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Dynamics of a predatorprey model with impulsive biological control and unilaterally impulsive diffusion
Advances in Difference Equations volume 2016, Article number: 150 (2016)
Abstract
In this paper, we establish a predatorprey model with impulsive biological control and unilaterally impulsive diffusion. This predatorprey model for two regions, which are connected by diffusion of predator population, portrays the evolution of the population. We study the model for biological pest control in which a pest population is controlled by a program of periodic releases of a fixed yield of predators that prey on the pest population. We prove that there exists a globally asymptotically stable preyextinction boundary periodic solution. The condition for permanence is also obtained. Simulations are also employed to verify our results. We conclude that the impulsive diffusion and releasing predator provide reliable tactic basis for pest management.
Introduction
The warfare between humans and pests has persisted for thousands of years. In the past few decades, man has adopted some advanced and modern weapons for instance chemical pesticides, biological pesticides, remote sensing and measure, computers, atomic energy etc. Some brilliant achievements have been obtained. However, the warfare will never be over. Although a great deal of pesticides were used to control pests, the insect pests impairing crops are increasing because of the resistance to the pesticide. With pesticides employed, the residual pests breed a large number of pests with resistance to pesticides. So the pesticide is invalid in a sense. Moreover, insect pests will remain. On the other hand, the chemical pesticides kill not only pests but also their natural enemies. Therefore, insect pests are rampant again. Then the effect of chemical control was challenged. Furthermore, the practice proves that longterm adopting chemical control may give rise to disastrous results, for example, we witness environmental contamination and toxicosis of man and animal, and so on.
The use of a natural enemy to suppress pests is one of the most important approaches in pest control. Biological control [1–8] is one of the reductions in pest populations from the actions of other living organisms, often called natural enemies or beneficial species. It is the purposeful introduction and establishment of one or more natural enemies from the region of origin of an exotic pest, specifically for the purpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economic damage. Jiao et al. [9] analyzed the dynamics of a stagestructured Holling mass defence predatorprey model with impulsive perturbations on predators,
where \(x_{1}(t)\), \(x_{2}(t)\) represent the immature and mature pest densities, respectively. \(x_{3}(t)\) denotes the density of nature enemy. The biological meanings of the parameters can be found in [9].
The dispersal is a ubiquitous phenomenon in the natural world. It is important for us to understand the ecological and evolutionary dynamics of populations mirrored by the large number of mathematical models devoted to it in the scientific literature [10–13]. In recent years, the analysis of these models focused on the coexistence of populations and the local (or global) stability of the equilibria [14–20]. Spatial factors play a fundamental role on the persistence and stability of the population, although the complete results have not yet been obtained even in the simplest onespecies case. If the population dynamics with the effects of spatial heterogeneity is modeled by a diffusion process, most previous papers focused on the population dynamical system modeled by the ordinary differential equations. But in practice, it is often the case that diffusion occurs in a regular pulse. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs at a fixed period of time every year. Thus impulsive diffusion provides a more natural description. Lately theories of impulsive differential equations [21] have been introduced into population dynamics. Jiao et al. [22] proposed and investigated the dynamical behaviors of a stagestructured predatorprey model with prey impulsively diffusing between two patches
where we suppose that the system is composed of two patches connected by diffusion and occupied by a single species \(x_{i}\) (\(i = 1, 2\)) is the density of species in the ith patch. \(y_{1}(t)\), \(y_{2}(t)\) represent the densities of the immature individual predator and mature individual predator at time t in the second patch. The biological meanings of the parameters can be found in [22].
Theories of impulsive differential equations have been introduced into population dynamics lately [23–28]. Impulsive equations are found in almost every domain of applied science and have been studied in many investigations [28–33], they generally describe phenomena which are subject to steep or instantaneous changes. The theories of population dynamical systems and their applications have achieved many good results. However, the oasis vegetation degradation combined with a dynamical system has been considered very little. In this paper, we will investigate an impulsive dispersal on SIR model on restricting infected individuals boarding transports. We expect to obtain some dynamical properties of the investigated system. We also expect that impulsive dispersal will provide a reliable tactic for controlling epidemic.
