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Dynamical analysis of almost periodic solution for a multispecies predator-prey model with mutual interference and time delays
Advances in Difference Equations volume 2017, Article number: 393 (2017)
Abstract
In this paper, we build a multispecies predator-prey model with mutual interference and time delays. By means of the comparison theorem, Ascoli theorem and Lebesgue dominated convergence theorem, we establish the sufficient conditions of permanence and investigate the existence of a unique almost periodic solution. By constructing a suitable Lyapunov function, we obtain that the positive almost periodic solution is globally attractive. Finally, we give numerical simulations to indicate the complex dynamical behaviors of this system.
1 Introduction
In population dynamics, the linkages between predator and prey are usually expressed by different functional response functions, which reflect different dynamical behaviors. Holling [1] carried out a large number of experiments on predator and prey and got some different functional response functions. For example, the mathematical expression of Holling \(x_{i}\) (\(i = 1,2 \)) model is as follows [2]:
Besides, in ecosystems, mutual interference between species is always present. The authors [3] proposed a mutual interference factor that tended to leave when the host or parasite met. A lot of articles studied the ecosystem with interference factors. Their obtained results showed that the effect of this factor should not be ignored [4–7]. For example, Wang et al. [6] concluded that mutual interference had great effect on the relative properties of predator-prey models.
In real life, time delay always exists. Food digestion time, resource regeneration time, mature time, pregnancy period and so on, these all can be expressed by time delay. Usually time delay plays a key role in many systems. For example, time delay can destroy the stability of the positive equilibrium. The obtained results showed that delayed differential equations exhibited more complex dynamical properties than ordinary differential equations [8–14]. Du et al. [10] gave the following model:
where all parameter meanings can be seen in [10]. The time delay of system (1.1) made the system very unstable and led to more complex dynamical behaviors. At the same time, the research methods were also very different from other systems.
From the point of view of the interaction between biology and environment, Darwin thought that biological variation, heredity and natural selection could lead to the adaptive change of organisms. We know that natural environment is not a constant, and organisms can change their habits to adapt to the new environment, which is called adaptive control. In recent years, adaptive control has been widely used in biological control systems, aerospace systems, satellite tracking systems, and so on [15, 16].
On the other hand, in Ref. [10], the authors assume that the coefficients \(r_{1} ( t ) \), \(b_{1} ( t ) \), \(\tau ( t ) \), \(c _{1} ( t ) \), \(r_{2} ( t ) \), \(b_{2} ( t ) \), \(c _{2} ( t ) \) of system (1.1) are continuous positive almost periodic functions. It is well known that the assumption of almost periodicity of the coefficients in (1.1) is a way of incorporating the time-dependent variability of the environment, especially when the factors of the environment exhibit periodical changes with not necessarily commensurate periods, such as weather, food, mating habits, harvest, etc. In view of these factors, it is necessary to study the relevant properties of ecosystems by using almost periodic coefficients. Recently, many scholars have studied the almost periodic solution and got some nice results, which showed that the almost periodic solution of a population dynamical system with mutual interference and time delay had wider application value [10, 17–19].
However, in the actual ecosystem, predator and prey always coexist, which is a common and widespread phenomenon. The dynamical property of a multispecies predator-prey system is much more complex than the system with only two or three species, and the analytical methods are very different [11, 20–22].
Based on the above discussion, we establish a multispecies predator-prey model with almost periodic coefficients, mutual interference and time delays. The corresponding mathematical model is as follows:
with the initial conditions
where \(\tau = \max_{t \in R} \{ \tau_{k} ( t ) ,k = 1,2,\ldots,m \} \), \(\tau_{k} ( t ) \) is a nonnegative and continuously differentiable almost periodic function on R and \(\min_{t \in R} \{ 1 - \tau '_{k} ( t ) \} > 0\). \(r _{i} ( t ) \), \(b_{ik} ( t ) \), \(c_{ik} ( t ) \), \(f _{ik} ( t ) \), \(r_{j} ( t ) \), \(p_{jk} ( t ) \), \(q _{kj} ( t ) \), \(f_{kj} ( t ) \) are all continuous positive almost periodic functions on R and the brief description about other parameters used in system (1.2) is presented in Table 1.
