Multi-State Dependent Impulsive Control for Pest Management

Definition 2.1. A triple (X, Π, R⁺) is said to be a semidynamical system if X is a metric space, R⁺ is the set of all nonnegative real, and Π(P, t) : X × R⁺ → X is a continuous map such that:

• (i)

  Π(P, 0) = P for all P ∈ X;

• (ii)

  Π(P, t) is continuous for t and s;

• (iii)

  Π(Π(P, t)) = Π(P, t + s) for all P ∈ X and t, s ∈ R⁺. Sometimes a semi-dynamical system (X, Π, R⁺) is denoted by (X, Π).

Definition 2.2. Assuming that

• (i)

  (X, Π) is a semi-dynamical system;

• (ii)

  M is a nonempty subset of X;

• (iii)

  function I : M → X is continuous and for any P ∈ M, there exists a ε > 0 such that for any 0<|t | < ε, Π(P, t) ∉ M.

Then, (X, Π, M, I) is called an impulsive semi-dynamical system.

For any P, the function Π_P : R⁺ → X defined as Π_P(t) = Π(P, t) is continuous, and we call Π_P(t) the trajectory passing through point P. The set C⁺(P) = {Π(P, t)/0 ≤ t < +∞} is called positive semitrajectory of point P. The set C⁻(P) = {Π(P, t)/−∞ < t ≤ 0} is called the negative semi-trajectory of point P.

Definition 2.3. One considers state-dependent impulsive differential equations:

Definition 2.4. Suppose that the impulse set M and the phase set N are both lines, as shown in Figure 1. Define the coordinate in the phase set N as follows: denote the point of intersection Q between N and x-axis as O, then the coordinate of any point A in N is defined as the distance between A and Q and is denoted by y_A. Let C denote the point of intersection between the trajectory starting from A and the impulse set M, and let B denote the phase point of C after impulse with coordinate y_B. Then, we define B as the successor point of A, and then the successor function of point A is that f(A) = y_B − y_A.

Definition 2.5. A trajectory $$ is called order one periodic solution with period T if there exists a point P₀ ∈ N and T > 0 such that P = Π(P₀, t) ∈ M and P⁺ = I(P) = P₀.

Lemma 2.6. Successor function defined in Definition 2.1 is continuous.

Lemma 2.7. In system (1.4), if there exist A ∈ N, B ∈ N satisfying successor function f(A)f(B) < 0, then there must exist a point P (P ∈ N) satisfying f(P) = 0 the function between the point of A and the point of B, thus there is an order one periodic solution in system (1.4).

Next, we consider the model (1.4) without impulse effects:

• (I)

  two steady states O(0,0)-saddle point, and R(d/b(λ − dh), a/b) = R(x^*, y^*)(λ > dh)-stable centre;

• (II)

  a unique closed trajectory through any point in the first quadrant contained inside the point R.

Case 1 (0 < h1 < d/b(λ − dh)). In this case, set M₁ and N₁ are both in the left side of stable center R (as shown in Figure 3). Take a point B₁(h₁, (a/b) + ε) ∈ N₁ above A, where ε > 0 is small enough, then there must exist a trajectory passing through B₁ which intersects with M₁ at point $$, we have $$. Since p₁ ∈ M₁, pulse occurs at the point P₁, the impulsive function transfers the point P₁ into $$ and $$ must lie above B₁, therefore inequation $$ holds, thus the successor function of B₁ is $$.

On the other hand, the trajectory with the initial point $$ intersects M₁ at point $$, in view of vector field and disjointness of any two trajectories, we know $$. Supposing the point P₂ is subject to impulsive effects to point $$, where $$, the position of $$ has the following two cases.

Subcase 1.1 ($$). In this case, the point $$ lies above the point A and below $$, then we have $$.

By Lemma 2.7, there exists an order one periodic solution of system (1.4), whose initial point is between B₁ and $$ in set N₁.

Subcase 1.2 ($$ (as shown in Figure 4)). The point $$ lies below the point A, that is, $$, then pulse occurs at the point $$, the impulsive function transfers the point $$ into $$.

