to Theorem 5, we can conclude that the periodic solution is asymptotic stable.

For any trajectory starting from [mathematical expression not reproducible], if [v.sup.+.sub.0] [member of] [0, [v.sub.H]], then [v.sup.+.sub.0] < [phi]([v.sup.+.sub.0]) < [v.sub.H]. After n times pulses, [[phi].sup.n]([v.sup.+.sub.0]) monotonically increases, so [lim.sub.n[right arrow]+[infinity]] [[phi].sup.n]([v.sup.+.sub.0]) = [v.sub.H].

In contrary, if [v.sup.+.sub.0] > [v.sub.H], we need to discuss this according to different cases. On the one hand, if [[phi].sup.n]([v.sup.+.sub.0]) > [v.sub.H] is always holding, we can conclude that [[phi].sup.n]([v.sup.+.sub.0]) is monotonically decreasing because [[phi].sup.n]([v.sup.+.sub.0]) < [v.sup.+.sub.0] and [lim.sub.n[right arrow]+[infinity]] [[phi].sup.n]([v.sup.+.sub.0]) = [v.sub.H]. On the other hand, [[phi].sup.n]([v.sup.+.sub.0]) > [v.sub.H] is not true for all n. We make [n.sub.0] the smallest which satisfies [mathematical expression not reproducible]. Then, there must be a positive integer [n.sub.1] > [n.sub.0] and [mathematical expression not reproducible] monotonically increases as [n.sub.1] increases, so [mathematical expression not reproducible]. Therefore, there is a globally asymptotically stable order-1 periodic solution of system (5).

Theorem 7. If [phi]([v.sub.H]) > [v.sub.H], [[phi].sup.2]([v.sub.H]) [greater than or equal to] [v.sub.H], and there are no fixed points on [0, [v.sub.H]] of [phi]; then, system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution.

Proof. If there are no fixed points on [0, [v.sub.H]] of [phi], then there is a positive constant i which makes [v.sup.+.sub.i] [less than or equal to] [v.sub.H] and [v.sup.+.sub.i+1] [greater than or equal to] [v.sub.H]. Based on the definition of the Poincare map [phi], we get [v.sup.+.sub.i+1] = [phi]([v.sup.+.sub.i]) [less than or equal to] [phi]([v.sub.H]) and [v.sup.+.sub.i+1] [member of] [[v.sub.H], [phi]([v.sub.H])]. Because the [phi] is decreasing on [[v.sub.H], +[infinity]) for any [v.sup.+.sub.1] > [v.sub.H], after one pulse, there is [v.sup.+.sub.2] = [phi]([v.sup.+.sub.1]) [less than or equal to] [phi]([v.sub.H]). Therefore, [v.sup.+.sub.i+1] [member of] [[v.sub.H], [phi]([v.sub.H])] holds for all i [greater than or equal to] 2. Also, [[phi].sup.2] is monotonically increasing on [[v.sub.H], +[infinity]) so that

[phi]([[v.sub.H], [phi]([v.sub.H])]) = [[[phi].sup.2]([v.sub.H]), [phi]([v.sub.H])] [subset] [phi]([v.sub.H], [phi]([v.sub.H])]. (48)

Next, the existence of the periodic solution is discussed. First of all, for any [v.sup.+.sub.0] [member of] [[v.sub.H], [phi]([y.sub.H])] and make [v.sup.+.sub.1] = [phi]([v.sup.+.sub.0]) [not equal to] [v.sup.+.sub.0] and [v.sup.+.sub.2] = [[phi].sup.2]([v.sup.+.sub.0]) [not equal to] [v.sup.+.sub.0]. If [v.sup.+.sub.1] = [phi]([v.sup.+.sub.0]) = [v.sup.+.sub.0] and [v.sup.+.sub.2] = [[phi].sup.2]([v.sup.+.sub.0]) = [v.sup.+.sub.0], then [v.sup.+.sub.0] is the fixed point of [phi] which proves system (5) either has a stable order-1 periodic solution or a stable order-2 periodic solution. So the relations about [v.sub.H], [phi]([v.sub.H]), [v.sup.+.sub.0], [v.sup.+.sub.1], and [v.sup.+.sub.2] are needed to be discussed.

