 Research
 Open Access
 Published:
Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations
Advances in Difference Equations volume 2017, Article number: 201 (2017)
Abstract
In this paper, a generalized nonautonomous stochastic competitive system with impulsive perturbations is studied. By the theories of impulsive differential equations and stochastic differential equations, we have established some asymptotic properties of the system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on. In order to show the correctness and feasibility of the theoretical results, several numerical examples are presented. Finally, the effects of different white noise perturbations and different impulsive perturbations are discussed and illustrated.
Introduction
It is well known that there are four kinds of relationships between the species in the population ecological systems, that is, competition, predation, mutualism and parasitism. Among these relationships, competition can always ensure the survival of species and make effective use of resources, maintain the permanence of a ecological system and keep the healthy development of the population. Thus, a competitive system has received great interest by many mathematical and ecological researchers in the last decades (see [1–10]). As far as the competition is concerned, there are usually two kinds of competitive relationship, i.e. one is the interspecific competition and the other is the intraspecific competition.
The basic twospecies competitive system is governed by the following coupled differential equations:
where \(a_{ii}\) is the intraspecific competition coefficient, while \(a_{ij}\) (\(i\neq j\), \(i,j=1,2\)) is the interspecific competition coefficient.
Based on the classic competitive system, Gopalsamy proposed a series of generalized competitive systems in the monograph [11], and one of the generalized competitive systems is as follows (see p. 168, [11]):
which means two species are allowed to cohabit in a common community, and each species inhibits the average growth rate of the other.
Recently, Wang and Liu (see [12]) studied the following nonautonomous competitive system:
in which the existence and global asymptotic stability of positive almost periodic solutions is obtained. More references related to these generalized competitive systems can be also seen in [11, 13, 14].
However, most of the above mentioned references focused on the deterministic models, while the growth of the species is often affected by the interferences of the environmental noises in the real world. Thus, it is more reasonable to study ecological models. The dynamical behavior of the ecological system, and whether it will make a change to the existing results, has received wide attention in the recent several years (see references [4, 15–20] etc.).
Enlightened by the above mentioned references, we suppose that the random fluctuations of the environment will mainly affect the intrinsic growth rate \(r_{i}(t)\) of the species, and they are estimated by the following form:
where \(B_{i}(t)\) is Brownian motion, \(\sigma_{i}(t)\) is a continuous and bounded function on \(t\geq0\) and \(\sigma^{2}_{i}(t)\) represents the intensity of the white noise, \(i=1,2\).
On the other hand, many natural or manmade factors, such as cropdusting, planting, hunting, harvesting, drought, flooding and so on, will lead to sudden changes to the system. From the viewpoint of mathematical modeling, these sudden changes could be described by impulsive effects or perturbations to the models (see [21, 22]). Thus, if we introduce both impulsive perturbations and stochastic perturbations of white noises on the previous system (3), we can obtain the following system:
where \(x_{i}(t)\) is the population density of the ith population, \(r_{i}(t)\) and \(a_{i}(t)\) are the intrinsic growth rate and the intraspecific competing rate, respectively, and \(c_{i}(t)\) represents the interspecific competing rate. \(r_{i}(t)\), \(a_{i}(t)\), \(c_{i}(t)\), \(t\in R^{+}=[0,\infty)\) are positive, continuous and bounded. \(0< t_{1}< t_{2}<\cdots\), \(\lim_{k\rightarrow+\infty}t_{k}=+\infty\). \(h_{ik}>1\), \(i = 1,2\), \(k\in N\), when \(h_{ik}>0\), the impulsive effects represent planting, while \(h_{ik}<0\) denote harvesting.
Throughout the present paper, we denote
for any positive, bounded function \(f(t)\) defined on \(R^{+}=[0,+\infty)\).
The rest of this paper is organized as follows. In Section 2 we demonstrate and prove the main results of the paper, such as the existence of a unique positive solution of the system, sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. In Section 3, several numerical examples are presented to support the theoretical results. Moreover, effects on the impulsive and stochastic perturbations are also analyzed and discussed at the end of the paper.
