Dynamic analysis of a prey–predator model with state-dependent control strategy and square root response function
- Hongxia Liu^{1, 2}Email author and
- Huidong Cheng^{1}
https://doi.org/10.1186/s13662-018-1507-0
© The Author(s) 2018
Received: 23 October 2017
Accepted: 27 January 2018
Published: 20 February 2018
Abstract
In this work, a prey-predator model with square root response function under a state-dependent impulse is proposed. Firstly, according to the differential equation geometry theory and the method of successor function, the existence, uniqueness and attractiveness of the order-1 periodic solution are analyzed. Then the stability of the order-1 periodic solution is discussed by the Poincaré criterion for impulsive differential equations. Finally, we show a specific example and carry out numerical simulations to verify the theoretical results.
Keywords
MSC
1 Introduction
The rest of the paper is organized as follows. Section 2 provides some basic definitions and lemmas as preparation. In Section 3, according to the differential equation geometry theory and the method of successor function, we analyze the existence and attractiveness of the order-1 periodic solution of system (2). In addition, sufficient conditions of the stability of the order-1 periodic solution is obtained by analogy of the Poincaré criterion. In Section 4, we show a specific example and carry out numerical simulations. Finally, we conclude our work.
2 Preliminaries
Some basic definitions and lemmas are provided in this section which are necessary for the following discussion.
Definition 2.1
([17])
For any \(Q\in\Omega\), the map \(\Gamma_{Q}: R^{+}\rightarrow\Omega\) defined as \(\Gamma_{Q}(t)=\Gamma(Q,t)\) is continuous and we call \(\Gamma_{Q}(t)\) the orbit passing through point Q. The set \(C^{+}(Q)=\{\Gamma(Q,t)\mid0\leq t<+\infty\}\) and the set \(C^{-}(Q)=\{ \Gamma(Q,t)\mid-\infty< t \leq0\}\) is called positive semi-orbit and the negative semi-orbit of point Q, respectively. For convenience, if \(Q\in\Omega-M\), \(g(Q)\) is called the first point of intersection of \(C^{+}(Q)\) and M. For any point \(B\in\Omega-N\), we define \(\Pi(B)\) as the first point of \(C^{-}(Q)\) and N.
Remark 2.1
Based on system (2), we get \(M=\{(x, y)\mid x=h, y\geq0\}\), \(N=\{(x, y)\mid x=(1-a)h, y\geq0\}\), for any point \((x,y)\in M\), when \(x=h\), we get \(I: (h, y)\in M\rightarrow ((1-a)h,(1+b)y+c)\in N\).
Definition 2.2
([18])
Definition 2.3
([19])
An orbit \(\widetilde{\Gamma }(Q_{0},T)\) is called order-1 periodic solution with period T if there exist a point \(Q_{0}\in N\) and \(T>0\) such that \(Q=\Gamma (Q_{0},T)\in M\) and \(Q^{+}={I(Q)=Q_{0}}\).
Lemma 2.1
([20])
In system (3), if there exist \(A\in N\), \(B\in N\) satisfying the successor function \(f(A)f(B)<0\), then there must exist a point S (\(S\in N\)) satisfying S between point A and point B such that \(f(S)=0\), then system (3) has an order-1 periodic solution.
Lemma 2.2
([21, 22], Analogue of the Poincaré criterion)
3 Dynamical analysis of system (2)
Theorem 3.1
The positive equilibrium E of system (4) is a stable focus if and only if the conditions \((H_{1})\) and \((H_{2})\) hold.
Proof
3.1 Existence of order-1 periodic solution of system (2)
On the basis of the ecological significance, system (2) should meet \(0 <(1-a)h<h<K\). In this paper, the coordinate of arbitrary point \(B\in R_{2}^{+}\) is denoted \((x_{B},y_{B})\). By Theorem 3.1, we know the x-isoline \(L_{1}\) intersects y-isoline \(L_{2}\) at point \(E(x_{E},y_{E})\). For different h, we discuss two cases as follows.
Case I. \(0<(1-a)h<h\leq x_{E}<K\).
