Dynamics of a single population system with impulsively unilateral diffusion and impulsive input toxins in polluted environment
- Shaohong Cai^{1},
- Jianjun Jiao^{1}Email author and
- Limei Li^{2}
https://doi.org/10.1186/s13662-015-0600-x
© Cai et al. 2015
Received: 10 February 2015
Accepted: 10 August 2015
Published: 6 October 2015
Abstract
In this paper, we consider a single population system with impulsively unilateral diffusion and impulsive input toxins in a polluted environment. All solutions of the investigated system are proved to be uniformly bounded. By mathematical analysis methods and the theory of impulsive differential equations, the condition of the globally asymptotically stable population-extinction solution of the investigated system is obtained. The permanent condition of the investigated system is also obtained. Finally, numerical analysis is carried out to illustrate our results. Our results provide a reliable theory basis for exploring biological resource management in a polluted environment.
Keywords
1 Introduction
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 devoting to it in the scientific literature [1–5]. In recent years, the analysis of these models focus on the coexistence of population and local (or global) stability of equilibria [6, 7]. Spatial factors play a fundamental role on the persistence and stability of the population, and the complete results have not yet been obtained even in the simplest one-species case. 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 regular pulses. 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 to occur at fixed periods of time every year, thus impulsive diffusion provides a more natural description. Jiao et al. [8] presented a delayed predator-prey model with impulsive diffusion between two patches. They obtained the permanent condition of the system by the theory of impulsive delay differential equation.
The most threatening problem to society is the change in both terrestrial and aquatic environment caused by the different kinds of stresses (temperature, toxicants/pollutants, etc.) affecting the long term survival of species, human life, and biodiversity of the habitat [9–11]. The presence of a toxicant in the environments decreases the growth rate of species and its carrying capacity. In recent years, some investigations have been carried out to study the effect of toxicant on a single species population [12–15], and a lot of scholars have adopted a mathematical modeling approach to study the influence of environmental pollution on the surviving of a biological population [16, 17]. Most of the previous work assumed that input of toxicant was continuous. The toxicants, however, are often emitted to the environment with regular pulse [18]. A lot of data have indicated that the use of agriculture chemicals may cause potential harm to the health of both human beings and other living beings. If the spraying of agriculture chemicals can be regarded as time pulse discharge, modeling by the continuous input of toxin can be regarded as obsolete and should be replaced by impulsive perturbations. In this case, though the discharge of toxin is transient, the influence of the toxin will last long. Therefore, it is very important how to control the pulse input cycle of toxin to protect the population’s persistent existence.
Theories of impulsive differential equations have been introduced into population dynamics lately [19–24]. Impulsive equations are found in almost every domain of applied science and have been studied in many investigations [19–33], they generally describe phenomena which are subject to steep or instantaneous changes. Especially, Jiao et al. [24] suggested releasing pesticides is combined with transmitting infective pests into an SI model. This may be accomplished by avoiding periods when the infective pests would be exposed or placing the pesticides in a location where the transmitting infective pests would not contact it. So we address an impulsive differential equation modeling the process of releasing infective pests and spraying pesticides at different fixed moments.
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. In Section 4, we give the globally asymptotically stable condition of population-extinction solution of system (2.1) and the population permanent condition of system (2.1). In Section 5, a numerical analysis and a brief discussion are given.
2 The model
- (A1)
the population diffuses periodically from patch 1 to patch 2 for dodging polluted environment in patch 1;
- (A2)
the toxicants are emitted to the environment with regular pulse in patch 1, and patch 2 has no pollution.
3 Preliminary lemmas
Before discussing the main results, we will give some definitions, notations, and lemmas. Denote by \(f=(f_{1},f_{2},f_{3},f_{4})\) the map defined by the right hand of system (2.1). The solution of system (2.1), denoted by \(z(t)=(x(t),y(t),c_{o}(t), c_{e}(t))^{T}\), is a piecewise continuous function z: \(R_{+}\rightarrow R_{+}^{4}\), where \(R_{+}=[0,\infty)\), \(R_{+}^{4}=\{z\in R^{4}:z>0\}\). \(z(t)\) is continuous on \((n\tau, (n+l)\tau]\) and \(((n+l)\tau, (n+1)\tau]\) (\(n \in Z_{+}\), \(0 \leq l \leq1\)). According to [21], the global existence and uniqueness of solutions of system (2.1) is guaranteed by the smoothness properties of f, which denotes the mapping defined by the right-hand side of system (2.1).