The organization of this paper is as follows. In the next section, we introduce the model and background concepts. In Section 3, some important lemmas are presented. We give the globally asymptotically stable conditions of the preyextinction boundary periodic solution of System (2.1), and the permanent condition of System (2.1). In Section 4 is a simulation analysis, and a brief discussion are given in the last section to conclude this work.
The model
In this paper, we establish a predatorprey model with impulsive biological control and unilaterally impulsive diffusion.
where we suppose that the system is composed of two patches connected by diffusion. These two patches are separated by rivers or highways or railways. The predator population can transverse the rivers or highways or railways, while the prey population cannot. \(x_{1}(t)\) and \(y_{1}(t)\) represent the numbers of preys and predators in the populations in patch 1 at time t. \(y_{2}(t)\) represents the number of predators in the population in patch 2 at time t. \(a_{1}>0\) represents the intrinsic growth rate of the prey population in patch 1. \(b_{1}>0\) represents the coefficient of the intraspecific competition of the prey population in patch 1. \(a_{2}>0\) represents the intrinsic growth rate of the predator population in patch 2. \(b_{2}>0\) represents the coefficient of the intraspecific competition of the predator population in patch 2. The predator consumes the prey according to \(\beta_{1}x_{1}(t)\), in patch 1 at time t. \(k_{1}\) is the rate of conversion of nutrients into the reproduction of the predator in patch 1. \(d_{1}\) represents the death rate of the predator population in patch 1. The pulse diffusion occurs every τ period (τ is a positive constant), the system evolves from its initial state without being further affected by diffusion until the next pulse appears; \(\triangle y_{i}((n+l)\tau) = y_{i}((n+l)\tau^{+})y_{i}((n+l)\tau)\) where \(y_{i}((n+l)\tau^{+})\) represents the density of population in the ith patch immediately after the nth diffusion pulse at time \(t = (n+l)\tau\), while \(y_{i}((n+l)\tau)\) represents the density of population in the ith patch before the nth diffusion pulse at time \(t = (n+l)\tau\), \(0< l<1\), \(n\in Z_{+}\). \(0< D<1\) represents the diffusive rate between two patches. \(\triangle y_{1}((n+1)\tau)=y_{1}((n+1)\tau^{+})y_{1}((n+1)\tau)\) and μ represent the releasing amount of predator population at \(t=(n+1)\tau\), \(n\in Z_{+}\), in patch 1.
The lemmas
The solution of (2.1), denoted by \(X(t)=(x_{1}(t),y_{1}(t),y_{2}(t))^{T}\), is a piecewise continuous function \(X : R_{+}\rightarrow R_{+}^{3}\), \(X(t)\) is continuous on \((n\tau,(n+l)\tau]\) and \(((n+l)\tau,(n+1)\tau ]\), \(n\in Z_{+}\) and \(X(n\tau^{+})=\lim_{t\rightarrow n\tau^{+} }X(t)\), \(X((n+l)\tau ^{+})=\lim_{t\rightarrow(n+l)\tau^{+} }X(t)\) exist. Obviously the global existence and uniqueness of solutions of (2.1) is guaranteed by the smoothness properties of f, which denotes the mapping defined by right side of System (2.1) [21].
Let \(V:R_{+}\times R^{3}_{+}\rightarrow R_{+}\), then V is said to belong to class \(V_{0}\), if:

(i)
V is continuous in \((n\tau, (n+l)\tau]\times R_{+}^{3}\) and \(((n+l)\tau, (n+1)\tau]\times R_{+}^{3}\), for each \(z\in R^{3}_{+}\), \(n\in Z_{+}\), \(V(n\tau^{+},z)=\lim_{(t,y)\rightarrow(n\tau^{+},z) }V(t,y)\), \(V((n+l)\tau^{+},z) =\lim_{(t,y)\rightarrow((n+l)\tau^{+},y) }V(t,y)\) exist.

(ii)
V is locally Lipschitzian in z.
Definition 3.1
\(V\in V_{0}\), then, for \((t,z)\in(n\tau, (n+l)\tau]\times R_{+}^{3}\) and \(((n+l)\tau, (n+1)\tau]\times R_{+}^{3}\), the upper right derivative of \(V(t,z)\) with respect to the impulsive differential System (2.1) is defined as
Since \(\frac{dx_{i}(t)}{dt}=0\), when \(x_{i}(t)=0\); \(\frac {dy_{i}(t)}{dt}=0\), when \(y_{i}(t)=0\), and \(\triangle y_{1}(t)=\mu>0\), when \(t=(n+1)\tau\), we can easily obtain the following lemma.