(i) [v.sub.H] [less than or equal to] [v.sup.+.sub.2] < [v.sup.+.sub.0] < [v.sup.+.sub.1] [less than or equal to] [phi]([v.sub.H]) (Figure 2(a)). Then, [v.sup.+.sub.3] = [phi]([v.sup.+.sub.2]) > [phi]([v.sup.+.sub.0]) = [v.sup.+.sub.1] and [v.sup.+.sub.4] = [phi]([v.sup.+.sub.3]) > [phi]([v.sup.+.sub.1]) = [v.sup.+.sub.2]. It can be obtained by mathematical induction that

[v.sub.H] [less than or equal to] ... < [v.sup.+.sub.2n+2] < [v.sup.+.sub.2n] < ... < [v.sup.+.sub.2] < [v.sup.+.sub.0]

< [v.sup.+.sub.1] < ... < [v.sup.+.sub.2n-1] < [v.sup.+.sub.2n+1] < ... [less than or equal to] [phi]([v.sub.H]). (49)

(ii) [v.sub.H] [less than or equal to] [v.sup.+.sub.0] < [v.sup.+.sub.2] < [v.sup.+.sub.1] [less than or equal to] [phi]([v.sub.H]) (Figure 2(b)). Then, [v.sup.+.sub.1] = [phi]([v.sup.+.sub.0]) > [phi]([v.sup.+.sub.2]) = [v.sup.+.sub.3] > [v.sup.+.sub.2] = [phi]([v.sup.+.sub.1]) and [v.sup.+.sub.2] = [phi]([v.sup.+.sub.1]) < [phi] ([v.sup.+.sub.3]) = [v.sup.+.sub.4] < [v.sup.+.sub.3] = [phi]([v.sup.+.sub.2]) < [v.sup.+.sub.1]; i.e., it can be obtained by mathematical induction that

[v.sub.H] [less than or equal to] [v.sup.+.sub.0] < [v.sup.+.sub.2] < ... [v.sup.+.sub.2n] < [v.sup.+.sub.2n+2]

< ... < [v.sup.+.sub.2n+1] < [v.sup.+.sub.2n-1] < ... < [v.sup.+.sub.1] [less than or equal to] [phi]([v.sub.H]). (50)

(iii) [v.sub.H] [less than or equal to] [v.sup.+.sub.1] < [v.sup.+.sub.2] < [v.sup.+.sub.0] [less than or equal to] [phi]([v.sub.H]) (Figure 2(c)). In the same way, about case (ii), we can obtain

[v.sub.H] [less than or equal to] [v.sup.+.sub.1] < ... < [v.sup.+.sub.2n-1] < [v.sup.+.sub.2n+1]

< ... < [v.sup.+.sub.2n+2] < [v.sup.+.sub.2n] < [v.sup.+.sub.2] < [v.sup.+.sub.0] [less than or equal to] [phi]([v.sub.H]). (51)

(iv) [v.sub.H] [less than or equal to] [v.sup.+.sub.1] < [v.sup.+.sub.0] < [v.sup.+.sub.2] [less than or equal to] [phi]([v.sub.H]) (Figure 2(d)). By using the same method as case (i), we can obtain

[v.sub.H] [less than or equal to] ... < [v.sup.+.sub.2n+1] < [v.sup.+.sub.2n-1] < ... < [v.sup.+.sub.1] < [v.sup.+.sub.0]

< [v.sup.+.sub.2] < ... < [v.sup.+.sub.2n] < [v.sup.+.sub.2n+2] < ... [less than or equal to] [phi]([v.sub.H]). (52)

