Uniform ultimate boundedness of solutions of predator-prey system with Michaelis-Menten functional response on time scales
- Yang-Yang Yu^{1},
- Lin-Lin Wang^{1}Email author and
- Yong-Hong Fan^{1}
https://doi.org/10.1186/s13662-016-1037-6
© The Author(s) 2016
Received: 28 June 2016
Accepted: 18 November 2016
Published: 7 December 2016
Abstract
In this paper, a predator-prey system with Michaelis-Menten functional response on time scales is investigated. First of all, we generalize the semi-cycle concept to time scales. Second, we obtain the uniformly ultimate boundedness of solutions of this system. Our results demonstrate that when the death rate of the predator is rather small without prey, whereas the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run. In particular, if \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\), some well-known results have been generalized. In addition, for the continuous case, we provide a new idea to prove its permanence. Finally, a numerical simulation is given to support our main results.
Keywords
MSC
1 Introduction
The permanence is based on a global criterion for the coexistence of species, which describes a numerical technique for assembly of ecological communities of Lotka-Volterra form [1]. During the last few years, the permanence of ecological models has been discussed by many authors [2–15]. In [4, 5, 10, 11], the comparison method was used, and some sufficient conditions of permanence of biological systems were established. In [6], by using the semi-cycle and related concepts, they discussed the permanence of a discrete biological system.
To the best of our knowledge, the permanence of biological system on time scales was first discussed by Zhang and Zhang [8]. Using the theory of differential inequality, they obtained the permanence of a cooperation system with feedback controls on time scales. But for the system itself, we could verify that some additional conditions in [8] may not be necessary. For further study, in [15], Li and Wang obtained a permanence result for a multispecies Lotka-Volterra mutualism system by establishing some dynamic inequalities on time scales.
Remark 1.1
For any \(t>l\), \(t\in\mathbb{T}\), we have \(t-\rho (t ) \leq\omega\).
Proof
For any \(t>l\), \(\rho ( t ) \in\mathbb{T}\), then \(\rho ( t ) +\omega\in\mathbb{T}\), if \(t-\rho ( t ) >\omega\), then \(\rho ( t ) +\omega< t\), this implies \(\omega=0\), we arrive at a contradiction. Thus \(t-\rho ( t ) \leq\omega\) holds. □
As is well known, predator-prey systems play an important role in ecosystems [2–7, 9–14, 16–27]. We find that there are few papers discussing the permanence of these systems on time scales.
Theorem 1.1
- (C1)
\(\overline{a}>m\overline{c}\);
- (C2)
\(k\overline{e}>\overline{d}\)
Theorem 1.2
- (B1)
\(2m\overline{b_{1}}>\overline{\alpha_{1}}\);
- (B2)
\(\overline{\alpha_{2}}>\overline{b_{2}}\);
- (B3)
\((b_{2}(k))^{M}<1\)
In fact, (B3) is not necessary. In [6], by using the semi-cycle and related concepts, Fan and Li considered permanence of the system (4), and obtained the following.
Theorem 1.3
- (H1)
\(\overline{a}>\overline{c/\gamma}\);
- (H2)
\(\overline{f}>\overline{d}\)
In recent years, the existence of periodic solutions of predator-prey systems on time scales has been obtained by coincidence degree theory in many articles [22–26], since the existence result could be obtained by coincidence degree theory both in the continuous case and the discrete case. Fazly and Hesaaraki [22] obtained the existence of periodic solutions of nonautonomous predator-prey dynamical system with Beddington-DeAngelis functional response by coincidence degree theory. Tong et al. [23] investigated the existence of periodic solutions of a predator-prey system with sparse effect and Beddington-DeAngelis or a Holling III functional response. By using a continuation theorem based on coincidence degree theory, they obtained sufficient criteria for the existence of periodic solutions for the system.
Since the permanence of this system has been obtained by the comparison theorem in the continuous case, while it has been obtained by semi-cycle concept in discrete case. Also, in order to delete the additional condition (B3) and the additional conditions of Theorem 3.1 in [8], we need to extend the semi-cycle concept of the discrete case to that of time scales.
Noticing that system (2) and (4) is derived from (1) by exponential transformations, \(x ( t ) =\exp \{ x_{1} ( t ) \} \), \(y ( t ) =\exp \{ x_{2} ( t ) \} \), and \(N_{1} ( k ) =\exp \{ x_{1} ( k ) \} \), \(N_{2} ( k ) =\exp \{ x_{2} ( k ) \} \), respectively, when \(\mathbb{T}=\mathbb{R}\) and \(\mathbb{T}=\mathbb{Z}\). Obviously, when solutions of system (1) are uniformly ultimate bounded, systems (2) and (4) are both permanent, and the contrary also holds true. Thus, our aim is to prove the uniform ultimate boundedness of solutions of system (1) by using the semi-cycle concept on time scales instead of comparison theorems. Our result is an unification and extension of a continuous and discrete analysis.