- (i)
V is continuous in \((n\tau, (n+l)\tau]\times R^{4}_{+}\) and \(((n+l)\tau, (n+1)\tau]\times R^{4}_{+}\), for each \(z\in R^{4}_{+}\), \(n \in Z_{+}\). \(\lim_{(t, u)\rightarrow((n+l)\tau^{+}, z)}V(t,u)=V((n+l)\tau^{+}, z)\), and \(\lim_{(t,u)\rightarrow ((n+1)\tau^{+}, z)}V(t, u)=V((n+1)\tau^{+}, z)\) exist;
- (ii)
V is locally \(Lipschitzian\) in z.
Definition 3.1
Now, we show that all solutions of system (2.1) are uniformly ultimately bounded.
Lemma 3.2
There exists a constant \(M>0\) such that \(x(t)\leq M\), \(y(t)\leq M\) and \(c_{o}(t)\leq M\), \(c_{e}(t)\leq M\) for each solution \((x(t),y(t),c_{o}(t),c_{e}(t) )\) of system (2.1) with all t large enough.
Proof
So \(V(t)\) is uniformly ultimately bounded. Hence, by the definition of \(V(t)\), there exists a constant \(M>0\) such that \(x(t)\leq M\), \(y(t)\leq M\), \(c_{o}(t)\leq M\), and \(c_{e}(t)\leq M\) for t large enough. The proof is complete. □
Lemma 3.3
[33]
Remark 3.4
From Lemma 3.3, we can obtain \(m_{o}-\varepsilon\leq c_{o}(t)\leq M_{o}+\varepsilon\) and \(m_{e}-\varepsilon\leq c_{e}(t)\leq M_{e}+\varepsilon\) for t large enough, where \(m_{o}=\frac{\mu f(e^{-(g+m)\tau}-e^{-h\tau})}{(h-g-m)(e^{(g+m)\tau}-1)(1-e^{-h\tau})}>0\), \(M_{o}=\frac{\mu f(e^{-(g+m)\tau}-e^{-h\tau})}{(h-g-m)(1-e^{-(g+m)\tau})(1-e^{-h\tau })}+\frac{\mu f}{\vert h-g-m\vert (1-e^{-h\tau})}>0\), \(m_{e}=\frac{\mu e^{-h\tau}}{1-e^{-h\tau}}>0\) and \(M_{e}=\frac{\mu}{1-e^{-h\tau}}>0\) for a sufficiently small \(\varepsilon>0\).
4 The dynamics
In this section, we firstly prove that system (2.1) is permanent. For system (2.1) obviously exists a population-extinction boundary periodic solution \((0,0,\widetilde {c_{o}(t)},\widetilde{c_{e}(t)})\). Then we prove that the population-extinction boundary periodic solution \((0,0,\widetilde{c_{o}(t)}, \widetilde{c_{e}(t)})\) of system (2.1) is globally asymptotically stable.
4.1 The permanence of (2.1)
Theorem 4.1
Proof
Similarly to the methods in [32], the following lemma can easily be proved.