Lemma 3.2
Suppose \(X(t)\) is a solution of (2.1) with \(X(0^{+})\geq0\), then \(X(t)\geq0\) for \(t\geq0\). Furthermore, \(X(t)> 0\) (\(t\geq0\)) for \(X(0^{+})> 0\).
Lemma 3.3
[21]
Let the function \(m\in PC'[R^{+},R]\) satisfy the inequalities
where \(p,q\in PC[R^{+},R]\) and \(d_{k}\geq0\), \(b_{k}\) are constants; then
Now, we show that all solutions of (2.1) are uniformly ultimately bounded.
Lemma 3.4
There exists a constant \(M>0\) such that \(x_{1}(t)\leq M\), \(y_{1}(t)\leq M\), \(y_{2}(t)\leq M \) for each solution \((x_{1}(t),y_{1}(t),y_{2}(t))\) of (2.1) with all t large enough.
Proof
Define
then \(t\neq n\tau\), \(t\neq(n+l)\tau\), we have
When \(t=(n+l)\tau\), \(V((n+l)\tau^{+})= k_{1}x_{1}((n+l)\tau^{+})+y_{1}((n+l)\tau ^{+})+y_{2}((n+l)\tau^{+})=k_{1}x_{1}((n+l)\tau)+y_{1}((n+l)\tau )+Dy_{2}((n+l)\tau) +(1D)y_{2}((n+l)\tau)=V((n+l)\tau)\). When \(t=(n+1)\tau\), \(V((n+1)\tau^{+})= k_{1}x_{1}((n+1)\tau^{+})+y_{1}((n+1)\tau ^{+})+y_{2}((n+1)\tau^{+}) =k_{1}x_{1}((n+1)\tau)+y_{1}((n+1)\tau)+\mu+y_{2}((n+1)\tau)= V((n+1)\tau)+\mu\). By Lemma 3.3, for \(t\in(n\tau,(n+1)\tau]\), we have
So \(V(t)\) is uniformly ultimately bounded. Hence, by the definition of \(V(t)\), we see that there exists a constant \(M>0\) such that \(x_{1}(t)\leq M\), \(y_{1}(t)\leq M\), \(y_{2}(t)\leq M \) for t large enough. The proof is complete. □
If \(x_{1}(t)=0\), we have the following subsystem of (2.1):
We can easily obtain the analytic solution of (3.2) between pulses as follows:
Considering the third and fourth equations of (3.2), we have
Considering the fifth and sixth equations of (3.2), we also have
Substituting (3.4) into (3.5), we have the stroboscopic map of (3.2),
Choosing notations \(A=e^{d_{1}\tau}<1\), \(B_{1}=Da_{2}e^{[a_{2}ld1(1l)]\tau}>0\), \(B_{2}=(1D)a_{2}e^{a_{2}\tau}>0\), \(C_{1}=b_{2}(e^{al\tau}1)>0\), \(C_{2}=b_{2}(e^{al\tau }1)[1+(1D)e^{a_{2}l\tau}]>0\), and
we can rewrite (3.6) as
Equation (3.7) has two fixed points \((y_{1}^{\ast},0)\) and \((y_{1}^{\ast\ast},y_{2}^{\ast\ast})\), where
Lemma 3.5

(i)
If \(1e^{a_{2}\tau}< D<1\) holds, the fixed point \((y_{1}^{\ast}, 0)\) of (3.6) or (3.7) is globally asymptotically stable.

(ii)
If \(0< D<1e^{a_{2}\tau}\) holds, the fixed point \((y_{1}^{\ast\ast}, y_{2}^{\ast\ast})\) of (3.6) or (3.7) is globally asymptotically stable.
