Permanence for a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and feedback controls
- Qin Yue^{1}Email author
https://doi.org/10.1186/s13662-015-0426-6
© Yue; licensee Springer. 2015
Received: 31 July 2014
Accepted: 25 February 2015
Published: 7 March 2015
Abstract
A modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response and feedback controls is studied. By applying the differential inequality theory, sufficient conditions which guarantee the permanence of the system are obtained. Our results improve the main results of Zhang et al. (Abstr. Appl. Anal. 2014:252579, 2014). One example is presented to verify our main results.
Keywords
1 Introduction
Theorem A
([10])
Theorem A shows that feedback control variables play important roles in the persistent property of system (1.4). But the question is whether or not the feedback control variables have influence on the permanence of the system. Many papers (see [11–13] and the references cited therein) have showed that feedback control variables have no influence on the permanent property of continuous system with feedback control. Thus, in this paper, we will apply the analysis technique of Chen et al. [11] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system. In fact, we obtain the following main result.
Theorem B
Comparing with Theorem A, it is easy to see that (H_{2}) in Theorem B is weaker than (H_{1}) in Theorem A, and feedback control variables have no influence on the permanent property of system (1.4), so our results improve the main results in Zhang et al. [10].
The organization of this paper is as follows. In Section 2, we introduce several lemmas, and the permanence of system (1.4) is then studied in this section. In Section 3, a suitable example together with its numerical simulations is given to illustrate the feasibility of the main results.
2 Permanence
Now let us state several lemmas which will be useful in proving the main results of this section.
Lemma 2.1
([14])
Lemma 2.2
([11])
- (i)then for all \(t\geq s\),$$\dot{x}(t)\leq -ax(t)+b(t), $$Especially, if \(b(t)\) is bounded above with respect to M, then$$x(t)\leq x(t-s)\exp\{-as\}+ \int_{t-s}^{t}b(\tau) \exp\bigl\{ a(\tau-t)\bigr\} \, d\tau. $$$$\limsup_{t\rightarrow +\infty}x(t)\leq \frac{M}{a}. $$
- (ii)then for all \(t\geq s\),$$\dot{x}(t)\geq -ax(t)+b(t), $$Especially, if \(b(t)\) is bounded above with respect to m, then$$x(t)\geq x(t-s)\exp\{-as\}+ \int_{t-s}^{t}b(\tau) \exp\bigl\{ a(\tau-t)\bigr\} \, d\tau. $$$$\liminf_{t\rightarrow +\infty}x(t)\geq \frac{m}{a}. $$
The following lemma is a direct conclusion of [10].
Lemma 2.3
Lemma 2.4
Proof
Lemma 2.5
Proof
The proof of Lemma 2.5 is similar to the proof of Lemma 2.4. However, for the sake of completeness, we give the complete proof here.
3 Examples and numeric simulations
Declarations
Acknowledgements
The author would like to thank the two anonymous referees for their constructive suggestions on improving the presentation of the paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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