•  

  If $$, like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4).

•  

  If $$, that is, $$, then we repent the above process until there exists k ∈ Z₊ such that $$ jumps to $$ after k times’ impulsive effects which satisfies $$. Like the analysis of Subcase 1.1, there exists an order one periodic solution of system (1.4).

Now, we can summarize the above results as the following theorem.

Theorem 3.1. If λ > dh, 0 < h₁ < d/b(λ − dh), then there exists an order one periodic solution of the system (1.4).

Remark 3.2. It shows from the proved process of Theorem 3.1 that the number of natural enemies should be selected appropriately, which aims to reduce releasing impulsive times to save manpower and resources.

Case 2 (h2 < d/b(λ − dh)). In this case, set M₂ and N₂ are both in the left side of stable center R, in the light of the different position of the set N₂, we consider the following two cases.

Subcase 2.1 (0 < h₁ < (1 − α)h₂ < h₂ < d/b(λ − dh)). In this case, the set N₂ is in the right side of M₁ (as shown in Figure 5). The trajectory passing through point A which tangents to N₂ at point A intersects with M₂ at point $$. Since the point P₀ ∈ M₂, then impulse occurs at point P₀. Supposing the point P₀ is subject to impulsive effects to point $$, where $$, the position of $$ has the following three cases:

(1) ($$). Take a point B₁((1 − α)h₂, ε + a/b) ∈ N₂ above A, where ε > 0 is small enough. Then there must exist a trajectory passing through the point B₁ which intersects with the set M₂ at point $$. In view of continuous dependence of the solution on initial value and time, we know $$ and the point P₁ is close to P₀ enough, so we have the point $$ is close to $$ enough and $$, then we obtain $$.

On the other hand, the trajectory passing through point B tangents to N₁ at point B. Set $$. Denote the coordinates of impulsive point $$ corresponding to the point $$.

•  

  If $$ then $$. So we obtain $$. There exists an order one periodic solution of system (1.4), whose initial point is between the point B₁ and the point S in set N₂ (Figure 5).

•  

  If $$ and $$, we have $$, we conclude that there exists an order one periodic solution of system (1.4).

•  

  If $$ and $$, from the vector field of system (1.4), we know the trajectory of system (1.4) with any initiating point on the N₂ will ultimately stay in Ω₁ = {(x, y)/0 ≤ x ≤ h₁, y ≥ 0} after one impulsive effect. Therefore there is no an order one periodic solution of system (1.4).

(2) ($$ (as shown in Figure 6)). In this case, the point $$ lies below the point A, that is, $$, thus the successor function of the point A is $$.

Take another point B₁((1 − α)h₂, ε) ∈ N₂, where ε > 0 is small enough. Then there must exist a trajectory passing through the point B₁ which intersects M₂ at point $$. Suppose the point $$ is subject to impulsive effects to point $$, then we have $$. So we have $$.

From Lemma 2.7, there exists an order one periodic solution of system (1.4), whose initial point is between B₁ and A in set N₂.

(3) ($$). $$ coincides with A, and the successor function of A is that f(A) = 0, so there exists an order one periodic solution of system (1.4) which is just a part of the trajectory passing through the A.

Theorem 3.3. Assuming that λ > dh and 0 < h₁ < (1 − α)h₂ < h₂ < d/b(λ − dh).

•  

  If $$, there exists an order one periodic solutions of the system (1.4).

•  

  If $$, if $$ or $$, and $$, there exists an order one periodic solutions of the system (1.4).

•  

  If $$, $$ and $$, there is no an order one periodic solutions of the system (1.4). The trajectory of system (1.4) with any initiating point on the N₂ will ultimately stay in Ω₁ = {(x, y)/0 ≤ x ≤ h₁, y ≥ 0} after one impulsive effect.

Theorem 3.4. If λ > dh, 0 < (1 − α)h₂ < h₁ < h₂ < d/b(λ − dh), there is no an order one periodic solutions to the system (1.4), and the trajectory with initial point $$ will stay in the region Ω₁ = {(x, y)∣x ≥ 0, y ≥ 0, x ≤ h₁}.