Proof
For convenience, we take the notation \((y^{n}_{1},y^{n}_{2})=(y_{1}(n\tau^{+}),y_{2}(n\tau^{+}))\). The linear form of (3.5) can be written as
Obviously, the near dynamics of \((y_{1}^{\ast}, 0)\) and \((y_{1}^{\ast \ast}, y_{2}^{\ast\ast})\) are determined by the linear System (3.6) or (3.7). The stabilities of \((y_{1}^{\ast}, 0)\) and \((y_{1}^{\ast\ast}, y_{2}^{\ast\ast})\) are determined by the eigenvalue of M less than 1. If M satisfies the Jury criterion [34], we can know the eigenvalue of M less than 1,

(i)
If \(1e^{a_{2}\tau}< D<1\) holds, that is, \(\frac{B_{2}}{a_{2}}<1\), we can easily see that \((y_{1}^{\ast},0)\) is a unique fixed point of (3.6) or (3.7), and
$$ \left . \textstyle\begin{array}{@{}l@{}} M= \left( \textstyle\begin{array}{@{}c@{\quad}c@{}} A&\frac{B_{1}}{a_{2}} \\ 0&\frac{B_{2}}{a_{2}} \end{array}\displaystyle \right). \end{array}\displaystyle \right . $$(3.11)For
$$\begin{aligned} 1 \operatorname {tr}M+ \det M&=1\biggl(A+\frac{B_{2}}{a_{2}}\biggr)+\biggl(A \times \frac{B_{2}}{a_{2}}\biggr) \\ &=(1A) \biggl(1\frac{B_{2}}{a_{2}}\biggr)>0. \end{aligned}$$From the Jury criterion, \((y_{1}^{\ast}, 0)\) is locally stable, then it is globally asymptotically stable.

(ii)
If \(0< D<1e^{a_{2}\tau}\) holds, that is, \(\frac {a_{2}}{B_{2}}<1\), we can easily see that \((y_{1}^{\ast\ast},y_{2}^{\ast \ast})\) is a unique fixed point of (3.6) or (3.7), and
$$ \left . \textstyle\begin{array}{@{}l@{}} M= \left( \textstyle\begin{array}{@{}c@{\quad}c@{}} A&\frac{B_{1}a_{2}}{(a_{2}+C_{1}y_{2}^{\ast\ast})^{2}} \\ 0&\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}} \end{array}\displaystyle \right). \end{array}\displaystyle \right . $$(3.12)For
$$\begin{aligned} 1 \operatorname {tr}M+ \det M &=1\biggl(A+\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast })^{2}}\biggr)+\biggl(A \times \frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}}\biggr) \\ &=(1A)\biggl[1\frac{B_{2}a_{2}}{(a_{2}+C_{2}y_{2}^{\ast\ast})^{2}}\biggr] \\ &=(1A) \biggl(1\frac{a_{2}}{B_{2}}\biggr)>0. \end{aligned}$$From the Jury criterion, \((y_{1}^{\ast\ast},y_{2}^{\ast\ast})\) is locally stable, then it is globally asymptotically stable. This completes the proof.
□
Lemma 3.6

(i)
If \(0< D<1e^{a_{2}\tau}\) holds, the periodic solution \((\widetilde{y_{1}(t)},\widetilde{y_{2}(t)} )\) of System (3.2) is globally asymptotically stable, where
$$ \left \{ \textstyle\begin{array}{@{}l@{}} \left . \textstyle\begin{array}{@{}l@{}} \widetilde{y_{1}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} y_{1}^{\ast\ast}e^{d_{1}(tn\tau)}, & t\in[n\tau,(n+l)\tau), \\ y_{1}^{\ast\ast\ast}e^{d_{1}(t(n+l)\tau)}, &t\in[(n+l)\tau,(n+1)\tau), \end{array}\displaystyle \right . \\ \left . \textstyle\begin{array}{@{}l@{}} \widetilde{ y_{2}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} \frac{a_{2}y_{2}^{\ast\ast}e^{a_{2}(tn\tau)}}{a_{2}+b_{2}y_{2}^{\ast \ast}(e^{a_{2}(tn\tau)}1)}, & t\in[n\tau,(n+l)\tau), \\ \frac{a_{2}y_{2}^{\ast\ast\ast}e^{a_{2}(t(n+l)\tau )}}{a_{2}+b_{2}y_{2}^{\ast\ast\ast}(e^{a_{2}(t(n+l)\tau)}1)}, &t\in[(n+l)\tau,(n+1)\tau), \end{array}\displaystyle \right . \end{array}\displaystyle \right . $$(3.13)where \(y^{\ast\ast}_{1}\) and \(y^{\ast\ast}_{2}\) are determined as (3.8), \(y_{1}^{\ast\ast\ast}\) and \(y_{2}^{\ast\ast\ast}\) are defined as