In this article, we aim to investigate the dynamical properties of almost periodic system (1.2), which can greatly enrich the biological background.
The structure of the article as follows. In Section 2, we introduce several important definitions and lemmas. We discuss the permanence of the system in Section 3. Next, we prove the global attractivity of system (1.2) in Section 4. In Section 5, we give conditions of the existence and uniqueness of almost periodic solutions for the system. We put numerical simulations in Section 6. In Section 7, we give a brief conclusion to this paper.
2 Main descriptions
In this part, we give some definitions and lemmas.
For continuous and bounded f on R, we denote \(f^{u} = \sup_{t \in R}f ( t ) \), \(f^{l} = \inf_{t \in R}f ( t ) \).
Definition 2.1
The positive solution \(( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y_{1} ( t ) ,y _{2} ( t ) ,\ldots, y_{m} ( t ) ) ^{T}\) of system (1.2) is said to be globally attractive if, for any other positive solution \(( \bar{x} ( t ) ,\bar{y} ( t ) ) ^{T} = ( \bar{x}_{1} ( t ) ,\bar{x}_{2} ( t ) ,\ldots, \bar{x}_{n} ( t ) ,\bar{y}_{1} ( t ) ,\bar{y}_{2} ( t ) ,\ldots,\bar{y}_{m} ( t ) ) ^{T}\) of (1.2), the following condition holds:
Definition 2.2
([23])
A function \(f ( t,x ) \) is said to be almost periodic in t uniformly with respect to \(x \in X\) if \(f ( t,x ) \) is continuous and, for \(\forall \varepsilon > 0\), it is possible to find a constant \(I ( \varepsilon ) > 0\) such that, for any interval of length \(I ( \varepsilon ) \), there exists Ï„ such that
where the number τ is called an ε-translation number of \(f ( t,x ) \).
By the continuity of almost periodic functions, we obtain that the almost periodic coefficients satisfy \(\min_{ i = 1,2,\ldots,n ; j = 1,2,\ldots,m} \{ r_{i}^{l},r_{j}^{l},b_{ik}^{l},p_{jk}^{l},c_{ik}^{l},q_{kj} ^{l} \} > 0\) and \(\max_{i = 1,2,\ldots,n ;j = 1,2,\ldots,m}\{ r_{i}^{u},r_{j}^{u},b_{ik}^{u}, p_{jk}^{u},c_{ik} ^{u},q_{kj}^{u} \} < + \infty \). For, the characteristics and relevant definitions of almost periodic functions, the reader may refer to [10, 17, 24].
Definition 2.3
([25])
An almost periodic function \(f:R \to R\) is said to be asymptotic if there exist an almost periodic function \(q ( t ) \) and a continuous function \(r ( t ) \) such that
Lemma 2.1
([26])
If the function \(f ( t ) \) is nonnegative, integral and uniformly continuous on \([ 0, + \infty )\), then \(\lim_{t \to \infty } f ( t ) = 0\).
Lemma 2.2
The set \(\{ ( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y _{1} ( t ) ,y_{2} ( t ) ,\ldots,y_{m} ( t ) ) ^{T} \in R^{n + m}| x_{i} ( t_{0} ) > 0,i = 1,2,\ldots,n;y_{j} ( t_{0} ) > 0,j = 1,2,\ldots, m,\exists t_{0} \in R \}\) is positive invariant with respect to system (1.2).
Proof
For \(x_{i} ( t_{0} ) > 0\), \(y_{j} ( t_{0} ) > 0\), we have
Then Lemma 2.2 is obtained. □
Lemma 2.3
([27])
Suppose that the continuous operator A maps the closed and bounded convex set \(Q \subset R^{n}\) onto itself, then the operator A has at least one fixed point in the set Q.
Lemma 2.4
([28])
If \(x' \mathbin{{\ge} {( {\le} )}} x ( b - ax^{\alpha } ) \), where \(a > 0\), \(b > 0\) and α is a positive constant, then
Lemma 2.5
([29])
If \(x' \mathbin{{\ge} ( {\le} )} x^{m} ( t ) ( b - ax^{1 - m} ( t ) ) \), \(x ( 0 ) > 0\), \(a > 0\), \(b > 0\), then \(\forall t \ge 0\), we have
3 Permanence of system (1.2)
Theorem 3.1
If the following condition holds:
then system (1.2) is permanent, that is, there exists \(T>0\), for \(t > T > 0\), the solution \(( x ( t ) ,y ( t ) ) ^{T}\) of (1.2) satisfies \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n _{j} \le y_{j} ( t ) \le N_{j}\), where
for \(i = 1,2,\ldots,n\); \(j = 1,2,\ldots,m\). In this article, the values of i, j are no longer repeated.