Case 3 (d/b(λ − dh) < h2). In this case, the set M₂ is on the right side of stable center R. In the light of the different position of N₂, we consider the following two subcases.

Subcase 3.1 (h₁ < (1 − α)h₂ < d/b(λ − dh) < h₂). In this case, the set M₂ is in the right side of R. Then there exists a unique closed trajectory Γ₁ of system (1.4) which contains the point R and is tangent to M₂ at the point A.

Since Γ₁ is closed trajectory, we take their the minimal value δ_min of abscissas at the trajectory Γ₁, namely, δ_min ≤ x holds for any abscissas of Γ₁.

(1) (h₁ < (1 − α)h₂ < δ_min < d/b(λ − dh)   < h₂). In this case, there is a trajectory, which contains the point R(d/b(λ − dh), a/b) and is tangent to the N₂ at the point B intersects M₂ at a point $$. Suppose point P₁ is subject to impulsive effects to point $$, here $$. The position of $$ has the following three sub-cases.

•  

  If $$ (Figure 7), the point $$ lies below the point B. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case.

•  

  If $$, the point $$ lies above the point B; the trajectory from initiating point $$ intersects with the line L₁ at point C. If h₁ ≤ y_C (Figure 8), we have $$ and $$, then the successor function of $$ is that $$. Then, we know that there exists an order one periodic solution of system (1.4), whose initial point is between the point $$ and B in set N₂. If h₁ > y_C (Figure 9), there is a trajectory which is tangent to the N₁ at a point D intersects with M₂ at a point $$, P₃ jumps to $$ after the impulsive effects. If $$, we can easily know that there exists an order one periodic solution of system (1.4). If $$, by the qualitative analysis of the system (1.4), we know that trajectory with any initiating point on the N₂ will ultimately stay in Γ₁ after a finite number of impulsive effects.

•  

  If $$, the point $$ coincides with the point B, and the successor function of the point B is that f(B) = 0; then there exists an order one periodic solution which is just a part of the trajectory passing through the point B.

Theorem 3.5. Assuming that λ > dh and h₁ < (1 − α)h₂ < δ_min < d/b(λ − dh) < h₂.

•  

  If $$, there exists an order one periodic solution to the system (1.4).

•  

  If $$ and h₁ ≤ y_C, then there exists an order one periodic solution to the system (1.4).

•  

  If $$, h₁ > y_C and $$, then there exists an order one periodic solution to the system (1.4).

(2)  (h₁ < δ_min < (1 − α)h₂ < d/b(λ − dh) < h₂). In this case, denote the closed trajectory Γ₁ of system (1.4) intersects with N₂ two points $$ and $$ (as shown in Figure 10). Since A ∈ M₂, impulse occurs at the point A. Suppose point A is subject to impulsive effects to point $$, here $$.

•  

  If $$ or $$, then $$ coincides with A₁ or $$ coincides with A₂, and the successor function of A₁ or A₂ is that f(A₁) = 0 or f(A₂) = 0. So, there exists an order one periodic solution of system (1.4) which is just a part of the trajectory Γ₁.

•  

  If $$, the point $$ lies below the point A₂. Like the analysis of Subcase 2.1(2), we can prove there exists an order one periodic solution to the system (1.4) in this case.

•  

  If $$ (as shown in Figure 11), the point $$ is above the point A₁. Like the analysis of Subcase 3.1(1), we obtain sufficient conditions of existence of order one periodic solution to the system (1.4).

Theorem 3.6. Assuming that λ > dh, h₁ < (1 − α)h₂ < δ_min < d/b(λ − dh) < h₂.

•  

  If $$, there exists an order one periodic solution to the system (1.4).

•  

  If $$ and h₁ ≤ y_C, then there exists an order one periodic solution to the system (1.4).

•  

  If $$, and $$, then there exists an order one periodic solution to the system (1.4).