$$ \left \{ \textstyle\begin{array}{@{}l@{}} y_{1}^{\ast\ast\ast}=e^{d_{1}l\tau}y_{1}^{\ast\ast}+D\times\frac {a_{2}y_{2}^{\ast\ast}e^{a_{2}l\tau}}{a_{2}+b_{2}y_{2}^{\ast\ast }(e^{a_{2}l\tau}1)}, \\ y_{2}^{\ast\ast\ast}=(1D)\times\frac{a_{2}y_{2}^{\ast\ast}e^{a_{2}l\tau }}{a_{2}+b_{2}y_{2}^{\ast\ast}(e^{a_{2}l\tau}1)}. \end{array}\displaystyle \right . $$(3.14) 
(ii)
If \(1e^{a_{2}\tau}< D<1\) holds, the periodic solution \((\widehat{y_{1}(t)},0 )\) of System (3.2) is globally asymptotically stable, where
$$ \left . \textstyle\begin{array}{@{}l@{}} \left . \textstyle\begin{array}{@{}l@{}} \widehat{y_{1}(t)}= \end{array}\displaystyle \right . \left\{ \textstyle\begin{array}{@{}l@{\quad}l@{}} y_{1}^{\ast}e^{d_{1}(tn\tau)},& t\in[n\tau,(n+l)\tau), \\ y_{1}^{\ast\ast\ast\ast}e^{d_{1}(t(n+l)\tau)},& t\in[(n+l)\tau ,(n+1)\tau), \end{array}\displaystyle \right . \end{array}\displaystyle \right . $$(3.15)where \(y_{1}^{\ast\ast\ast\ast}=e^{d_{1}\tau}y^{\ast}_{1}\), and \(y^{\ast}_{1}\) is determined as (3.8).
Therefore, if \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(d_{1}=0.6\), \(a_{2}=0.1\), \(b_{2}=0.2\), \(\mu=0.4\), \(l=0.25\), \({\tau=1}\), \(D=0.15\), then \(0.15=D>D^{\ast} =1e^{a_{2}\tau }\doteq0.0952\); the predator population \(y_{2}(t)\) will go into extinction. Its dynamical behaviors can be seen in Figure 1. If \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(d_{1}=0.6\), \(a_{2}=0.1\), \(b_{2}=0.2\), \(\mu=0.4\), \(l=0.25\), \(\tau=1\), \(D=0.05\), then \(0.05=D< D^{\ast} \doteq0.0952\); the predator population are permanent. Its dynamical behaviors can be seen in Figure 2. Obviously, the diffusion rate between the patches affects the dynamics of System (3.2).
The dynamics
Theorem 4.1
If
and
hold, the preyextinction boundary periodic solution \((0,\widehat {y_{1}(t)},0)\) of (2.1) is globally asymptotically stable, where \(y^{\ast}_{1}\) and \(y^{\ast\ast\ast\ast}_{1}\) are defined as (3.8) and (3.15).
Proof
First, we prove the local stability of the preyextinction boundary periodic solution \((0,\widehat{y_{1}(t)},0)\) of (2.1). Defining \(x_{1}(t)=x_{1}(t)\), \(y_{11}(t)=y_{1}(t)\widehat{y_{1}(t)}\), \(y_{2}(t)=y_{2}(t)\), then we have the following linearly similar system for (2.1) which concerns one periodic solution \((0,\widehat{y_{1}(t)},0)\):
It is easy to obtain the fundamental solution matrix
The linearization of the fourth, fifth, and sixth equations of (2.1) is
The linearization of the seventh, eighth, and ninth equations of (2.1) is
The stability of the periodic solution \((0,\widehat{y_{1}(t)},0)\) is determined by the eigenvalues of
where
and
According to condition (4.1), (4.2), and the Floquet theory [21], i.e.
and
we have
and
the preyextinction boundary periodic solution \((0,\widehat {y_{1}(t)},0)\) of (2.1) is locally stable.