Preliminaries
In this section, based on the methods proposed by Yan and Zhao (see [23]), the corresponding stochastic differential equations without impulses are studied, and we will discuss the existence of a positive solution of above system (4) firstly. Further, by the definitions proposed by Liu and Wang (see [18]), we will derive some asymptotic behavior of this system, such as the extinction, nonpersistence and persistence in the mean, weak persistence and stochastic permanence and so on.
Theorem 2.1
For any initial conditions \((x_{10}, x_{20})^{T}\in R_{+}^{2}=\{(x,y)^{T}\in R^{2} x>0,y>0\}\), system (4) has a unique positive solution \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) on \([0, +\infty)\), and the solution will remain in \(R^{2}_{+}\) almost surely.
Proof
Consider the following stochastic differential equations (SDEs) without impulses:
with the initial value \((y_{10}, y_{20})^{T}=(x_{10}, x_{20})^{T}\).
It is easy to prove that there is a unique global positive solution \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) of system (5) by the theory of nonimpulsive stochastic differential equations (see [18]).
Denote \(x_{i}(t)= \prod_{0< t_{k}< t}(1+h_{1k})y_{i}(t)\) (\(i=1,2\)), then we claim that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is the solution of system (4) with the initial data \((x_{10}, x_{20})^{T}\).
In fact, since \(x_{1}(t)\) is continuous on \((0,t_{1})\) and each interval \((t_{k},t_{k+1})\subset[0,+\infty)\) and for \(t\neq t_{k}\), \(k\in N\),
Similarly, we can check that
And for every \(t_{k} \in R^{+}\), \(k\in N\),
These mean that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is the unique global positive solution of system (4), so we complete the proof of this theorem. □
In Theorem 2.1, we can see that solutions of system (4) will remain in the first quadrant, but how do they vary in this quadrant? In the following part, we will discuss the sufficient conditions for several cases, such as extinction and weak persistence, nonpersistence and persistence in the mean and so on.
Theorem 2.2
Denote by \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) a solution of system (4), then
where \(b_{i}(t)=r_{i}(t)0.5\sigma^{2}_{i}(t)\), \(i=1,2\).
Proof
For the nonimpulsive system (5), by Itô’s formula, we obtain
where \(j=1,2\), \(j\neq i\), and this leads to
Integrating both sides of inequality (10) on the interval \([0,t]\) yields
where \(M_{i}(t)= \int^{t}_{0} \sigma_{i}(s)\,dB_{i}(s)\).
Thus,
which yields
Note that \(M_{i}(t)\) is a local martingale whose quadratic variation is
By the strong law of large numbers for local martingale (see [24]), we have
If we multiply \(\frac{1}{t}\) on each side of inequality (13) and take superior limit on both sides of it, we can obtain
□
Corollary 2.1
If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]<0\), then the ith species of system (4) is extinct.
Theorem 2.3
Suppose that \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is a solution of system (4), then
Proof
By the definition of the limit, for \(\forall\varepsilon_{i}>0\), there exists \(T_{1}>0\) such that
for \(t>T_{1}\).
Combining inequality (13) and the above inequality, we have
for \(\forall t>T_{1}\) a.s., where \(\lambda_{i}=b^{*}_{i}+\varepsilon_{i}\).
If we denote \(h_{i}(t)= \int^{t}_{0} x_{i}(s)\,ds\), then \(h_{i}'(t)=x_{i}(t)\). Then it follows from inequality (14) that
Integrating inequality (15) from \(T_{1}\) to t, we have
Thus,
If we multiply \(\frac{1}{t}\) on each side of inequality (17) and take superior limit on both sides of it, we can obtain
By L’Hospital’s rule we have
This means that we have completed the proof. □
If \(b^{*}_{i}=0\), it is easy to obtain \(\lim_{t\rightarrow+\infty} \frac{\int^{t}_{0} x_{i}(s)\,ds}{t}=0\), and we can obtain the following Corollary 2.2.
Corollary 2.2
If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]=0\), then system (4) is nonpersistent in the mean.
