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Dynamic behaviors of an obligate GilpinAyala system
Advances in Difference Equations volume 2016, Article number: 270 (2016)
Abstract
In this paper, a nonautonomous obligate GilpinAyala system is proposed and studied. The persistence and extinction of the system are discussed by using the comparison theorem of differential equations. The results show that, depending on the cooperation intensity between the species, the first species will be driven extinct or be permanent. After that, by using the Lyapunov function method, series of sufficient conditions are obtained which ensure the global attractivity of the system. Finally, two examples are given to illustrate the feasibility of the main results.
Introduction
During the last decade, the dynamic behaviors of the mutualism model have been extensively investigated [1–8] and many excellent results were obtained, which concern the persistence, existence of a positive periodic solution, and stability of the system. However, there are only a few scholars to study the commensal symbiosis model.
In 2003, Sun and Wei [6] proposed the following system to describe the interspecific commensal relationship:
They investigated the local stability of all equilibrium points and showed that there was only one local stable equilibrium point in the system.
Zhu et al. [7], probably because of the work of [6], proposed the following obligate LotkaVolterra model:
where \(a_{1}<0\), \(a_{2}>0\), \(b_{1}<0\), \(c_{1}>0\), \(c_{2}<0\) are all constants. They conducted a qualitative analysis on the model, and studied the thresholds of persistence and extinction of two species in a polluted environment.
As is well known, the living environment of the species is constantly changing over time, so the nonautonomous model is more realistic. Yang et al. [8] proposed the following system:
where \(a_{i}(t)\), \(b_{i}(t)\), \(i=1,2\), \(c_{1}(t)\), are continuous and positive functions with upper and lower bound. They investigated the persistence, extinction, and global attractivity of the system. For more work on the commensal symbiosis system, one could refer to [6–12].
On the other hand, Ayala et al. [13] conducted experiments on fruit fly dynamics to test the validity of competitions, and one of the models accounting best for the experimental results is given by
where \(r_{i}\) is the intrinsic rate of growth of species, \(k_{i}\) is the carrying capacity of species i, \(\theta_{i}\) provides a nonlinear measure of interspecific interference, and \(\alpha_{ij}\) provides a measure of interspecific interference. The GilpinAyala competition system also has been studied from different aspects by many researchers (see [13–19]). Chen [17] proved that system (1.4) is global stable while \(\theta_{i}\geq1\) and \(\theta_{i}<1\) (\(i=1,2\)). Chen et al. [18] proposed a two dimensional GilpinAyala competition system with infinite delay. They completely discussed the structure of equilibria and proved the global stability of the system. After that, Wang et al. [19] discussed the extinction to the same system as studied in [18], and their results supplemented the previous work.
The GilpinAyala competition system has been extensively studied, and many excellent results as regards the system have been obtained. However, to this day, still no scholar considered the GilpinAyala system with commensalism symbiosis.
Simulated by the work of [8], in this paper, we propose the following commensalism system:
where \(x_{i}(t)\) is the population density of the ith species. In this system, the second species is favorable to the first species; while the first species has no influence on the second one, the first species cannot live without the second one. The typical relationship like epiphyte and plants with epiphyte can be described by this system.
The aim of the paper is, by using Lemma 2.3 and developing the analysis technique of Chen [20] to obtain a set of sufficient conditions to ensure the extinction, persistence, and global attractivity of system (1.5).
The paper is organized as follows: In Section 2, the necessary preliminaries are presented. In Section 3, the dynamic behaviors such as the permanence, extinction, and the globally attractivity of the system are investigated. In Section 4, some examples are given to illustrate the feasibility of main results. In the end, we finish this paper by a brief conclusion.
Throughout the paper, we let \(f^{l}=\inf_{t\in R}f(t)\) and \(f^{u}=\sup_{t\in R}f(t)\), where \(f(t)\) is a continuous and bounded function. In system (1.5), we always assume that \(\theta_{i}\) (\(i=1,2\)) are positive constants, \(r_{i}(t)\), \(k_{i}(t)\), \(\alpha_{ij}(t)\) (\(i=1,2\)) are continuous and strictly positive functions, which satisfy
Preliminaries
Now we will state three lemmas which will be useful in proving the main theorems.
Lemma 2.1
If \(a>0\), \(b>0\) and \(\frac{dx(t)}{dt}\geq{x(t)(bax^{\alpha}(t))}\), where α is a positive constant, when \(t\geq0\), \(x(0)>0\), we have
If \(a>0\), \(b>0\) and \(\frac{dx(t)}{dt}\leq{x(t)(bax^{\alpha}(t))}\), where α is a positive constant, and when \(t\geq0\), \(x(0)>0\), we have
Lemma 2.1 is a direct corollary of Lemma 2.2 of Chen [20].
Lemma 2.2
Let h be a real number and f be a nonnegative function defined on \([h;+\infty)\) such that f is integrable on \([h;+\infty)\) and is uniformly continuous on \([h;+\infty)\), then \(\lim_{t\rightarrow+\infty}f(t)=0\).
Lemma 2.2 is Lemma 2.4 of Chen [20].
Lemma 2.3
If \(m\leq x,y\leq M \), where \(m \leq M\) are positive constants, when \(\theta\geq1\), we have \(\frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\), and when \(0\leq\theta< 1\), we have \(\frac{1}{\theta m^{\theta 1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\).
Proof
Let \(f(t)=t^{\theta}\), \(t\in[m,M]\), for any \(m\leq x,y\leq M \), by the Lagrange theorem, we can obtain
where ξ is between x and y.