Proof
By the first equation of (1.2), we get
Integrating (3.1), we have \(x_{i} ( t ) \le x_{i} ( t - \tau ) \exp ( r_{i}^{u}\tau ) \), \(t > \tau \), that is,
Combining (3.2) and the first equation of (1.2), we have
By applying Lemma 2.4 to (3.3), we obtain
By (3.4), there exists \(T_{1} > \tau \), when \(t \ge T_{1}\) and \(T_{1} \to \infty \), then
By (3.5), there also exists \(T_{2} = T_{1} + \tau \), when \(t \ge T _{2}\), then
Combining (3.6) and the second equation of (1.2), we have
Using Lemma 2.5 to (3.7), then
Therefore, there exists \(T_{3} > 0\) such that
Combining (3.5), (3.6), (3.9) and the first equation of (1.2), we get
Suppose \(x_{i} ( \tilde{t} ) \) is any local minimal value of \(x_{i} ( t ) \), then we have
Let
From (3.11) and (3.12), we have
Integrating (3.10) on \([ \tilde{t} - \tau_{i} ( \tilde{t} ) , \tilde{t} ] \) and noticing that \(\hat{g}_{i} - b_{ii}^{u}x_{i} ( \tilde{t} - \tau_{i} ( \tilde{t} ) ) \le 0\), we obtain
Hence, for \(T_{4} > 0\) and \(t > T_{4}\), we have
Combining (3.9), (3.16) and the second equation of (1.2), when \(T_{5} \ge \max \{ T_{3},T_{4} \} > 0\), for \(t > T_{5}\), we get
It follows from Lemma 2.5 that there exists \(T_{6} > 0\) such that
Make \(T \ge \max \{ T_{2},T_{5},T_{6} \} > 0\), for \(t > T\), we get \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\).
Therefore system (1.2) is permanent.
Next, we prove that system (1.2) has at least one bounded positive solution for \(t \ge 0\). Define \(\Omega = \{ ( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y _{1} ( t ) ,y_{2} ( t ) ,\ldots,y_{m} ( t ) ) ^{T}\in R^{n + m}| ( x ( t ) ,y ( t ) ) ^{T}\text{ is the solution of system (1.2), satisfying }m_{i} \le x_{i} ( t ) \le M_{i},n_{j} \le y_{j} ( t ) \le N_{j},t \in R \}\). □
Theorem 3.2
For system (1.2), the set \(\Omega \ne \emptyset\).
Proof
According to the characteristics of an almost periodic function, for a sequence of \(\{ t_{\gamma } \} \), \(t_{\gamma } \to \infty \) as \(\gamma \to \infty \), then \(r_{i} ( t + t_{\gamma } ) \to r_{i} ( t ) \), \(r_{j} ( t + t_{\gamma } ) \to r_{j} ( t ) \), \(b_{il} ( t + t_{\gamma } ) \to b_{il} ( t ) \), \(p_{jk} ( t + t_{\gamma } ) \to p_{jk} ( t ) \), \(c_{ik} ( t + t_{\gamma } ) \to c_{ik} ( t ) \), \(q_{lj} ( t + t_{\gamma } ) \to q_{lj} ( t ) \), \(\tau_{i} ( t + t_{\gamma } ) \to \tau_{i} ( t ) \), \(f_{ij} ( t + t_{\gamma } ) \to f_{ij} ( t ) \) (\(i,l = 1,2,\ldots,n\); \(j,k = 1,2,\ldots,m\)) uniformly on R as \(\gamma \to \infty \). By Lemma 2.3, system (1.2) has at least one solution \(z ( t ) = ( x ( t ) ,y ( t ) ) ^{T}\) satisfying \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\) when \(t > T\).