Theorem 2.4
If \(b^{*}_{i}= \limsup_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln(1+h_{ik})+\int^{t}_{0}b_{i}(s)\,ds ]>0\), then at least one of the species in system (4) is weakly persistent.
Proof
It follows from (9) that
If we integrate on each side of the above equation, we have
Set \(S= \{ \lim_{t\rightarrow+\infty}\sup x_{i}(t)=0 \}\), if the assertion of this theorem is not true, then \(\mathscr{P}(S)>0\), and for \(\omega\in S\), \(\lim_{t\rightarrow+\infty}x_{i}(t,\omega)=0\).
Note that \(\limsup_{t\rightarrow+\infty}\frac{M_{i}(t)}{t} =0\). Further, it follows from the boundedness of \(a_{i}(t)\) and \(c_{i}(t)\) that
Thus,
which is a contradiction, and this completes the proof of this theorem. □
Theorem 2.5
Denote \(b_{i*}= \liminf_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t}\ln (1+h_{ik})+ \int^{t}_{0} \bar{b_{i}}(s)\,ds ]\), then the solution of system (4) satisfies
where \(\bar{b_{i}}(t)=r_{i}(t)c_{i}(t)0.5\sigma^{2}_{i}(t)\), \(i=1,2\).
Proof
It follows from (9) again that
Integrating both sides of inequality (20) from 0 to t yields
Then
Note the definition of \(M_{i}(t)\) and \(b_{i*}\), according to the property of the limit again, for any \(\epsilon_{i}>0\), \(i=1,2\), there exists \(T_{2}>0\) such that, for \(t>T_{2}\),
Substituting above inequalities into (22) yields
for all \(t>T_{1}\) almost surely, where \(\mu_{i} =b_{i*}\epsilon_{i}\).
Note that \(h_{i}(t)= \int^{t}_{0} x_{i}(s)\,ds\) and \(\frac {dh_{i}(t)}{dt}=x_{i}(t)\), then from (23) we have
If we integrate the above inequality on the interval \([T_{2},t]\), then we have
Thus,
Taking inferior limit on (26) yields
By L’Hospital’s rule again, we have
Thus, we complete the proof of the above theorem. □
Further, if \(b_{i*}>0\), we have the following Corollary 2.3.
Corollary 2.3
If \(b_{i*}= \liminf_{t\rightarrow+\infty} \frac{1}{t} [ \sum_{0< t_{k}< t} \ln(1+h_{ik})+ \int^{t}_{0}\bar{b_{i}}(s)\,ds ]>0\), then system (4) is persistent in the mean a.s.
Theorem 2.6
If system (4) satisfies the following two conditions:

(H1)
there exist positive constants \(m_{i}\) and \(M_{i}\) such that \(m_{i}< \prod_{0<t_{k}<t}(1+h_{ik})<M_{i}\);

(H2)
\((\sigma^{u}_{i})^{2}<\bar{b}^{l}_{i}\);
then system (4) is stochastically permanent, where \(\bar{b}_{i}(t)=r_{i}(t)c_{i}(t)0.5\sigma^{2}_{i}(t)\), \(i=1,2\).
Proof
Applying Itô’s integration by parts formula, we can derive that
Integrating the above inequality on the interval \([0, t]\), we have
Taking expectations on both sides of (29) and making some estimations lead to
Thus, by the maximum principle, we have
where \(L_{i}= \frac{(1+2r^{u}_{i}+(\sigma^{u}_{i})^{2})^{3}}{27(m_{i}a^{l}_{i})^{2}}\), \(i=1,2\).
Thus,
which yields
Then, for any \(\xi_{i}>0\), set \(\beta_{i}=M_{i}\sqrt{\frac{L_{i}}{\xi_{i}}}\), and by Chebyshev’s inequality, we have
In other words,
Now we will prove that for \(\forall\xi_{i}>0\), \(\exists \eta_{i}>0\), s.t. \(\liminf_{t\rightarrow+\infty} \mathscr{P}\{x_{i}(t)\geq\eta_{i}\}\geq1\xi_{i}\).