For notation simplicity, let the intersection of the phase set N and the x-isoline \(L_{1}\) be \(A_{0}((1-a)h,y_{A_{0}})\). For system (2), there must exist an orbit Γ tangent to set N and intersect with set M at a point \(A_{1}(h,y_{A_{1}})\), namely, \(g(A_{0})=A_{1}\), then the point \(A_{1}\) jumps to a point \(A_{1}^{+}((1-a)h,y_{A_{1}^{+}}) \in N\) under the action of impulse, where \(y_{A_{1}^{+}}=(1+b)y_{A_{1}}+c\). Then the successor function of point \(A_{0}\) is \(f(A_{0})=y_{A_{1}^{+}}-y_{A_{0}}\). Consider the following three subcases based on the different position of point \(A_{1}^{+}\).
If \(y_{A_{1}^{+}}=y_{A_{0}}\), then \(f(A_{0})=0\), thus the orbit \(\widehat{A_{0}A_{1}}\) and segment \(\overline{A_{1}A_{0}}\) constitute an order-1 periodic solution.
If \(y_{A_{1}^{+}}>y_{A_{0}}\), then \(f(A_{0})>0\). Let \(g(A_{1}^{+})=A_{2}\in M\), under the action of impulse, \(A_{2}\) jumps to a point \(A_{2}^{+}\in N\). Because any two orbits are disjoint, then we get \(y_{A_{2}}< y_{A_{1}}\) and \(y_{A_{2}^{+}}< y_{A_{1}^{+}}\), thus \(f(A_{1}^{+})=y_{A_{2}^{+}}-y_{A_{1}^{+}}<0\). We can choose another point \(C_{0}((1-a)h,y_{A_{0}}+\delta)\in N\). Then \(g(C_{0})=C_{1}\in M\) and under the action of impulse, \(C_{1}\) jumps to a point \(C_{1}^{+}\in N\), then \(y_{C_{1}^{+}}=(1+b)y_{C_{1}}+c>\delta\). Due to any two orbits are disjoint, then \(y_{C_{1}}< y_{A_{1}}\) and point \(C_{1}\) is very close to point \(A_{1}\), thus \(y_{C_{1}^{+}}< y_{A_{1}^{+}}\) and point \(C_{1}^{+}\) is very close to point \(A_{1}^{+}\). Since \(y_{A_{1}^{+}}< y_{A_{0}}\), we have \(f(C_{0})=y_{C_{1}^{+}}-y_{C_{0}}>0\). Therefore, there must be a point \(B\in\overline{A_{0}A_{1}^{+}}\) such that \(f(B)=0\) (see Figure 3(b)).
Case II. \(0<(1-a)h<x_{E}<h<K\).
Let the impulsive set M intersect isocline \(L_{1}\) at point \(P_{1}(h,y_{P_{1}})\). The orbit staring from point \(P_{1}\) tangents to M at point \(P_{1}\), and intersects \(L_{1}\) and M at point \(P_{2}(x_{P_{2}},y_{P_{2}})\) and \(P_{0}(h,y_{P_{0}})\), respectively.
If \(x_{P_{2}}<(1-a)h\), the orbit passing through \(P_{1}\) intersects set N at points \(F_{1}((1-a)h,y_{F_{1}})\) and \(F_{2}((1-a)h,y_{F_{2}})\). Assume \(P_{1}\) jumps to \(P_{1}^{+}((1-a)h,y_{P_{1}^{+}})\in N\) under the action of impulse, there are three subcases to be discussed.
If \(y_{P_{1}^{+}}= y_{F_{1}}\) or \(y_{P_{1}^{+}}= y_{F_{2}}\), then the segment \(\overline{P_{1}F_{1}}\) and the orbit \(\widehat {F_{1}F_{2}P_{1}}\) or the segment \(\overline{P_{1}F_{2}}\) and the orbit \(\widehat{F_{2}P_{1}}\) constitute an order-1 periodic solution of system (2).
If \(y_{P_{1}^{+}}> y_{F_{1}}\) or \(y_{P_{1}^{+}}< y_{F_{2}}\), according to the analysis of Case I, system (2) exists an order-1 periodic solution (see Figure 4(b)).
If \(y_{F_{1}}< y_{P_{1}^{+}}< y_{F_{2}}\), due to any two orbits are disjoint, then the orbit passing through point \(P_{1}^{+}\) tangents to the phase set N at point \(P_{1}^{+}\), and does not intersect with the impulsive set M, thus the order-1 periodic solution is nonexistent (see Figure 4(c)). The reader can find the detailed proof in reference [23].
We obtain the following theorem by the above discussions.
Theorem 3.2
- (I)
If \(0<(1-a)h<h\leq x_{E}<K\), then the order-1 periodic solution is existent in system (2).