Obviously, the sequence \(z ( t + t_{\gamma } ) \) is uniformly bounded and equi-continuous on any bounded subset of R. By the Ascoli theorem, we know there exists a subsequence \(z ( t + t_{\lambda } ) \) which converges to a continuous function
as \(\lambda \to \infty \) uniformly on any bounded subset of R.
Make \(T_{7} \in R\), suppose \(T_{7} + t_{\lambda } \ge T\) for all λ. When \(t \ge 0\), we obtain
Letting \(\lambda \to \infty \) in (3.17) and (3.18), for \(\forall t \ge 0\), by the Lebesgue dominated convergence theorem, we get
Since \(T_{7} \in R\) is arbitrarily given, \(g ( t ) \) is a solution of system (1.2) on R.
It is easy to know \(m_{i} \le g_{i1} ( t ) \le M_{i}\), \(n_{j} \le g_{j2} ( t ) \le N_{j}\) for any \(t \in R\). Thus, the set \(\Omega \ne \emptyset \), that is, system (1.2) has at least one bounded positive solution. □
4 Global attractivity of system (1.2)
Theorem 4.1
If the parameters of system (1.2) satisfy condition \([H_{1}]\) and the following conditions:
where
and \(\varphi_{i} ^{ - 1}\) is the inverse function of \(\varphi_{i} ( t ) = t - \tau_{i} ( t ) \), then the solution of system (1.2) is globally attractive.
Proof
Let \(( x ( t ) ,y ( t ) ) ^{T}\), \(( \bar{x} ( t ) ,\bar{y} ( t ) ) ^{T}\) be any two solutions of system (1.2). From Theorem 3.1, for \(\forall t > T\), we get
Next, we set up several Lyapunov functions. Let
By calculating the upper right derivative of \(V_{i1} ( t ) \) along system (1.2), we have
where
Substituting (1.2) into (4.3), we get
Considering (4.1) and (4.4), for \(t \ge T + \tau \), we get
Define
Combining (4.5) and (4.6), for \(t \ge T + \tau \), we get
Next, we define
where
Considering (4.7)-(4.9), for \(t \ge T + \tau \), we get
Define
Calculating its Dini derivative along system (1.2), we get
Define the Lyapunov functional \(V ( t ) \) as follows:
Considering (4.10), (4.11) and (4.12), for \(t \ge T + \tau \), we have
where \(A_{i} ( t ) \), \(B_{j} ( t ) \) are given in Theorem 4.1.
From conditions \([H _{2} ]\) and \([H _{3} ]\), there exist \(\alpha_{i}, \beta_{j} > 0\) and \(T_{0} \ge T + \tau \) such that
Let \(\alpha_{0} = \min \{ \alpha_{1},\alpha_{2},\ldots,\alpha_{n}; \beta_{1},\beta_{2},\ldots,\beta_{m} \} \), combining (4.14) and (4.15), then
Integrating (4.16) on \([ T_{0},t ] \), we get
So, \(\int_{T_{0}}^{ + \infty } ( \sum_{i = 1}^{n} \vert \bar{x}_{i} ( u ) - x_{i} ( u ) \vert + \sum_{j = 1}^{m} \vert \bar{y}_{j} ( u ) - y_{j} ( u ) \vert ) \,du < + \infty \) and \(V ( t ) \) is bounded on the interval \([ T_{0}, + \infty ) \). Combining Theorem 3.1 and (1.2), we get \(\bar{x}_{i} ( t ) - x_{i} ( t ) \), \(\bar{y}_{j} ( t ) - y_{j} ( t ) \) and \(( \bar{x}_{i} ( t ) - x_{i} ( t ) ) ^{\prime }\), \(( \bar{y}_{j} ( t ) - y_{j} ( t ) ) ^{\prime }\) are bounded on the interval \([ T_{0}, + \infty ) \). Then \(\sum_{i = 1}^{n} \vert \bar{x}_{i} ( u ) - x_{i} ( u ) \vert + \sum_{j = 1}^{m} \vert \bar{y}_{j} ( u ) - y_{j} ( u ) \vert \) is uniformly continuous.
Using Lemma 2.1, we get
Therefore, system (1.2) is globally attractive. □
5 Existence of almost periodic solution
Theorem 5.1
Suppose \([H_{1}]\), \([H_{2}]\) and \([H_{3}]\) hold, then system (1.2) has a unique almost periodic solution.