(3)  ($$). In this case, we note that the point $$ must lie between the point A₁ and the point A₂ (As shown in Figure 12). Taking a point B₁ ∈ M₂ such that B₁ jumps to A₂ after the impulsive effect, denote $$. Since $$, we have $$. Let $$, take a point B₂ ∈ M₂ such that B₂ jumps to $$ after the impulsive effects, then we have $$, $$. This process continues until there exists a $$ (K ∈ Z₊) satisfying $$. So we obtain a sequence $$ of the set M₂ and a sequence {B_k} _k=1,2,…,K of set N₂ satisfying $$. In the following, we will prove the trajectory of system (1.4) with any initiating point of set N₂ will ultimately stay in Γ₁.

Theorem 3.7. Assuming that λ > dh, h₁ < δ_min < (1 − α)h₂ < d/b(λ − dh) < h₂, and $$, there is no periodic solution in system (1.4), and the trajectory with any initiating point on the set N₂ will stay in Γ₁ or in the region Ω₁ = {(x, y)∣x ≥ 0, y ≥ 0, x ≤ h₁}.

Theorem 4.1. Assuming that λ > dh, h₁ < d/(b(λ − dh)) and δ ≥ a/b.

If $$ or $$ (Figure 14), then

• (I)

  there exists a unique order one periodic solution of system (1.4),

• (II)

  if (1 − α)h₂ < h₁, order one periodic solution of system (1.4) is attractive in the region Ω₀ = {(x, y)∣x ≥ 0, y ≥ 0, x ≤ h₂}.

Proof. By the derivation of Theorem 3.1, we know there exists an order one periodic solution of system (1.4). We assume trajectory $$ and segment $$ formulate an order one periodic solution of system (1.4), that is, there exists a P⁺ ∈ N₂ such that the successor function of P⁺ satisfies f(P⁺) = 0. First, we will prove the uniqueness of the order one periodic solution.

We take any two points $$ satisfying $$, then we obtain two trajectories whose initiate points are C₁ and C₂ intersects the set M₁ two points $$ and $$, respectively, (Figure 13). In view of the vector field of system (1.4) and the disjointness of any two trajectories without impulse, we know $$. Suppose the points D₁ and D₂ are subject to impulsive effect to points $$ and $$, respectively, then we have $$ and $$, so we get f(C₁) − f(C₂) < 0, thus we obtain the successor function f(x) is decreasing monotonously of N₁, so there is a unique point P⁺ ∈ N₁ satisfying f(P⁺) = 0, and the trajectory $$ is a unique order one periodic solution of system (1.4).

Next, we prove the attractiveness of the order one periodic solution $$ in the region Ω₀ = {(x, y)∣x ≥ 0, y ≥ 0, x ≤ h₂}. We focus on the case $$; by similar method, we can obtain similar results about case $$ (Figure 14).

Take any point $$ above P⁺. Denote the first intersection point of the trajectory from initiating point $$ with the set M₁ at $$, and the corresponding consecutive points are $$, respectively. Consequently, under the effect of impulsive function, the corresponding points after pulse are $$.

Due to conditions $$, $$, δ ≥ a/b and disjointness of any two trajectories, then we get a sequence $$ of the set N₁ satisfying

So the successor function $$ and $$ hold. Series $$ increases monotonously and has upper bound, so $$ exists. Next, we will prove $$. Set lim _k→∞ P_2k−1 = C⁺, we will prove P⁺ = C⁺. Otherwise P⁺ ≠ C⁺, then there is a trajectory passing through the point C⁺ which intersects the set M₁ at point $$, then we have $$, $$. Since f(C⁺) ≥ 0 and P⁺ ≠ C⁺, according to the uniqueness of the periodic solution, then we have $$, thus $$ hold. Analogously, let trajectory passing through the point C⁺ which intersects the set M₁ at point $$, and the corresponding consecutive points is $$, then $$, then we have $$, this is, contradict to the fact that C⁺ is a limit of sequence $$, so we obtain P⁺ = C⁺. So, we obtain $$. Similarly, we can prove $$.

From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from initiating any point of the N₁ will ultimately tend to be order one periodic solution $$.