For case (ii), [[phi].sup.2n]([v.sup.+.sub.0]) = [v.sup.+.sub.2n] is monotonically increasing and [[phi].sup.2n+1]([v.sup.+.sub.0]) = [v.sup.+.sub.2n+1] is monotonically decreasing; for case (iii), [[phi].sup.2n]([v.sup.+.sub.0]) = [v.sup.+.sub.2n] is monotonically decreasing and [[phi].sup.2n+1]([v.sup.+.sub.0]) = [v.sup.+.sub.2n+1] is monotonically increasing. It is concluded that for case (ii) and case (iii), there exists either a unique fixed point [v.sub.a] such that

[mathematical expression not reproducible], (53)

or exists two distinct values [v.sub.a] and [v.sub.b] and [v.sub.a] [not equal to] [v.sub.b] such that

[mathematical expression not reproducible]. (54)

However, for cases (i) and (iv), only the later case can be true.

These results verify that there exists either an order-1 periodic solution or a periodic solution+ (5).

Theorem 8. If [phi]([v.sub.H]) > [v.sub.H], then [v.sup.+.sub.m] satisfies [v.sup.+.sub.m] = min{[v.sup.+] : [[phi].sub.m]([v.sup.+]) = [v.sup.+]}, and if there is no fixed point on (0, [v.sub.H]), when [[phi].sup.2]([v.sub.H]) [greater than or equal to] [v.sup.+.sub.m], then system (5) has an order- 3 periodic solution.

Proof. If [phi]([v.sub.H]) > [v.sub.H], and there is no fixed point on (0, [v.sub.H]), it can be seen from Theorem 1 that there is a unique order-1 period solution in ([v.sub.H], [phi]([v.sub.H])):

[phi]([??]) = [??], [??] [member of] ([v.sub.H], [phi]([v.sub.H])), (55)

because Poincare map [phi] is continuous on closed intervals [0, [??]] and

[phi](0) = [tau],

[phi]([??]) = [??]. (56)

According to the intermediate value theorem, there exists [v.sup.+.sub.m] [member of] (0, [??]), and [phi]([v.sup.+.sub.m]) = [u.sub.H].

Furthermore,

[[phi].sup.3]([v.sup.+.sub.m]) = [[phi].sup.2]([u.sub.H]) < [v.sup.+.sub.m],

[[phi].sup.3](0) > 0. (57)

According to the properties of continuous functions on closed intervals, there must be at least one value of [??] to enable

[[phi].sup.3]([??]) = [??]. (58)

This means that system (5) has an order-3 periodic solution.

If we replace condition [[phi].sup.2]([v.sub.H]) [greater than or equal to] [v.sup.+.sub.m] in Theorem 8 with condition [[phi].sup.k-1]([u.sub.H]) < [v.sup.+.sub.m], where [phi]([v.sup.+.sub.m]) = [u.sub.H], the order-k periodic solution of system (5) can be obtained by a similar method of Theorem 8.

5. Numerical Simulation

In the state impulse feedback control, we assign appropriate thresholds for TH and D (see Figures 3 and 4). In Figures 3 and 4, the red line shows the trajectory of the system without the impulse, and the green line shows the trajectory of the system with the impulse; this suggests that populations of predators and pests can be kept within a stable range.

It can be seen from Figure 5 that different initial points will eventually converge to the same order-1 periodic solution and tend to be stable; this indicates the global asymptotic stability of the order-1 periodic solution.

The above numerical simulation also shows that the number of pests can be controlled in the state pulse feedback control, which verifies the feasibility of state pulse feedback control.

In Section 4.1, when the predators disappear and the pests reach TH, we obtain the expression of the boundary period solution and the expression of the pesticide dose. Next, we discuss which key factors can affect the pesticide dose D. We gave some reasonable parameters, as shown in Figure 6. The results show that as [mathematical expression not reproducible] decreases, dose D must also be increased (Figure 6(a)). Furthermore, as the T of chemical control increases, the dose D increases (Figure 6(b)). Biologically, we need to consider both the threshold TH and the period T in the process of pest control.