Proof
From Theorem 3.2, we know \(( x ( t ) ,y ( t ) ) ^{T} \), \(t \in R\) is a bounded positive solution. Then there exists a sequence \(\{ t'_{\lambda } \} \), \(t'_{ \lambda } \to \infty \) as \(\lambda \to + \infty \) such that \(( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) is a solution of the following system (5.1):
From the above and Theorem 3.1, we know \(( x ( t + t'_{ \lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) and \(( x' ( t + t'_{\lambda } ) ,y' ( t + t'_{\lambda } ) ) ^{T}\) are uniformly bounded. Clearly, the sequence \(( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) is equi-continuous. By the Ascoli theorem, there exists a uniformly convergent subsequence \(\{ ( x ( t + t_{\lambda } ) ,y ( t + t_{\lambda } ) ) ^{T} \} \subseteq \{ ( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T} \} \) such that, for any \(\forall \varepsilon > 0\), there exists \(\lambda_{0} ( \varepsilon ) > 0\) with the property that if λ, \(\varpi > \lambda_{0} ( \varepsilon ) \), then
which indicates that \(( x ( t + t_{\lambda } ) ,y ( t + t_{\lambda } ) ) ^{T}\) is an almost periodic and asymptotic function. Then there exist functions \(g_{i1} ( t ) \), \(g_{j2} ( t ) \), \(h_{i1} ( t ) \), \(h_{j2} ( t ) \) such that
where
\(g_{i1} ( t ) \), \(g_{j2} ( t ) \) are almost periodic functions. It shows that
Besides, we still have
Therefore, the derivatives \(g_{i1}^{\prime } ( t ) \), \(g_{j2} ^{\prime } ( t ) \) exist.
Next, we prove \(g ( t ) = ( g_{1} ( t ) ,g _{2} ( t ) ) ^{T}\) is an almost periodic solution of system (1.2).
By the characteristics of almost periodic solution, there exists a sequence \(\{ t_{\gamma } \} \), \(t_{\gamma } \to \infty \) as \(\gamma \to + \infty \) such that \(r_{i} ( t + t_{\gamma } ) \to r_{i} ( t ) \), \(r_{j} ( t + t_{\gamma } ) \to r_{j} ( t ) \), \(b_{il} ( t + t_{\gamma } ) \to b_{il} ( t ) \), \(p_{jk} ( t + t_{\gamma } ) \to p_{jk} ( t ) \), \(c_{ik} ( t + t_{\gamma } ) \to c_{ik} ( t ) \), \(q_{lj} ( t + t_{\gamma } ) \to q_{lj} ( t ) \), \(\tau_{i} ( t + t_{\gamma } ) \to \tau_{i} ( t ) \), \(f_{ij} ( t + t_{\gamma } ) \to f_{ij} ( t ) \) (\(i,l = 1,2,\ldots,n\); \(j,k = 1,2,\ldots,m\)) as \(\gamma \to \infty \) uniformly on R. Obviously, \(\lim_{\gamma \to + \infty } x_{i} ( t + t_{\gamma } ) = g _{i1} ( t ) \), \(\lim_{\gamma \to + \infty } y_{j} ( t + t _{\gamma } ) = g_{j2} ( t ) \). So we have
From the above, we know \(g ( t ) \) satisfies (1.2), that is, it is a positive almost periodic solution.
Next, we prove that the positive almost periodic solution of system (1.2) is unique.
Let \(g ( t ) > 0\) and \(\bar{g} ( t ) > 0\) be any two almost periodic solutions of system (1.2), then we claim that \(g_{1} ( t ) \equiv \bar{g}_{1} ( t ) \) and \(g_{2} ( t ) \equiv \bar{g}_{2} ( t ) \) for \(\forall t \in R\). Otherwise, there is at least one \(\xi \in R\) such that \(g _{i1} ( \xi ) \ne \bar{g}_{i1} ( \xi ) \), that is, \(\vert g_{i1} ( \xi ) - \bar{g}_{i1} ( \xi ) \vert : = \delta > 0\). Then
which is a contradiction to (4.18). Thus \(\forall t \in R\), \(g_{1} ( t ) \equiv \bar{g}_{1} ( t ) \) holds. By the same method, we can prove that \(\forall t \in R\), \(g_{2} ( t ) \equiv \bar{g}_{2} ( t ) \). □
Remark 5.1
If \(\tau_{i} ( t ) \equiv \tau_{i}\), where \(\tau_{i}\) (\(i = 1,2,\ldots,n \)) is a nonnegative constant, then assumptions \([H _{2} ] \) and \([H _{3} ] \) can be redefined. So, we give the following Corollary 5.1.