In fact, it follows from condition (H2) that \(b^{l}_{i}>(\sigma^{u}_{i})^{2}\). If we define
where \(0< k<2[b^{l}_{i}(\sigma^{u}_{i})^{2}]\).
Then it follows from Itô’s formula again that
which yields
Similarity, if we apply Itô’s integration by parts formula on \(V_{2}(t)\), then
which yields
where
Since \(0< k<2[b^{l}_{i}(\sigma^{u}_{i})^{2}]\), then \(J(y_{i})\) is upper bounded, and we denote \(J_{i}:= \sup_{y_{i}\in R^{+}}J(y_{i})<+\infty\), then
Integrating (37) from 0 to t, then multiplying \(e^{kt}\) and taking expectations on each side of it, we can obtain
which yields
Thus,
Then, for any \(\xi_{i}>0\), set \(\eta_{i}=m_{i} \sqrt{\frac{k\xi_{i}}{J_{i}}}\), applying Chebyshev’s inequality again, we have
In other words,
From (34) and (42), the stochastic permanence of system (4) is obtained. This completes the proof of this theorem. □
Remark
In fact, ‘persistence in the mean’ in this section is not a good definition of persistence for stochastic population models. Some authors have introduced some more appropriate definitions of permanence for stochastic population models. For example, stochastic persistence in probability (see [25, 26]) or a new definition of stochastic permanence (see [27]).
Numerical simulations and discussions
In this paper, a stochastic nonautonomous competitive system with impulsive perturbations is proposed and studied. We establish sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. Furthermore, the critical value between extinction, nonpersistence and weak persistence of at least one species in the system is obtained.
In order to verify the correctness and the feasibility of the derived conditions in the theoretical results, we will give a series of numerical examples to illustrate them by using the extension of Milstein’s method (see [28]) in this section. Furthermore, we will show the effects of different white noises or impulsive perturbations to the dynamics of the system, and by the figures of corresponding simulations, one can observe the population fluctuation of the species in the competitive system more intuitively.
In the following, we choose the same initial value \((x_{10},x_{20})=(0.5,0.2)\) and parameters \(a_{1}(t)=0.1+0.01\sin(t)\), \(a_{2}(t)=0.1+0.01\cos(t)\), \(c_{1}(t)=0.22+0.02\sin(t)\), \(c_{2}(t)=0.22+0.02\cos(t)\), \(\Delta t=0.01\).
Example 3.1
For system (4), we set the following choice of parameters:
By a simple computation, we have \({\bar{b}}_{1}(t)=0.050.01\sin(t)\), \({\bar{b}}_{2}(t)=0.10.01\cos(t)\), then \(b^{*}_{1}=0.04<0\), \(b^{*}_{2}=0.09<0\), which satisfies the condition of Corollary 2.1, then both of the species are extinct (see Figure 1).
If we decrease the white noises of the species \(x_{1}\) and let \(\sigma ^{2}_{1}(t)=0.2+0.04\sin(t)\), while the values of other parameters are the same as above, then \(b^{*}_{1}=0.11>0\), at the moment the species \(x_{2}\) will still be extinct, while the species \(x_{1}\) can survive (see Figure 2).
Example 3.2
For system (4), if we set the following choice of parameters:
It is easy to calculate that \({\bar{b}}_{1}(t)={\bar{b}}_{2}(t)=0\), then \(b^{*}_{1}=b^{*}_{2}=0\), which satisfies the condition of Corollary 2.2, then the species of system (4) is nonpersistent in the mean (see Figure 3).
If we increase the intrinsic growth rate of the species as \(r_{1}(t)=0.18+0.02\sin(t)\), \(r_{2}(t)=0.28+0.03\cos(t)\), while the values of other parameters are the same as above, then one can calculate \(b^{*}_{1}=0.02 >0\), \(b^{*}_{2}=0.04>0\) at this time, which satisfies the condition of Theorem 2.3. According to the theorem, at least one of the two species will be weakly persistent in the mean. On the other hand, from the stochastic simulation of this case (see Figure 4), we can observe that \(x_{1}\) is weakly persistent.