In the following, we will prove the global attraction. From condition (4.2), we can choose an \(\varepsilon>0\) such that
From the second equation of (2.1), we notice that \(\frac{dy_{1}(t)}{dt}\geqd_{1}y_{1}(t)\). Then we consider the following impulsive comparative differential equation:
From Lemma 3.6 and the comparison theorem of impulsive equations (see Theorem 3.1.1 in [29]), we have \(y_{1}(t)\geq y_{21}(t)\), \(y_{2}(t)\geq y_{22}(t)\), and \(y_{21}(t)\rightarrow\widehat{y_{1}(s)}\), \(y_{22}(t)\rightarrow0\), as \(t\rightarrow\infty\). Then
for all t large enough. For convenience, we may assume (4.4) holds for all \(t\geq0\). From (2.1) and (4.4), we get
So \(x_{1}((n+1)\tau)\leq x_{1}(n\tau^{+})\exp[\int^{(n+1)\tau}_{n\tau}(a_{1}\beta_{1}(\widehat{y_{1}(s)} \varepsilon))\,ds]\). Hence, \(x_{1}(n\tau)\leq x_{1}(0^{+})\rho^{n}\) and \(x_{1}(n\tau )\rightarrow0\) as \(n\rightarrow\infty\), therefore \(x_{1}(t)\rightarrow0\) (\(i=1,2\)) as \(t\rightarrow\infty\).
Next, we will prove that \(y_{1}(t)\rightarrow\widehat{y_{1}(t)}\) and \(y_{2}(t)\rightarrow0\) as \(t\rightarrow\infty\). For \(0<\varepsilon_{1}<\frac{d_{1}}{k_{1}\beta _{1}}\) small enough, there must exist a \(t_{0}>0\) such that \(0< x_{1}(t)<\varepsilon_{1} \) for all \(t\geq t_{0}\). Without loss of generality, we may assume that \(0< x_{1}(t)<\varepsilon _{1} \) for all \(t\geq{0}\). For System (2.1) we have
then we have \(y_{21}(t)\leq y_{1}(t)\leq y_{31}(t)\), \(y_{22}(t)\leq y_{2}(t)\leq y_{32}(t)\), and \(y_{21}(t)\rightarrow\widehat{y_{1}(t)}\), \(\widehat {y_{22}(t)}\rightarrow0\), \(y_{31}(t)\rightarrow\widehat{y_{31}(t)}\), \(\widehat{y_{32}(t)}\rightarrow 0\), as \(t\rightarrow\infty\). Meanwhile \((y_{21}(t),y_{22}(t))\) and \((y_{31}(t),y_{32}(t))\) are the solutions of (4.3) and
respectively. We have
where \(y_{31}^{\ast\ast\ast\ast}=e^{(d_{1}k_{1}\beta_{1}\varepsilon _{1})\tau}y^{\ast}_{31}\), and \(y^{\ast}_{31}\) is determined as
For any \(\varepsilon_{2}>0\), there exists a \(t_{1}\), \(t>t_{1}\) such that
and
Let \(\varepsilon_{1}\rightarrow0\), so we have
and
for t large enough, which implies \(y_{1}(t)\rightarrow \widehat{y_{1}(t)}\) and \(y_{2}(t)\rightarrow 0\) as \(t\rightarrow\infty\). This completes the proof. □
We can easily prove Theorem 4.2 similar to Theorem 4.1.
Theorem 4.2
If
and
hold, the preyextinction boundary periodic solution \((0,\widetilde {y_{1}(t)},\widetilde{y_{2}(t)})\) of (2.1) is globally asymptotically stable, where \(y^{\ast\ast}_{1}\), \(y^{\ast\ast\ast}_{1}\), \(y^{\ast\ast}_{2}\), and \(y^{\ast\ast\ast}_{2}\) are defined as (3.8) and (3.14).
The next task is to investigate the permanence of System (2.1).