- (II)If \(0<(1-a)h<x_{E}<h<K\), there are the following three subcases:
- (i)
If \(x_{P_{2}}\geq(1-a)h\), the order-1 periodic solution is existent in system (2).
- (ii)
If \(x_{P_{2}}< (1-a)h\), \(y_{P_{1}^{+}}\geq y_{F_{1}}\) or \(y_{P_{1}^{+}}\leq y_{F_{2}}\), the order-1 periodic solution is existent in system (2).
- (iii)
If \(x_{P_{2}}< (1-a)h\) and \(y_{F_{1}}< y_{P_{1}^{+}}< y_{F_{2}}\), the order-1 periodic solution is nonexistent.
- (i)
3.2 Attractiveness of order-1 periodic solution of system (2)
Based on the conditions of Theorem 3.2, let the initial value \(x_{0}< h\), the attractiveness of order-1 periodic solution of system (2) is discussed in this subsection. We mainly discuss Case I, similarly, we can get similar conclusions about Cases II.
Theorem 3.3
If the conditions of \(0<(1-a)h<h\leq x_{E}<K\) and \(y_{A_{1}^{+}}\leq y_{A_{0}}\) hold, and system (2) exists a unique order-1 periodic solution, then the periodic solution in region \(\Omega_{1}=\{(x,y)\mid(x,y)\in R^{2}_{+}, x_{0}< h\}\) is attractive.
Proof
Firstly, we choose a point \(B_{0}^{+}((1-a)h,\delta)\in N\) which satisfies \(\delta< c\) and \(y_{B_{0}^{+}}< y_{B^{+}}\). Let \(g(B_{0}^{+})=B_{1}\in M\), under the action of impulse, \(B_{1}\) jumps to \(B_{1}^{+}\in N\), then \(y_{B_{1}}< y_{B}\), thus we have \(y_{B_{1}^{+}}< y_{B^{+}}\). Since \(y_{B_{1}^{+}}=(1+b)y_{B_{1}}+c>\delta \), \(f(B_{0}^{+})=y_{B_{1}^{+}}-y_{B_{0}^{+}}>0\). Let \(g(B_{1}^{+})=B_{2}\in M\), then \(B_{2}\) jumps to \(B_{2}^{+}\in N\). Since \(y_{B_{0}^{+}}< y_{B_{1}^{+}}< y_{B^{+}}\), we have \(y_{B_{1}}< y_{B_{2}}< y_{B}\), then \(y_{B_{1}^{+}}< y_{B_{2}^{+}}< y_{B^{+}}\) and \(f(B_{1}^{+})=y_{B_{2}^{+}}-y_{B_{1}^{+}}>0\). Repeating the process, then we obtain a sequence \(\{B_{i}^{+}\}_{i=0,1,2,\ldots}\in N\) such that \(y_{B_{0}^{+}}< y_{B_{1}^{+}}<\cdots<y_{B_{i}^{+}}<\cdots <y_{B^{+}}\) and \(f(B_{i}^{+})=y_{B_{i+1}^{+}}-y_{B_{i}^{+}}>0\). Thus, \(\{y_{B_{i}^{+}}\}_{i=0,1,2,\ldots}\) is monotonically increasing. Hence, \(\lim_{i\rightarrow+\infty}y_{B_{i}^{+}}\) exists. Next we prove \(\lim_{i\rightarrow+\infty}y_{B_{i}^{+}}=y_{B^{+}}\). Assume \(B_{\ast }^{+}=\lim_{i\rightarrow+\infty}B_{i}^{+}\), then we prove \(B_{\ast }^{+}=B^{+}\). Otherwise, \(B_{\ast}^{+}\neq B^{+}\). Let \(g(B_{\ast }^{+})=\bar{B}\in M\), then B̄ jumps to \({\bar{B}}^{+}\in N\) by impulsive effect. Then \(y_{\bar{B}}< y_{B}\), \(y_{{\bar {B}}^{+}}< y_{B^{+}}\). Because \(f(B_{\ast}^{+})\geq0\), \(B_{\ast }^{+}\neq B^{+}\), and the periodic solution \(\widehat{B^{+}BB^{+}}\) is unique, \(f(B_{\ast}^{+})=y_{{\bar{B}}^{+}}-y_{B_{\ast}^{+}}>0\), thus \(y_{B_{\ast}^{+}}< y_{{\bar{B}}^{+}}< y_{B^{+}}\). Let \(g({\bar {B}}^{+})=\bar{\bar{B}}\in M\), then \(\bar{\bar{B}}\) jumps to \({\bar{\bar {B}}}^{+}\) by impulsive effect. It is easy to know, \(y_{\bar{B}}< y_{\bar {\bar{B}}}< y_{B}\) and \(y_{{\bar{B}}^{+}}< y_{{\bar{\bar {B}}}^{+}}< y_{B^{+}}\), then \(f({\bar{B}}^{+})=y_{{\bar{\bar {B}}}^{+}}-y_{{\bar{B}}^{+}}>0\), this is contradictory to the fact that \(B_{\ast}^{+}=\lim_{i\rightarrow+\infty}B_{i}^{+}\), therefore, \(B_{\ast }^{+}=B^{+}\) and \(\lim_{i\rightarrow+\infty}y_{B_{i}^{+}}=y_{B^{+}}\).