The rest of the paper is organized as follows. In Section 2, we state some basic properties about time scales, and generalize the semi-cycle concept to time scales. Section 3 is devoted to the uniform ultimate boundedness of solutions of system (1). A discussion is presented in Section 4. The final section of the paper contains a numerical example supporting the result.
2 Preliminary
First we will give some definitions about time scales before presenting our main result (see [28, 29]).
Definition 2.1
A time scale is an arbitrary nonempty closed subset \(\mathbb{T}\) of the real number \(\mathbb{R}\).
Definition 2.2
Definition 2.3
The following lemmas will be useful to prove our main result. Their proofs are similar to [6], we omit them here.
Lemma 2.1
Lemma 2.2
Similar to the definition of semi-cycle in discrete case (see [30]), we give the definition of a semi-cycle on time scales.
Definition 2.4
A positive semi-cycle of a rd-continuous function \(f: \mathbb{T}\rightarrow\mathbb{R}\) consists of a ‘string’ of terms \(\{ f ( t ) ,t\in [ s,t ] ,s,t\in\mathbb{T} \} \), all greater than or equal to 0. A negative semi-cycle of a rd-continuous function \(f: \mathbb{T}\rightarrow \mathbb{R}\) consists of a ‘string’ of terms \(\{ f ( t ) ,t\in [ p,q ] ,p,q\in\mathbb{T} \} \), all less than or equal to 0.
3 The uniform ultimate boundedness
Before giving our main result, we list the definition of uniform ultimate boundedness.
Definition 3.1
Here we say ‘uniformly’, because \(\lambda_{1}\) and \(\lambda_{2}\) are independent on \(( x_{1} ( 0 ) ,x_{2} (0 ) ) ^{\mathbb{T}}\).
We now state the main result as follows.
Theorem 3.1
- (A1)
\(\overline{a}>\overline{c/\gamma}\);
- (A2)
\(\overline{d}<\overline{f/\beta}\)
Proof
We will prove \(\lambda_{1}\leq\lim\inf_{t\rightarrow \infty}x_{i} ( t ) \leq\lim\sup_{t\rightarrow\infty }x_{i} ( t ) \leq\lambda_{2}\), \(\lambda_{1}\) and \(\lambda_{2}\) are constants, \(i=1,2\). Thus, we divide the proof into four parts.
Part 1. \(\lim\sup_{t\rightarrow\infty}x_{1} ( t ) \leq K_{1}\), that is to say, \(x_{1} ( t ) \) is uniformly ultimate bounded above.
- (a1)
For any \(\alpha,\beta\in\mathcal{L}\), if \(\alpha\neq \beta\), \([ s_{\alpha},t_{\alpha} ] \cap [ s_{\beta },t_{\beta} ] =\varnothing\).
- (b1)
If \(s_{\alpha}\) is left-scattered, then \(u_{1} ( \rho ( s_{\alpha} ) ) <0\).
- (c1)
If \(s_{\alpha}\) is left-dense, then there exists a hollow left neighborhood \(\mathring{U}_{-} ( s_{\alpha} ) \) of \(s_{\alpha}\) such that \(u_{1} ( t ) <0\), for \(t\in\mathring{U}_{-} ( s_{\alpha} ) \).
- (d1)
If \(t_{\alpha}\) is right-scattered, then \(u_{1} ( \sigma ( t_{\alpha} ) ) <0\).
- (e1)
If \(t_{\alpha}\) is right-dense, then there exists a hollow right neighborhood \(\mathring{U}_{+} ( t_{\alpha} ) \) of \(t_{\alpha}\) such that \(u_{1} ( t ) <0\), for \(t\in\mathring{U}_{+} ( t_{\alpha} ) \).
Therefore from (10), (11), and (15), \(x_{1} ( t ) \) is uniformly ultimate bounded above.
Part 2. \(k _{1}\leq\lim\inf_{t\rightarrow\infty}x_{1} ( t ) \), that is to say, \(x_{1} ( t ) \) is uniformly ultimate bounded below.
- (a2)
For any \(\alpha,\beta\in\mathcal{L}\), if \(\alpha\neq \beta\), \([ p_{\alpha},q_{\alpha} ] \cap [ p_{\alpha },q_{\alpha} ] =\varnothing\).