Definition 4.3
System (2.1) is said to be permanent if there are constants \({m,M >0} \) (independent of the initial value) and a finite time \(T_{0}\) such that for all solutions \((x_{1}(t), y_{1}(t), y_{2}(t))\) with all initial values \(x_{1}(0^{+})>0\), \(y_{1}(0^{+})>0\), \(y_{2}(0^{+})>0\), \(m\leq x_{1}(t)\leq M\), \(m\leq y_{1}(t)\leq M\), \(m\leq y_{2}(t)\leq M\) hold for all \(t\geq T_{0}\). Here \(T_{0}\) may depend on the initial values \((x_{1}(0^{+}),y_{1}(0^{+}), y_{2}(0^{+}))\).
Theorem 4.4
If
and
hold, System (2.1) is permanent, where \(y^{\ast\ast}_{1}\) and \(y^{\ast \ast\ast}_{1}\) are defined as (3.8) and (3.14).
Proof
Suppose \((x_{1}(t), y_{1}(t), y_{2}(t))\) is a solution of (2.1) with \(x_{1}(0)>0\), \(y_{1}(0)>0\), \(y_{2}(0)>0\). By Lemma 3.4, we have proved there exists a constant \(M >0\) such that \(x_{1}(t)\leq M\), \(y_{1}(t)\leq M\), \(y_{2}(t)\leq M\), for t large enough. From (2.1), Theorem 4.1, and condition (4.11), we have
for \(\varepsilon_{2}\) small enough. So we only need to find \(m_{3}>0\) and \(\varepsilon_{3}\) such that \(x_{1}(t)> m_{3}\) for t large enough. Otherwise, we can select \(m_{4}>0\) small enough satisfying \(m_{4}<\frac{d_{1}}{k_{1}\beta_{1}}\), and we prove \(x_{1}(t)< m_{4}\) cannot hold for \(t\geq 0\). By condition (4.12) and choosing \(\varepsilon_{3}\) small enough, we can obtain
with \(y_{41}^{\ast\ast}\) and \(y_{41}^{\ast\ast\ast}\) defined as (4.17) and (4.18). Then
By Lemma 3.6, we have \(y_{1}(t)\leq y_{41}(t)\), \(y_{2}(t)\leq y_{42}(t)\) and \(y_{41}(t)\rightarrow\overline{y_{41}(t)}\), \(y_{42}(t)\rightarrow \overline{y_{42}(t)}\), \(t\rightarrow\infty\), where \((y_{41}(t),y_{42}(t))\) is the solution of
with
where \(y^{\ast\ast}_{41}\) and \(y^{\ast\ast}_{42}\) are determined as
with \(A_{4}=e^{(d_{1}k_{1}\beta_{1}m_{4})\tau}<1\), \(B_{41}=Da_{2}e^{a_{2}l\tau}>0\), \(B_{42}=(1D)a_{2}e^{a_{2}\tau}>0\), \(C_{41}=b_{2}(e^{al\tau}1)>0\), \(C_{42}=b_{2}(e^{al\tau }1)[1+(1D)e^{a_{2}l\tau}]>0\), and \(y_{1}^{\ast\ast\ast}\) and \(y_{2}^{\ast\ast\ast}\) are defined as
Therefore, there exist \(T_{1}>0\) and \(\varepsilon_{3}>0\) such that
and
Then
for \(t\geq T_{1}\), let \(N_{1}\in N\) and \(N_{1}\tau> T_{1}\). Integrating (4.19) on \((n\tau,(n+1)\tau)\), \(n\geq N_{1}\), we have
then \(x_{1}((N_{1}+k)\tau)\geq x_{1}(N_{1}\tau^{+})e^{k\sigma }\rightarrow\infty\), as \(k\rightarrow\infty\), which is a contradiction to the boundedness of \(x_{1}(t)\). Hence, there exists a \(t_{1}>0\) such that \(x_{1}(t)\geq m_{3}\). This completes the proof. □
Discussion
In this paper, we establish a predatorprey model with impulsive diffusion and releasing on predator population. This predatorprey model for two regions, which are connected by diffusion of the predator population, portrays the evolution of the population. We prove that all solutions of the investigated system are uniformly ultimately bounded. From Theorem 4.1, the preyextinction periodic solution \((0,\widehat{y_{1}(t)},0)\) of System (2.1) is globally asymptotically stable. From Theorem 4.2, the preyextinction periodic solution \((0,\widetilde{y_{1}(t)},\widetilde {y_{2}(t)})\) of System (2.1) is globally asymptotically stable. From Theorem 4.4, System (2.1) is permanent. It is assumed that \(x_{1}(0)=0.5\), \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(a_{1}=2.4\), \(b_{1}=0.1\), \(a_{2}=1.5\), \(b_{2}=0.21\), \(\beta_{1}=0.3\), \(k_{1}=0.5\), \(\mu=0.86\), \(d_{1}=0.1\), \(\tau=1\), \(l=0.25\), \(D=0.78\); the