On the other hand, according to the analysis of Case I, we know \(y_{B^{+}}< y_{A_{2}^{+}}< y_{A_{1}^{+}}< y_{A_{0}}\), \(y_{B}< y_{A_{2}}< y_{A_{1}}\) and \(f(A_{1}^{+})=y_{A_{2}^{+}}-y_{A_{1}^{+}}<0\). Let \(g(A_{2}^{+})=A_{3}\in M\), under the action of impulse, \(A_{3}\) jumps to \(A_{3}^{+}\in N\), then \(y_{B^{+}}< y_{A_{3}^{+}}< y_{A_{2}^{+}}\), \(f(A_{2}^{+})=y_{A_{3}^{+}}-y_{A_{2}^{+}}<0\). Repeat the process, we obtain a sequence \(\{A_{k}^{+}\}_{k=1,2,\ldots}\in N\) satisfying \(y_{A_{1}^{+}}>y_{A_{2}^{+}}>\cdots>y_{A_{k}^{+}}>\cdots>y_{S^{+}}\) and \(f(A_{k}^{+})=y_{A_{k+1}^{+}}-y_{A_{k}^{+}}<0\). Thus, \(\{y_{A_{k}^{+}}\} _{k=1,2,\ldots}\) is monotonically decreasing. Hence, \(\lim_{k\rightarrow+\infty}y_{A_{k}^{+}}\) is existent. Similarly, we can prove \(\lim_{k\rightarrow+\infty}y_{A_{k}^{+}}=y_{B^{+}}\).
Due to the orbit with arbitrary point of \(\Omega_{1}\) will intersect with N, next we just prove arbitrary orbit that passes through N eventually is attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\).
Choose an arbitrary point \(R\in N\) below \(A_{0}\) such that \(y_{R}\in [y_{A_{k+1}^{+}},y_{A_{k}^{+}})_{k=1,2,\ldots}\). The orbit starting from R moves between orbit \(\widehat{A_{k}^{+}A_{k+1}}\) and \(\widehat {A_{k+1}^{+}A_{k+2}}\) intersects with M at a point in segment \(\overline{A_{k+2} A_{k+1}}\), then jumps to a point of N in segment \(\overline{A_{k+2}^{+} A_{k+1}^{+}}\), the orbit continues to move between \(\widehat {A_{k+1}^{+}A_{k+2}}\) and \(\widehat{A_{k+2}^{+}A_{k+3}}\). Repeat the process indefinitely, because \(\lim_{k\rightarrow+\infty }y_{A_{k}^{+}}=y_{B^{+}}\), the intersection sequence of orbit which passes through R and the phase set N will be attracted to point \(B^{+}\) eventually. Similarly, if \(y_{R}\in [y_{B_{i}^{+}},y_{B_{i+1}^{+}})_{i=0,1,2,\ldots}\), we also can obtain the intersection sequence of orbit which passes through R and the phase set N will be attracted to point \(B^{+}\) eventually. Therefore, the orbit starting from arbitrary point below \(A_{0}\) eventually is attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\).
The orbit with arbitrary point above \(A_{0}\) of the phase set N will intersect with N at some point below \(A_{0}\) as time goes on, similar to the above discussion, the orbit with arbitrary point above \(A_{0}\) will be eventually attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\).