Any trajectory from initial point $$ will intersect with N₁ at some point with time increasing on the condition that (1 − α)h₂ < h₁ < h₂ < d/b(λ − dh); therefore the trajectory from initial point on N₁ ultimately tends to be order one periodic solution $$. Therefore, order one periodic solution $$ is attractive in the region Ω₀. This completes the proof.

Remark 4.2. Assuming that λ > dh, h₁ < h₂ < d/b(λ − dh) and δ ≥ a/b, if $$ or $$ then the order one periodic solution is unattractive.

Theorem 4.3. Assuming that λ > dh, h₁ < (1 − α)h₂ < h₂ < d/b(λ − dh) and $$ (as shown in Figure 15), then

• (I)

  There exists an odd number of order one periodic solutions of system (1.4) with initial value between $$ and A in set N₂.

• (II)

  If the periodic solution is unique, then the periodic solution is attractive in region Ω₂, here Ω₂ is open region which is constituted by trajectory $$, segment $$, segment $$, and segment $$.

Proof. (I) According to the Subcase 2.1(2), f(A) < 0 and $$, and the continuous successor function f(x), there exists an odd number of root satisfying f(x) = 0, then we can get there exists an odd number of order one periodic solutions of system (1.4) with initial value between $$ and A in set N₂.

(II) By the derivation of Theorem 3.3, we know there exists an order one periodic solution of system (1.4) whose initial point is between $$ and $$ in the set N₂. Assume trajectory $$ and segment $$ formulate the unique order one periodic solution of system (1.4) with initial point P⁺ ∈ N₂.

On the one hand, take a point $$ satisfying $$ and $$. The trajectory passing through the point $$ which intersects with the set M₂ at point $$, that is, $$, then we have $$, thus $$, since $$. So, we obtain $$; Set $$, because $$, we know $$, then we have $$ and $$. This process is continuing, then we get a sequence $$ of the set N₂ satisfying

On the other hand, set $$, then D₁ jumps to $$ under the impulsive effects. Since $$, we have $$, thus we obtain $$. Set $$, then D₂ jumps to $$ under the impulsive effects. We have $$; this process is continuing, we can obtain a sequence $$ of the set N₂ satisfying

Any point Q ∈ N₂ below A must be in some interval $$, $$, $$, $$. Without loss of generality, we assume the point $$. The trajectory with initiating point Q moves between trajectory $$ and $$ and intersects with M₂ at some point between D_k+2 and D_k+1; under the impulsive effects, it jumps to the point of N₂ which is between $$, then trajectory $$ continues to move between trajectory $$ and $$. This process can be continued unlimitedly. Since $$, the intersection sequence of trajectory $$, and the set N₂ will ultimately tend to the point P⁺. Similarly, if $$, we also can get the intersection sequence of trajectory $$ and the set N₂ will ultimately tend to point P⁺. Thus, the trajectory initiating any point below A ultimately tend to the unique order one periodic solution $$.

Denote the intersection of the trajectory passing through the point B which tangents to N₁ at the point B with the set N₂ at a point S((1 − α)h₂, y_S). The trajectory from any initiating point on segment $$ will intersect with the set N₂ at some point below A with time increasing. So like the analysis above, we obtain the trajectory from any initiating point on segment $$ will ultimately tend to be the unique order one periodic solution $$.

Since the trajectory with any initiating point of the Ω₂ will certainly intersect with the set N₂, then from the above analysis, we know the trajectory with any initiating point on segment $$ will ultimately tend to be order one periodic solution $$. Therefore, the unique order one periodic solution $$ is attractive in the region Ω₂. This completes the proof.

Remark 4.4. Assuming that λ > dh, h₁ < (1 − α)h₂ < h₂ < d/b(λ − dh), and $$, then the order one periodic solution with initial point between A and $$ is unattractive.

Theorem 4.5. Assuming that λ > dh, h₁ < (1 − α)h₂ < h₂ < d/b(λ − dh), $$ (Figure 16) then, there exists a unique order one periodic solution of system (1.4) which is attractive in the region Ω₂, here Ω₂ is open region which enclosed by trajectory $$, segment $$, segment $$ and segment $$.