It easily follows that

when \(\theta\geq1\), we have \(\frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\),

when \(0\leq\theta< 1\), we have \(\frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\).
This ends the proof of Lemma 2.3. □
Main results
Theorem 3.1
If
holds, for any positive solution \((x_{1}(t),x_{2}(t))\) of system (1.5), we have
where \(x_{2}^{*}(t)\) is the unique positive solution of system (1.5).
Proof
From the second equation of system (1.5), we have
From Lemma 2.1, we can obtain
For any positive constant ε small enough, it follows from (3.1) that there exists a large enough \(T_{1}>0 \) such that
From the above equality and the first equation of system (1.5), we have
Let \(\varepsilon\rightarrow0\), by simple calculation we have
From condition (H_{1}), we can obtain \(1+{\alpha_{12}}^{u}\cdot\frac{{k_{2}}^{u}}{k_{1}^{l}}<0\), it follows that
Now let us construct a Lyapunov function
Calculating the upper right derivative of \(V_{2}(t)\) along the solution of system (1.5), it follows that
Integrating the above inequality from T to t produces
Therefore
Hence,
From equality (3.1), we know that \(x_{2}(t)\) and \(x_{2}^{*}(t)\) all have bounded derivatives for \(t\geq T\). So it follows that \(x_{2}^{\theta_{2}}(t){x_{2}^{*}}^{\theta_{2}}(t)\) is uniformly continuous on \([T,+\infty)\). By Lemma 2.2, we have
From this, it easily follows that
This ends the proof of Theorem 3.1. □
Theorem 3.2
If
holds, then the system (1.5) is persistent. That is, for any positive solution \((x_{1}(t),x_{2}(t))\) of system (1.5), we have
where \(m_{i}\), \(M_{i}\), \(i=1,2\), are positive constants.
Proof
From the first equation of system (1.5) and equality (3.2), for ε and \(T_{1}\) in equality (3.2), we have
From (H_{2}), for the above ε, we can easily get
Let \(\varepsilon\rightarrow0\), by Lemma 2.1, we have
From the second equation of system (1.5), we have
From Lemma 2.1, we can obtain
From the above equality and the above ε, there exist \(T_{2}>T_{1}>0\), such that
From the first equation of system (1.5), we have
From (H_{2}) and Lemma 2.1, we can obtain
From inequalities (3.1), (3.3), (3.4), (3.5), we have
This ends the proof of Theorem 3.2. □
Theorem 3.3
If condition (H_{2}) holds, we see that, when
or
hold, system (1.5) is globally attractive.
Proof
Let \((x_{1}(t),x_{2}(t))\) with \(x_{1}(0)>0\), \(x_{2}(0)>0\) and \((x_{1}^{*}(t),x_{2}^{*}(t))\) with \(x_{1}^{*}(0)>0\), \(x_{2}^{*}(0)>0\) be any two positive solutions of (1.5).
Now let us construct a Lyapunov function
Calculating the upper right derivative of \(V(t)\) along the solution of system (1.5), it follows that
From Theorem 3.2, we can see that there exists a large enough \(T_{3}>0\) such that
From Lemma 2.3, we see that, if \({\theta_{2}}\geq1\), then
So, from inequality (3.6) we have
From (H_{3}), we have
So, it follows that there exists a positive constant \(\alpha>0\) and large enough \(T_{4}>T_{3}>0\) such that
Similar to the analysis in Theorem 3.1, we can get
It easily follows that
From Lemma 2.3, we see that, if \(0\leq{\theta_{2}}< 1\), then
So, from equality (3.6) we have
From (H_{4}), we have
So, it follows that there exists a positive constant \(\beta>0\) and large enough \(T_{5}>T_{3}>0\) such that
Similar to the analysis in Theorem 3.1, we can get
It easily follows that
This ends the proof of Theorem 3.3. □
Examples
The following two examples show the feasibility of our main results.
Example 4.1
Consider the following equations:
By simple calculation, we see that (H_{1}) holds, by Theorem 3.1, for any positive solution \((x_{1}(t),x_{2}(t))\) of system (4.1),
Figure 1 shows the dynamic behavior of system (4.1).
Example 4.2
Consider the following system:
By simple calculation, (H_{2}) and (H_{3}) hold, and by Theorem 3.3, system (4.2) is globally attractive. Figure 2 shows the dynamic behavior of system (4.2).
Conclusion
In this paper, a nonautonomous obligate GilpinAyala system is considered. Some results as regards extinction, persistence, and global attractivity of the system are obtained. Theorem 3.1 and Theorem 3.2 show that, when (H_{1}) holds, that is, \(\frac{\alpha_{12}^{u}}{k_{1}^{l}}< \frac{1}{k_{2}^{u}}\), the cooperation between the species is small, then the first species could be driven extinct, while the second species has stability; when (H_{2}) holds, that is, \(\frac{\alpha_{12}^{l}}{k_{1}^{u}}> \frac{1}{k_{2}^{l}}\), the cooperation between species is big, then the two species could be permanent. These results show that the extinction or permanence of the first species is depending on the cooperation intensity between the two species.
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Acknowledgements
The author is grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work was completed with the support of the Foundation of Fujian Provincial Department of Education (2015JA15431).
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Wang, D. Dynamic behaviors of an obligate GilpinAyala system. Adv Differ Equ 2016, 270 (2016). https://doi.org/10.1186/s1366201609655
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MSC
 34C25
 92D25
 34D20
 34D40
Keywords
 obligate GilpinAyala system
 extinction
 permanence
 Lyapunov function
 global attractivity