Based on the above analysis, the orbit with arbitrary point of N will eventually attracted to periodic solution \(\widehat{B^{+}BB^{+}}\). Thus, in the region \(\Omega_{1}\), the periodic solution \(\widehat {B^{+}BB^{+}}\) is attractive. This completes the proof. □
Theorem 3.4
If the conditions of \(0<(1-a)h<h\leq x_{E}<K\) and \(y_{A_{0}}< y_{G_{1}^{+}}< y_{A_{1}^{+}}\) hold, then system (2) exists a unique order-1 periodic solution in region \(\Omega_{1}\) which is attractive.
Proof
We still suppose that \(\widehat{B^{+}BB^{+}}\) is the order-1 periodic solution of system (2). Firstly, we analyze the uniqueness of \(\widehat{B^{+}BB^{+}}\).
Select two points \(I_{0}, J_{0}\in\overline{A_{0}A_{1}^{+}}\) such that \(y_{J_{0}}>y_{I_{0}}>y_{A_{0}}\), the orbits starting from points \(J_{0}\) and \(I_{0}\), respectively, intersect M at \(J_{1}\), \(I _{1}\), then jump to \(J_{1}^{+}, I _{1}^{+}\in N\) under the action of impulse, respectively (see Figure 6(b)). Because any two orbits are disjoint, then \(y_{I_{1}}>y_{J_{1}}\), \(y_{I_{1}^{+}}>y_{J_{1}^{+}}\), \(f(J_{0})=y_{J_{1}^{+}}-y_{J_{0}}\), \(f(I_{0})=y_{I_{1}^{+}}-y_{I_{0}}\), we get \(f(J_{0})-f(I_{0})=(y_{J_{1}^{+}}-y_{I_{1}^{+}})+(y_{I_{0}}-y_{J_{0}})<0\), thus, in the segment \(\overline{A_{0}A_{1}^{+}}\), the successor function f is monotonically decreasing, therefore, for system (2) there exists a unique point \(S^{+}\in\overline{A_{0}A_{1}^{+}}\) such that \(f(B^{+})=0\).
Next, in the region \(\Omega_{1}\), the attractiveness of the periodic solution \(\widehat{B^{+}BB^{+}}\) is proved. See Figure 6(a), Let \(g(G_{1}^{+})=G_{2}\in M\), then \(G_{2}\) jumps to point \(G_{2}^{+}\in N\). In view of any two orbits are disjoint and \(y_{A_{0}}< y_{G_{1}^{+}}< y_{A_{1}^{+}}\), then \(y_{G_{1}}< y_{G_{2}}< y_{A_{1}}\), \(y_{G_{1}^{+}}< y_{G_{2}^{+}}< y_{A_{1}^{+}}\). Let \(g(G_{2}^{+})=G_{3}\in M\), then \(G_{3}\) jumps to point \(G_{3}^{+}\in N\), we have \(y_{G_{1}}< y_{G_{3}}< y_{G_{2}}\), \(y_{G_{1}^{+}}< y_{G_{3}^{+}}< y_{G_{2}^{+}}\). Repeating the steps, we obtain two sequences \(\{G_{k}\}_{k=1,2,\ldots}\in M\) and \(\{G_{k}^{+}\} _{k=1,2,\ldots}\in N\) satisfying \(y_{G_{1}^{+}}<\cdots <y_{G_{2k-1}^{+}}<y_{G_{2k+1}^{+}}<\cdots<y_{G_{2k}^{+}} <y_{G_{2k-2}^{+}}<y_{G_{2}^{+}}\), then \(f(G_{2k-1}^{+})=y_{G_{2k}^{+}}-y_{G_{2k-1}^{+}}>0\), and \(f(G_{2k}^{+})=y_{G_{2k+1}^{+}}-y_{G_{2k}^{+}}<0\). By the proof of Theorem 3.3, we get \(\lim_{k\rightarrow+\infty }y_{G_{2k-1}^{+}}=\lim_{k\rightarrow+\infty}y_{G_{2k}^{+}}=y_{B^{+}}\).
The orbit with arbitrary point in segment \(\overline{A_{1}^{+}G^{+}}\) will intersect N as time goes on, under the action of impulses, it passes through a point in segment \(\overline{G_{2k-1}^{+}G_{2k+1}^{+}}\) or \(\overline{G_{2k}^{+}G_{2k-2}^{+}}\), here \(G_{0}^{+}=A_{1}^{+}\). Similar to the discussion of Theorem 3.3, the orbit with arbitrary point in segment \(\overline{A_{1}^{+}G^{+}}\) will be eventually attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\).