Proof. By the derivation of Theorem 3.3, we know there exists an order one periodic solution of system (1.4). We assume trajectory $$ and segment $$ formulate an order one periodic solution of system (1.4), that is, P⁺ ∈ N₂ is its initial point satisfying f(P⁺) = 0. Like the proof of Theorem 4.1, we can prove the uniqueness of the order one periodic solution of system (1.4).

Next, we prove the attractiveness of the order one periodic solution $$ in the region Ω₂.

Denote the first intersection point of the trajectory from initiating point $$ with the impulsive set M₂ at $$, and the corresponding consecutive points are $$ respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are $$, …. In view of $$ and disjointness of any two trajectories, we have

The trajectory from initiating point between $$ and $$ will intersect with impulsive set N₂ with time increasing, under the impulsive effects it arrives at a point of N₂ which is between $$ or $$. Then like the analysis of Theorem 4.3, we know the trajectory from any initiating point between $$ and $$ will ultimately tend to be order one periodic solution $$.

Denote the intersection of the trajectory passing through point B which tangents to N₁ at point B with the set N₂ at S. Since the trajectory from initiating any point below S of the set N₂ will certain intersect with set N₂, next we only need to prove the trajectory with any initiating point below S of the set N₂ will ultimately tend to be order one periodic solution $$.

Assume a point B₀ of set M₂ jumps to $$ under the impulsive effect. Set $$. Assume point B₁ of set N₂ jumps to $$ under the impulsive effect. Set $$. This process is continuing until there exists a $$ ($$) satisfying $$. So we obtain a sequence $$ of set M₂ and a sequence $$ of set N₂ satisfying $$. For any point of set N₂ below $$, it must lie between $$ and $$ here k = 1,2, …, K₀. After K₀ + 1 times’ impulsive effects, the trajectory with this initiating point will arrive at some point of the set N₂ which must be between $$ and $$, and then will ultimately tend to order one periodic solution $$. There is no order one periodic solution with the initial point below $$.

The trajectory with any initiating point in segment $$ will intersect with the set N₂ at some point below $$ with time increasing. Like the analysis above, we obtain the trajectory initiating any point on segment $$ will ultimately tend to be the unique order one periodic solution $$.

From above analysis, we know there exists a unique order one periodic solution in system (1.4), and the trajectory from any initiating point below S will ultimately tend to be order one periodic solution $$. Therefore, order one periodic solution $$ is attractive in the region Ω₂. This completes the proof.

Example 5.1. Existence and attractiveness of order one periodic solution.

We set h₁ = 1, α = 0.6, β = 0.8, q = 0.8, h₂ = 1.8, initiating points are (0.85,0.2) (red), (0.8,0.5) (green), and (0.75,0.11) (blue), respectively. Figure 17 shows that the conditions of Theorems 3.1 and 4.1 hold, system (5.1) exists order one periodic solution. The trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.

Example 5.2. Existence and attractiveness of positive periodic solution.

We set h₁ = 0.7, α = 0.6, β = 0.8, q = 0.8, h₂ = 1.8, h₁ < (1 − α)h₂ < x^*, initiating points are (0.8,0.1) (red), (0.7,0.5) (green), and (0.75,0.3) (blue), respectively. Figure 18 shows that the conditions of Theorems 3.3 and 4.3 hold, there exists order one periodic solution of the system (5.1), and the trajectory from different initiating must ultimately tend to be the order one periodic solution. Therefore, order one periodic solution is attractive.

Example 5.3. Existence and attractive of positive periodic solutions.

We set h₁ = 0.7, α = 0.6, β = 0.8, q = 0.8, h₂ = 3.5, h₁ < (1 − α)h₂ < x^* < h₂, initiating points are (1,0.7) (red), (1.4,0.5) (green), and (1.2,1) (blue) as shown in Figure 19. Therefore, the conditions of Theorems 3.6 and 4.5 hold, then system (5.1) exists order one periodic solution, and it is attractive.