If we go on increasing the intrinsic growth rate of the species as \(r_{1}(t)=0.40+0.02\sin(t)\), \(r_{2}(t)=0.48+0.03\cos(t)\), while the values of other parameters are the same as above, then one can calculate that \(b_{1*}= b_{2*}=0.01>0\) at this time, which satisfies the condition of Corollary 2.3, then system (4) should be persistent in the mean almost surely by this corollary. And this is also proved by our stochastic numerical simulation (see Figure 5).
Example 3.3
For system (4), we set another series of parameters as follows:
Further, by a simple computation, we can verify that \(e^{1}< \prod_{0<t_{k}<t}(1+h_{ik})<e^{0.75}\) and \(b_{1}(t)=r_{1}(t)c_{1}(t)0.5\sigma^{2}_{1}(t)=0.520.02\sin t\), \(b_{2}(t)=r_{2}(t)c_{2}(t)0.5\sigma^{2}_{2}(t)=0.640.03\cos t\), which leads to \((\sigma^{u}_{1})^{2}=0.14< b^{l}_{1}=0.5\), \((\sigma^{u}_{2})^{2}=0.12< b^{l}_{2}=0.61\). That is to say, conditions (H1) and (H2) in Theorem 2.5 hold, then the system is stochastic permanent, and this is also proved by our stochastic numerical simulation (see Figure 6).
However, if we suppose that the white noises are increased as \(\sigma ^{2}_{1}(t)=2.12+0.02\sin(t)\), \(\sigma^{2}_{2}(t)=1.92+0.02\cos(t)\), then it is obvious that both species of system (4) will be extinct rapidly by our stochastic simulation (see Figure 7). This means that the species in the ecological system might become extinct as the white noises increase.
On the other hand, in order to see how the impulsive perturbations will affect the system, we choose the same parameters as those in Example 3.2, but only change the intensity of the impulses to \(h_{1k}=h_{2k}=e^{0.2}1\), \(t_{k}=k=1,2,\ldots \) , then the condition \(b^{*}_{i}>0\) does not hold any more. At the moment, we can see that both species become extinct instead of being persistent in the mean by the stochastic simulation (see Figure 8).
Conclusions
From the above numerical simulations and discussions, we can conclude that both heavy intensity of environmental noises and large impulsive perturbations to the ecological system will lead to the extinction of the species. And this shows that the departments of environment protection should control the environmental noises and impulsive disturbance reasonably to protect the ecological balance.
In addition, as far as the study of population models is concerned, stability of the positive equilibrium state is one of the most interesting topics. For example, models with noise, some of the stochastic models do not keep the positive equilibrium state of the corresponding deterministic systems. And many authors have studied stability in distribution of several stochastic population models in recent years (see [29, 30] etc.). Thus, we could try to consider these aspects and get much more interesting results in the future investigation.
References
 1.
Ahmad, S, Stamova, IM: Asymptotic stability of competitive systems with delays and impulsive perturbations. J. Math. Anal. Appl. 334, 686700 (2007)
 2.
Hou, ZY: Permanence and extinction in competitive LotkaVolterra systems with delays. Nonlinear Anal., Real World Appl. 12, 21302141 (2011)
 3.
Lisena, B: Competitive exclusion in a periodic LotkaVolterra system. Appl. Math. Comput. 177, 761768 (2006)
 4.
Liu, M, Wang, K: Asymptotic behavior of a stochastic nonautonomous LotkaVolterra competitive system with impulsive perturbations. Math. Comput. Model. 57, 909925 (2013)
 5.
Qiu, JL, Cao, JD: Exponential stability of a competitive LotkaVolterra system with delays. Appl. Math. Comput. 201, 819829 (2008)
 6.
Samanta, GP: A twospecies competitive system under the influence of toxic substances. Appl. Math. Comput. 216, 291299 (2010)
 7.
Song, XY, Chen, LS: Global asymptotic stability of a two species competitive system with stage structure and harvesting. Commun. Nonlinear Sci. Numer. Simul. 6, 8187 (2001)
 8.