Assume a point \(H_{0}\in M\) jumps to point \(G^{+}\in N\) under the action of impulse. Let \(\Pi(H_{0})=H_{1}^{+}\). Assume a point \(H_{1}\in M\) jumps to point \(H_{1}^{+}\in N\) under the action of impulse. Let \(\Pi (H_{1})=H_{2}^{+}\). Repeat the process until the phase set N exists a \(H_{K_{0}}^{+}\) (\(K_{0}\in Z_{+}\)) such that \(y_{H_{K_{0}}^{+}}< c\). Then there are two sequences \(\{H_{k}\}_{k=1,2,\ldots K_{0}-1}\in M\) and \(\{ H_{k}^{+}\}_{k=1,2,\ldots K_{0}}\in N\) such that \(\Pi (H_{k-1})=H_{k}^{+}\), \(y_{H_{k}^{+}}< y_{H_{k-1}^{+}}\), here \(H_{0}^{+}=G^{+}\). For arbitrary point of N below \(G^{+}\), it must in segment \(\overline{H_{k}^{+}H_{k+1}^{+}}\), where \(k=1,2,\ldots, K_{0}\) and \(y_{H_{k+1}^{+}}=0\). Under \(k+1\) times the impulsive action, the orbit with arbitrary point that below \(G^{+}\) will passes through some point of segment \(\overline{A_{1}^{+}G^{+}}\) and will be attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\) eventually. Thus, the order-1 periodic solution with the initial point that below \(G^{+}\) is nonexistent.
The orbit with arbitrary point above \(A_{1}^{+}\) of N will intersect with N at some point below \(G^{+}\) as time goes on, then the orbit will be attracted to the periodic solution \(\widehat{B^{+}BB^{+}}\). Thus, the order-1 periodic solution with the initial point that above \(A_{1}^{+}\) is nonexistent.
Based on the above analysis, system (2) exists a unique order-1 periodic solution which is attractive. That completes the proof. □
Like by the discussions of Theorem 3.3 and Theorem 3.4, we find the following.
Theorem 3.5
If \(0<(1-a)h<x_{E}<h<K\), \(x_{P_{2}}\geq(1-a)h\) and \(y_{D^{+}}\leq y_{A_{0}}\), and system (2) exists a unique order-1 periodic solution, then the periodic solution in region \(\Omega_{2}\) is attractive, where \(\Omega_{2}=R_{+}-Q_{1}\) and \(Q_{1}\) is an open region enclosed by orbit \(\widehat{P_{0}P_{2}P_{1}}\) (see Figure 4(a)). Meanwhile if \(y_{D^{+}}> y_{A_{0}}\), then for system (2) there exists a unique order-1 periodic solution in region \(\Omega_{2}\) which is attractive.
Theorem 3.6
If (ii) of II in Theorem 3.2 is true, and \(y_{P^{+}_{1}}\leq y_{F_{2}}\) and system (2) exists a unique order-1 periodic solution, then the periodic solution in region \(\Omega_{3}\) is attractive, where \(\Omega_{3}=R_{+}-Q_{2}\) and \(Q_{2}\) is an open region enclosed by orbit \(\widehat{P_{0}F_{1}F_{2}P_{1}}\) (see Figure 4(b)). And if \(y_{P^{+}_{1}}> y_{F_{2}}\), then for system (2) also there exists a unique order-1 periodic solution in region \(\Omega_{3}\) which is attractive.
3.3 Stability of order-1 periodic solution of system (2)
Theorem 3.7
Proof
Let \(\Phi(x,y)=-ax\), \(\Psi(x,y)=by+c \), \(\eta(x,y)=x-h\).
4 Simulations and conclusion
In order to prevent the extinction of predator under the herd behavior of the prey (such as the drifting herbivores observed in the savanna), this paper presents a prey–predator system with square root response function under state-dependent control strategy. In different cases, we discuss the existence of the order-1 periodic solution by the successor function method. Then we analyze the uniqueness and attractiveness of the periodic solution. Furthermore, we prove order-1 periodic solution is stable under certain conditions. Numerical simulations with an example are carried out which illustrate that the state-dependent impulse control strategy is effective. Compared with the literature [16], our research is more comprehensive, which is an improvement and complement for the results of the above literature.
Declarations
Acknowledgements
The paper was supported by supported by the National Natural Science Foundation of China (No. 11371230).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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