conditions (4.1) and (4.2) are obviously satisfied, then the preyextinction periodic solution \((0,\widehat{y_{1}(t)},0)\) of System (2.1) is globally asymptotically stable (see Figure 3). It is assumed that \(x_{1}(0)=0.5\), \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(a_{1}=2.4\), \(b_{1}=0.1\), \(a_{2}=1.5\), \(b_{2}=0.21\), \(\beta_{1}=0.3\), \(k_{1}=0.5\), \(\mu=0.86\), \(d_{1}=0.1\), \(\tau=1\), \(l=0.25\), \(D=0.6\); the conditions (4.9) and (4.10) are obviously satisfied, then the preyextinction periodic solution \((0, \widetilde{y_{1}(t)},\widetilde {y_{2}(t)})\) of System (2.1) is globally asymptotically stable (see Figure 4). It is also assumed that \(x_{1}(0)=0.5\), \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(a_{1}=4\), \(b_{1}=0.1\), \(a_{2}=4\), \(b_{2}=0.21\), \(\beta_{1}=0.3\), \(k_{1}=0.1\), \(\mu=0.01\), \(d_{1}=0.1\), \(\tau=1\), \(l=0.25\), \(D=0.01\); the conditions (4.11) and (4.12) are obviously satisfied, then System (2.1) is permanent (see Figure 5). It is also assumed that \(x_{1}(0)=0.5\), \(y_{1}(0)=0.5\), \(y_{2}(0)=0.5\), \(a_{1}=2\), \(b_{1}=0.1\), \(a_{2}=1.5\), \(b_{2}=0.21\), \(\beta_{1}=0.3\), \(k_{1}=0.5\), \(\mu=0.1\), \(d_{1}=0.1\), \(\tau=1\), \(l=0.25\), \(D=0.99\); predator (\(y_{2}(t)\)) will go into extinction and \(x_{1}(t)y_{1}(t)\) will be permanent in System (2.1) (see Figure 6).
From the simulations, we can guess that there exist three controlling thresholds with D. It is always assumed that \(0< D^{\ast}< D^{\ast\ast }< D^{\ast\ast\ast}<1\). If \(0< D< D^{\ast}\) holds, System (2.1) is permanent. If \(D^{\ast }< D< D^{\ast\ast}\) holds, the preyextinction periodic solution \((0,\widetilde{y_{1}(t)},\widetilde {y_{2}(t)})\) of System (2.1) is globally asymptotically stable. If \(D^{\ast\ast}< D< D^{\ast\ast\ast}\) holds, the preyextinction periodic solution \((0,\widehat{y_{1}(t)},0)\) of System (2.1) is globally asymptotically stable. If \(D^{\ast\ast\ast}< D<1\) holds, the predator \(y_{2}(t)\) will go into extinction, prey \(x_{1}(t)\) and predator \(y_{1}(t)\) will be permanent. We can discuss parameter μ similar to parameter D. We discover that the diffusive rate of the predator population plays an important role in pest management. We conclude that the impulsive diffusion and the released predator provide reliable tactic bases for pest management.
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Acknowledgements
Article is supported by the National Natural Science Foundation of China (11361014, 10961008), and the project of high level creative talents in Guizhou province (No. 20164035).
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JJ carried out the main part of this article, LS corrected the manuscript, SH, LM, and YJ brought forward some suggestions on this article. All authors have read and approved the final manuscript.
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Jiao, J., Cai, S., Li, L. et al. Dynamics of a predatorprey model with impulsive biological control and unilaterally impulsive diffusion. Adv Differ Equ 2016, 150 (2016). https://doi.org/10.1186/s1366201608511
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Keywords
 predatorprey model
 unilaterally impulsive diffusion
 impulsive biological control
 extinction
 permanence