Theorem 4.2
- (i)Ifthe triviality periodic solution \((0,0)\) of system (4.3) is globally asymptotically stable.$$(1-D)ae^{(a-\beta M_{o})\tau}< a-\beta M_{o}, $$
- (ii)Ifthe periodic solution \((\widetilde{x_{1}(t)},\widetilde{y_{1}(t)} )\) of system (4.3) is globally asymptotically stable, where$$(1-D)ae^{(a-\beta M_{o})\tau}>a-\beta M_{o}, $$where \(x_{1}^{\ast}\) and \(y_{1}^{\ast}\) are determined as in (4.8).$$ \textstyle\begin{cases} \widetilde{x_{1}(t)}=\frac{ae^{(a-\beta(M_{o}+\varepsilon))(t-(n+l)\tau )}x_{1}^{\ast}}{(a-\beta(M_{o}+\varepsilon)) +b[e^{(a-\beta(M_{o}+\varepsilon))(t-(n+l)\tau)}-1]x_{1}^{\ast}}, &(n+l)\tau< t\leq(n+l+1)\tau,\\ \widetilde{ y_{1}(t)}=y_{1}^{\ast}e^{-d(t-(n+l)\tau)}, &(n+l)\tau< t\leq(n+l+1)\tau, \end{cases} $$(4.13)
From Theorem 4.2, we can obtain the following.
Remark 4.3
If \((1-D)ae^{(a-\beta M_{o})\tau}>a-\beta M_{o}\), for any sufficiently small \(\varepsilon_{1}>0\), there exists a \(T_{1}\) such that \(x_{1}(t)\geq \widetilde{x_{1}(t)}-\varepsilon_{1}\) and \(y_{1}(t)\geq \widetilde{y_{1}(t)}-\varepsilon_{1}\) for \(t>T_{1}\).
Theorem 4.4
- (i)Ifthe fixed point \(G'_{1}(0,0)\) of (4.15) is globally asymptotically stable.$$(1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}< a-\beta(m_{o}-\varepsilon), $$
- (ii)Ifthe fixed point \(G'_{2}(x_{2}^{\ast}, y_{2}^{\ast})\) of (4.15) is globally asymptotically stable.$$(1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}>a-\beta(m_{o}-\varepsilon), $$
Theorem 4.5
- (i)
If \((1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}< a-\beta(m_{o}-\varepsilon )\), the triviality periodic solution \((0,0)\) of system (4.4) is globally asymptotically stable.
- (ii)If \((1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}>a-\beta (m_{o}-\varepsilon)\) the periodic solution \((\widetilde{x_{2}(t)},\widetilde{y_{2}(t)} )\) of system (4.4) is globally asymptotically stable, whereHere \(x_{2}^{\ast}\) and \(y_{2}^{\ast}\) are determined as in (4.16).$$ \textstyle\begin{cases} \widetilde{x_{2}(t)}=\frac{ae^{(a-\beta(m_{o}-\varepsilon))(t-(n+l)\tau )}x_{2}^{\ast}}{(a-\beta(m_{o}-\varepsilon)) +b[e^{(a-\beta(m_{o}-\varepsilon))(t-(n+l)\tau)}-1]x_{2}^{\ast}},& (n+l)\tau< t\leq(n+l+1)\tau,\\ \widetilde{y_{2}(t)}=y_{2}^{\ast}e^{-d(t-(n+l)\tau)}, &(n+l)\tau< t\leq(n+l+1)\tau. \end{cases} $$(4.17)
Remark 4.6
If \((1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}< a-\beta (m_{o}-\varepsilon)\), for any sufficiently small \(\varepsilon_{2}>0\), there exists a \(T_{2}\) such that \(x_{2}(t)\leq \varepsilon_{2}\) and \(y_{2}(t)\leq \varepsilon_{2}\) for \(t>T_{2}\).
From Theorem 4.2, Remark 4.3, Theorem 4.5, and Remark 4.6, we present an important theorem in this paper.
Theorem 4.7
Proof
(i) According to the impulsive comparative theorem [18] and the condition \((1-D)ae^{(a-\beta m_{o})\tau}< a-\beta m_{o}\), there exists a sufficiently small \(\varepsilon>0\) such that \((1-D)ae^{(a-\beta(m_{o}-\varepsilon))\tau}< a-\beta(m_{o}-\varepsilon )\), and, from Remark 4.6, for any sufficiently small \(\varepsilon_{2}>0\), there exists a \(T_{2}>0\) such that \(x(t)\leq x_{2}(t)\leq\varepsilon_{2}\) and \(y(t)\leq y_{2}(t)\leq \varepsilon_{2}\) for \(t>T_{2}\). That is to say, for any sufficiently small \(\varepsilon_{2}>0\), there exists a \(T_{2}>0\) such that \(x(t)\leq\varepsilon_{2}\) and \(y(t)\leq\varepsilon_{2}\) for \(t>T_{2}\). These show that the population of system (2.1) will go extinct.