Corollary 5.1
Make \(\tau_{i} ( t ) \equiv \tau _{i}\), where \(\tau_{i} \ge 0\). If system (1.2) satisfies both \([H _{1} ]\) and the following two conditions:
then system (1.2) has a unique positive almost periodic solution which is globally attractive.
6 Model simulation
We give examples to verify the correctness of our theoretical results in this part.
Example 6.1
Consider the following system:
with the initial conditions \(( \phi ( 0 ) ,\psi ( 0 ) ) = ( 1,1 ) \) and \(( \phi ( 0 ) , \psi ( 0 ) ) = ( 2.5,2 ) \).
By calculation, the parameters of (6.1) meet the conditions of Theorem 3.1 and Corollary 5.1. Using MATLAB, by simulation, time series diagrams of (6.1) are shown in Figure 1. Figure 1 indicates that (6.1) is persistent and has a unique positive almost periodic solution which is globally attractive.
In order to demonstrate the dynamical behaviors of a multispecies predator-prey system, we give the time series diagrams with only three species in system (1.2).
Example 6.2
Consider the following system:
with the initial conditions \(( \phi_{1} ( 0 ) ,\phi_{2} ( 0 ) ,\psi ( 0 ) ) = ( 2,2,2 ) \) and \(( \phi_{1} ( 0 ) ,\phi_{2} ( 0 ) ,\psi ( 0 ) ) = ( 0.1, 0.1,0.1 ) \).
By calculation, these parameters of (6.2) meet the conditions of Theorem 3.1 and Corollary 5.1. Using MATLAB, by simulation, time series diagrams of (6.2) are shown in Figure 2. Figure 2 shows that (6.2) is persistent and has a unique positive almost periodic solution which is globally attractive.
7 Conclusion
We construct a multispecies predator-prey model with mutual interference and time delays in this article. We obtain the conditions of permanence, global attractivity and uniqueness of positive almost periodic solutions of the system by using the Ascoli theorem, Lebesgue dominated convergence theorem, Lyapunov functions and comparison theorem. Finally, simulation results indicate the correctness of the theoretical results and demonstrate the complex dynamical behaviors of the system.
Compared with Ref. [10], Du only considered the basic dynamics of two species, which can not be effectively promoted and applied in actual production and life. However, we comprehensively integrate the universal phenomenon of multispecies coexistence in the real ecosystem. By studying the dynamics of multispecies predator-prey system, we can better protect the ecosystem and practice the concept of green development.
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Acknowledgements
This paper is supported by the Natural Science Foundation of Guangxi (2016GXNSFAA380194), National Natural Science Foundation of China (11161015).
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QL is now studying for the M.S. degree at Guilin University of Technology, Guilin, China. His research interest is the study of dynamical behaviors and simulation analysis of periodic impulsive differential equations. YS received Ph.D. degree in mathematics from Central South University, Changsha, China. He is a professor of School of Science, Guilin University of Technology, Guilin, China. His research interests are differential equations and dynamical systems. He is mainly engaged in qualitative studies of complex dynamical systems. SZ, ZW and HC are now studying for the M.S. degree at Guilin University of Technology, Guilin, China. Their research interests are stability analysis and numerical simulation of impulsive systems.
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Liu, Q., Shao, Y., Zhou, S. et al. Dynamical analysis of almost periodic solution for a multispecies predator-prey model with mutual interference and time delays. Adv Differ Equ 2017, 393 (2017). https://doi.org/10.1186/s13662-017-1447-0
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DOI: https://doi.org/10.1186/s13662-017-1447-0
Keywords
- almost periodic solution
- global attractivity
- mutual interference
- numerical simulation