Wei, FY, Lin, YR, et al.: Periodic solution and global stability for a nonautonomous competitive LotkaVolterra diffusion system. Appl. Math. Comput. 216, 30973104 (2010)
 9.
Zhao, JD, Zhang, ZC, Ju, J: Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive LotkaVolterra system. Appl. Math. Comput. 230, 587596 (2014)
 10.
Zhang, L, Teng, ZD: Permanence for a class of periodic timedependent competitive system with delays and dispersal in a patchyenvironment. Appl. Math. Comput. 188, 855864 (2007)
 11.
Gopalsamy, K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and Its Applications. Kluwer Academic, Dordrecht (1992)
 12.
Wang, QL, Liu, ZJ: Existence and global asymptotic stability of positive almost periodic solutions of a twospecies competitive system. Int. J. Biomath. 7, Article ID 1450040 (2014)
 13.
Chen, YP, Tan, RH, Liu, ZJ: Global attractivity in a periodic delayed competitive system. Appl. Math. Sci. 34, 16751684 (2007)
 14.
Qin, WJ, Liu, ZJ: Permanence and positive periodic solutions of a discrete delay competitive system. Discrete Dyn. Nat. Soc. 2010, Article ID 381750 (2010)
 15.
Liu, M, Wang, K: Global stability of a nonlinear stochastic predatorprey system with BeddingtonDeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 11141121 (2011)
 16.
Liu, M, Wang, K: Persistence and extinction in stochastic nonautonomous logistic systems. J. Math. Anal. Appl. 375, 443457 (2011)
 17.
Liu, M, Wang, K: Persistence and extinction of a singlespecies population system in a polluted environment with random perturbations and impulsive toxicant input. Chaos Solitons Fractals 45, 15411550 (2012)
 18.
Liu, M, Wang, K: On a stochastic logistic equation with impulsive perturbations. Comput. Math. Appl. 63, 871886 (2012)
 19.
Wu, RH, Zou, XL, Wang, K: Asymptotic properties of a stochastic LotkaVolterra cooperative system with impulsive perturbations. Nonlinear Dyn. 77, 807817 (2014)
 20.
Wu, RH, Zou, XL, Wang, K: Asymptotic behavior of a stochastic nonautonomous predatorprey model with impulsive perturbations. Commun. Nonlinear Sci. Numer. Simul. 20, 965974 (2015)
 21.
Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993)
 22.
Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)
 23.
Yan, JR, Zhao, AM: Oscillation and stability of linear impulsive delay differential equations. J. Math. Anal. Appl. 227, 187194 (1998)
 24.
Mao, XR: Stochastic Differential Equations and Applications, 2nd edn. Horwood, Chichester (2007)
 25.
Schreiber, SJ, Benam, M, Atchadé, KAS: Persistence in fluctuating environments. J. Math. Biol. 62, 655683 (2011)
 26.
Liu, M, Bai, CZ: Analysis of a stochastic tritrophic foodchain model with harvesting. J. Math. Biol. 73, 597625 (2016)
 27.
Zhu, Y, Liu, M: Permanence and extinction in a stochastic serviceresource mutualism model. Appl. Math. Lett. 69, 17 (2017)
 28.
Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525546 (2001)
 29.
Liu, M, Bai, CZ: Dynamics of a stochastic oneprey twopredator model with Lévy jumps. Appl. Math. Comput. 284, 308321 (2016)
 30.
Liu, M, Bai, CZ: Optimal harvesting of a stochastic delay competitive model. Discrete Contin. Dyn. Syst., Ser. B 22, 14931508 (2017)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11372294), Applied Basic Research Program of Sichuan Provincial Science and Technology Department in 2017, Scientific Research Fund of Sichuan Provincial Education Department (11ZB192, 14ZB0115) and the Doctorial Research Fund of Southwest University of Science and Technology (15zx7138).
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Yang, L., Tian, B. Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations. Adv Differ Equ 2017, 201 (2017). https://doi.org/10.1186/s1366201712565
Received:
Accepted:
Published:
Keywords
 competitive system
 impulsive perturbations
 stochastic perturbations
 extinction
 stochastic permanence