(ii) According to the impulsive comparative theorem [18] and the condition \((1-D)ae^{(a-\beta M_{o})\tau}>a-\beta M_{o}\), there exists a sufficiently small \(\varepsilon>0\) such that \((1-D)ae^{(a-\beta(M_{o}+\varepsilon))\tau}>a-\beta(M_{o}+\varepsilon )\), and from Remark 4.3, for any sufficiently small \(\varepsilon_{1}>0\), there exists a \(T_{1}>0\) such that \(x(t)\geq x_{1}(t)\geq\widetilde{x_{1}(t)}-\varepsilon_{1}\geq x_{1}^{\ast}e^{-(d_{1}+M_{o}+\varepsilon)\tau}-\varepsilon_{1}\stackrel {\Delta}{=}m_{1}\) and \(y(t)\geq y_{1}(t)\geq\widetilde{y_{1}(t)}-\varepsilon_{1}\geq y_{1}^{\ast}e^{-d_{2}\tau}-\varepsilon_{1}\stackrel{\Delta}{=}m_{2}\) for \(t>T_{1}\), where \(x^{\ast}_{1}\) and \(y^{\ast}_{1}\) are determined as (4.8). From Lemma 3.2, there exist \(M>0\) and \(T>0\) such that \(x(t)< M\) and \(y(t)< M\) for \(t>T\). From the above discussion, we know \(m_{1}< x(t)< M\) and \(m_{2}< y(t)< M\) for \(t>\max\{T,T_{1}\}\). That is to say, the population of system (2.1) is permanent. □
From Lemma 3.2, Remark 3.4, and Theorem 4.7, we have the following.
Theorem 4.8
4.2 The globally asymptotical stability of population-extinction boundary periodic solution of (2.1)
Theorem 4.9
Proof
5 Discussion
In this work, we consider a single population system with impulsively unilateral diffusion and impulsive input toxins in a polluted environment. We prove that all solutions of system (2.1) are uniformly ultimately bounded. The condition of the globally asymptotically stable population-extinction solution of system (2.1) is obtained, and the condition of the population permanence of system (2.1) is also obtained.
5.1 The simulation of system (2.1) affected by parameter μ
From the simulation experiments, the parameters μ obviously affect the dynamical behaviors of system (2.1). If all parameters of system (2.1) are fixed, when \(\mu=2\), the population of system (2.1) will go extinct, when \(\mu=1\), system (2.1) is permanent. From Theorem 4.7, we can easily deduce that there must exist a threshold \(\mu^{\ast}\). If \(\mu >\mu^{\ast}\), the population-extinction periodic solution of system (2.1) is globally asymptotically stable. If \(\mu<\mu^{\ast}\), system (2.1) is permanent.
5.2 The simulation of system (2.1) affected by parameter D
From the simulation experiments, the parameters D obviously affect the dynamical behaviors of system (2.1). If all parameters of system (2.1) are fixed, when \(D=0.95\), the population of system (2.1) will go extinct, when \(D=0.1\), system (2.1) is permanent. From Theorem 4.7, we can easily deduce that there must exist a threshold \(D^{\ast}\). If \(D >D^{\ast}\), the population-extinction periodic solution of system (2.1) is globally asymptotically stable. If \(D < D^{\ast}\), system (2.1) is permanent.
From the simulations of system (2.1), the diffusing parameter D of the population plays an important role in system (2.1). The environmental pollution will also reduce the biological diversity of the nature world. Our results also provide a reliable tactic basis for practical biological resource management.
Declarations
Acknowledgements
Supported by National Natural Science Foundation of China (11361014, 10961008), the Science Technology Foundation of Guizhou Education Department (2008038), and the Science Technology Foundation of Guizhou (2010J2130).
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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