According to condition (39), we can judge whether the chemical control can stabilize the boundary periodic solution alone. [R.sub.1] < 1 means that chemical control by D dose alone can control the pest population below the TH, and vice versa. Therefore, how much the dose D and the threshold TH can affect [R.sub.1] has drawn our attention. For these, we have carried out numerical simulations, as shown in Figure 7(a). The results show that when a single chemical control method is used, high dose D can control the pest population. In addition, as shown in Figure 7(b), for a relatively small TH, we have [R.sub.1] < 1, and once TH is greater than a certain value, [R.sub.1] > 1. The results show that under the fixed parameter values, the smaller the TH value, the better the prevention and control of pests. In the process of pesticide management, as long as we choose a reasonable threshold TH under pulse state feedback control, we can avoid excessive use of pesticides and reduce some negative effects of pesticides.

6. Conclusion

Compared with previous studies on state-dependent feedback control, we mainly do the following work: the global dynamics of complex models are studied according to the Poincaree map, and the main properties of Poincaree map are studied to prove the existence of fixed points and the existence of order-k (k [greater than or equal to] 1) periodic solutions. Besides, we study the effect of pesticide dose on single chemical control or chemical control combined with biological control. The results show that the pest population density can not only be controlled below the threshold under the state pulse feedback control but also avoid excessive application of pesticides and reduce some negative effects of pesticides.

https://doi.org/10.1155/2019/5308014

Data Availability

The data used to support the findings of this study are available upon request to the corresponding author.

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), SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001), Joint Innovative Center for Safe And Effective Mining Technology and Equipment of Coal Resources, SDUST Innovation Fund for Graduate Students (SDKDYC190351).

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Dezhao Li, (1) Huidong Cheng [ID], (1) and Yu Liu (2)

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

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

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

Received 4 September 2019; Revised 15 October 2019; Accepted 28 October 2019; Published 22 November 2019

Guest Editor: Alejandro Hossian

Caption: Figure 1: Poincare map [phi] related to the impulsive point series [v.sup.+.sub.k] with parameters fixed as s = 2, A = 0.5, [delta] = 1, d = 0.25, [k.sub.1] = 1, [k.sub.2] = 0.2, TH = 0.6, and [tau] = 0.3. (a) D = 1. (b) D = 0.5.

Caption: Figure 2: Four cases of existence of periodic solutions in Theorem 7.

Caption: Figure 3: State pulse feedback control strategy of system (5): s = 2, A = 0.5, S = 1, [delta] = 0.25, [k.sub.1] = 1, [k.sub.2] = 0.2, TH = 0.6, [tau] = 0.3, and D = 0.5.

Caption: Figure 4: State pulse feedback control strategy of system (5): s = 2, A = 0.5, S = 1, [delta] = 0.25, [k.sub.1] = 1, [k.sub.2] = 0.2, TH = 0.6, [tau] = 0.3, and D = 1.

Caption: Figure 5: State pulse feedback control strategy of system (5): s = 2, A = 0.5, [delta] = 1, d = 0.25, [k.sub.1] = 1, [k.sub.2] = 0.2, TH = 0.6, [tau] = 0.3, and D = 1, and with three different initial points.

Caption: Figure 6: Effects of parameters [mathematical expression not reproducible] and T on the D of chemical control: (a) T = 20, [k.sub.1] = 1.6; (b) TH = 0.22, [k.sub.1] = 1.6.

Caption: Figure 7: Effects of control parameters on the threshold condition [R.sub.1] and [k.sub.1] = -1.6, [k.sub.2] = -1.8, d = 0.25, and A = 0.5: (a) TH = 0.22; (b) D = 0.5.
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Author:Li, Dezhao; Cheng, Huidong; Liu, Yu
Publication:Complexity
Date:Dec 31, 2019
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