- (b2)
If \(p_{\alpha}\) is left-scattered, then \(u_{2} ( \rho ( p_{\alpha} ) ) >0\).
- (c2)
If \(p_{\alpha}\) is left-dense, then there exists a hollow left neighborhood \(\mathring{U}_{-} ( p_{\alpha} ) \) of \(p_{\alpha}\) such that \(u_{2} ( t ) >0\), for \(t\in\mathring{U}_{-} ( p_{\alpha} ) \).
- (d2)
If \(q_{\alpha}\) is right-scattered, then \(u_{2} ( \sigma ( q_{\alpha} ) ) >0\).
- (e2)
If \(q_{\alpha}\) is right-dense, then there exists a hollow right neighborhood \(\mathring{U}_{+} ( q_{\alpha} ) \) of \(q_{\alpha}\) such that \(u_{2} ( t ) >0\), for \(t\in\mathring{U}_{+} ( q_{\alpha} ) \).
Thus from (21), (22), and (23), \(x_{1} ( t ) \) is uniformly ultimate bounded below. Then Part 2 holds.
Therefore, from Part 1 and Part 2, \(x_{1} ( t ) \) is uniformly ultimate bounded, we can assume \(m\leq\exp \{ x_{1} ( t ) \}\leq M\) for any \(t\in{}[ T_{0},\infty)\cap\mathbb{T}\), where \(T_{0}\) is sufficiently large.
Part 3. \(k_{1}\leq\lim\inf_{t\rightarrow\infty}x_{2} ( t ) \), that is to say, \(x_{2} ( t ) \) is uniformly ultimate bounded below.
Now we assume \(v_{2} ( t ) \) oscillates about zero, by (26), we know that \(v_{2} ( t ) \leq0\) implies \(( v_{2} ( t ) ) ^{\Delta}\geq0\). Thus, by the semi-cycle concept, we let \(v_{2} ( t ) \leq0\), for \(t\in [ p_{\alpha},q_{\alpha } ] \), \(p_{\alpha},q_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, the interval \([ p_{\alpha },q_{\alpha} ]\) satisfies (a2)-(e2) by replacing \(u_{2}(t)\) in Part 2 with \(v_{2}(t)\).
Thus from (27) and (28), \(x_{2} ( t ) \) is uniformly ultimate bounded below.
Part 4. \(\lim\sup_{t\rightarrow\infty}x_{2} ( t ) \leq K_{2}\), that is to say, \(x_{2} ( t ) \) is uniformly ultimate bounded above.
Now we assume \(v_{1} ( t ) \) oscillates about zero, by (32), we know that \(v_{1} ( t ) \geq0\) implies \(( v_{1} ( t ) ) ^{\Delta}\leq0\). Thus, by the semi-cycle concept, we let \(v_{1} ( t ) \geq0\), for \(t\in [ s_{\alpha},t_{\alpha } ] \), \(s_{\alpha},t_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, the interval \([ s_{\alpha },t_{\alpha} ]\) satisfies (a1)-(e1) by replacing \(u_{1}(t)\) in Part 1 with \(v_{1}(t)\).
Thus from (34) and (37), \(x_{2} ( t ) \) is uniformly ultimate bounded above.
Therefore, from Part 3 and Part 4, \(x_{2} ( t ) \) is uniformly ultimate bounded.
Finally, we choose \(\lambda_{1}=\min\{k_{1},k_{2}\}\), \(\lambda_{2}=\max \{K_{1},K_{2}\}\). This completes the proof of Theorem 3.1. □
4 Discussion
In Theorem 3.1, if we let \(\mathbb{T}=\mathbb {R}\) and \(\mathbb{T}=\mathbb {Z}\), respectively, then the result is exactly changed into Theorem 1.1 (by the comparison theorem) and Theorem 1.3 (by the semi-cycle theory). That is, we provide a unified method to study the permanence for the continuous system and discrete system. In addition, we give a new method to investigate the permanence for the continuous system.
As is well known, the permanence of the periodic biological system is closely associated with the existence for the periodic solutions of the system, in general, when the periodic system is permanent, then there must exist at least one positive periodic solution. By a similar analysis to that in Fan and Wang [27], we can obtain the following remark.
Remark 4.1
Assume (A1), (A2) hold, then system (1) has at least one ω-periodic solution.
This shows that the conditions for uniform ultimate boundedness of solutions are the same as that for the existence of periodic solutions of the system.
5 Numerical example
Declarations
Acknowledgements
This work is supported by NSF of China (11201213, 11371183), NSF of Shandong Province (ZR2015AM026, ZR2013AM004) and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).
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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