Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control
Changyou Wang^{1, 2, 3 *}, Hao Liu^{1, 2}, Shuang Pan^{1, 2}, Xiaolin Su^{3}, Rui Li^{1, 2 *}
^{1}Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing, P.R. China
^{2}College of Automation, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
^{3}College of Science, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
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To cite this article:
Changyou Wang, Hao Liu, Shuang Pan, Xiaolin Su, Rui Li. Globally Attractive of a Ratio-Dependent Lotka-Volterra Predator-Prey Model with Feedback Control. Advances in Bioscience and Bioengineering. Vol. 4, No. 5, 2016, pp. 59-66. doi: 10.11648/j.abb.20160405.13
Received: September 15, 2016; Accepted: September 27, 2016; Published: October 19, 2016
Abstract: By constructing suitable Lyapunov function and developing some new analysis techniques, a Lotka-Volterrapredator-prey system with ratio-dependent functional responses and feedback controls is studied and a sufficient condition which guarantees the globally attractive of positive solution for the predator-prey model is obtained. Moreover, the numerical simulation to the system is given to illustrate our results.
Keywords: Predator-Prey System, Feedback Control,Ratio-Dependent, Globally Attractive
1. Introduction
As is well known, an ecosystem in the real world is continuously distributed by some forces, which can result in changes in the biological parameters such as survival rates. In ecology, we know that the practical question of interest is just whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. This is very significant question in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing feedback control variables. The feedback control mechanism might be implemented by means of some biological control schemes or by harvesting procedure. In fact, during the last decade, the qualitative behaviour of the population dynamics with feedback control has been studied extensively. In [1], Yin and Li proposed the following single species model with feedback regulation and distributed time delay of the following form
(1)
where denotes the density of species at time and is the regulator. By using the continuation theorem of coincidence degree theory, a criterion which guarantees the existence of positive periodic solution of system (1) is obtained. Furthermore, Chen [2] obtains sufficient condition which guarantees the global attractivity of the positive solution of system (1) by constructing a suitable Lyapunov function.
In 2009, Nie et al [3] consider the following non-autonomous predator-prey Lotka-Volterra system with feedback controls
(2)
where is the prey population density and is the predator population density, , , are the intrinsic growth rate and density-dependent coefficient of the prey, respectively; , , are the intrinsic growth rate and density-dependent coefficient of the predator, respectively; is the capturing rate of the predator and is the rate of conversion of nutrient into the reproduction of the predator; are control variables. Authors study whether or not the feedback controls have an influence on the permanence of a positive solution of the general non-autonomous predator-prey Lotka-Volterra type systems, and establish the general criteria on the permanence of system (2), which is independent of some feedback controls. In additional, by constructing a suitable Lyapunov function, some sufficient conditions are obtained for the global stability of any positive solution to system (2). More work on feedback controls can be found in [4-14] and the references cited therein. However, as far as we know, there is very little result about the Lotka–Volterra system with ratio-dependent functional responses and feedback controls.
The objective of this paper is to investigate the asymptotic behavior of the time-dependent solution for a 3-species predator-prey system with feedback control. The systems of equations under consideration are of ratio-dependent Lotka–Volterra type and are described as follows
(3)
where, andstand for the densities of the two preys and one predator, respectively, and , are the indirect control variables. The given coefficients, and are positive continuous bounded functions of for Model (3) describes the interaction between prey and predator species which is based on a ratio-dependent functional response, that is, the rate at which an individual predator species consumes two individual prey species, and two preys compete for the food resource. Thus system (3) is the so-called food web model with one predator and two competing preys. Here, and are the intrinsic growth rate in the absence of predator and the parameters and are the capturing (or catching efficiency) rates of the predator; andare the conversion rates (or maximum growth rates). is the death rate of the predator species. In [15], Wang et al establish the sufficient conditions which guarantee the permanence of systems (3) by developing a new analysis technique. When there are not feedback controls, the systems (3) are reduced to the following Lotka -Volterra model
(4)
where the meaning of the parameters of systems (4) is as same as those of (3). Wang et al [16] show that this system is permanent and globally asymptotically stable under some appropriate conditions by constructing suitable Lyapunov function. Comparing the systems (3) and (4), one could see that we introduce the control variables so as to implement a feedback control mechanism. Here, we will consider the Lotka-Volterra predator-prey system (3) with ratio-dependent functional responses and feedback controls. By constructing suitable Lyapunov function, the sufficient conditions are established for the globally attractive of positive solution for the model.
The plan of the paper is as follows: In Section 2, by constructing a non-negative Lyapunov function, we shall derive sufficient conditions for the globally attractive of positive solution for the predator-prey model (3). Some numerical simulations to the models are given in Section 3.
2. Globally Attractive
In order to establish a globally attractive result for the system (3), we need some preparations. Due to the biological interpretation of the model, it is reasonable to consider only positive solutions of (3), in other words, to take admissible initial conditions. Firstly, we introduce the following notations and definitions. Given a functiondefined on , we set
Definition 2.1. System (3) is called permanent, if there exist positive constants,, , and , such that for any positive solution of (3) as .
Definition 2.2. System (1.3) is said to be globally attractive, if there exists a positive solution of the system (3) such that
for any other positive solution of the system (3).
For the system (3), we let
, , , ,
, ,
, ,
, , ,
Lemma 2.1 (See [15], Theorem 1). Assume that the system (3) satisfies the following conditions
,
,
,,
, ,
then the system (1.3) is permanent.
Lemma 2.2 (See [17], Lemma 8.2). If the function is uniformly continuous, and the limitexists and is finite, then
Theorem 2.1. Assume that the system (3) satisfies and the following conditions
,
,
,
,
then the system (1.3) is globally attractive.
Proof. Let be a positive solution of the system (3) and be any positive solution of the system (1.3) with initial conditions,, , then from Lemma 2.1, there exist positive constants , , and , such that for all .
Set Lyapunov function
Computing the upper right derivative of along with the solution of the system (3), one can easily obtain
In view of conditions ，one has
.
Thus,
(5)
Integrating (5) from to (), one has
Therefore,
(6)
By (6), we have
From the uniformity permanence of the systems (3), is uniformity continuous. By Lemma 2.2, we can obtain that
This ends the proof of Theorem 2.1.
3. Numerical Simulation
In this section, we give some numerical simulations supporting our theoretical analysis. As an example, we consider the following two-prey one-predator model with ratio-dependent functional responses and feedback controls.
(7)
It is easy to show that the two-prey one-predator model (7) satisfy the conditions of Theorem 2.1. It follows from Theorem 2.1 that the predator-prey model (7) is permanent and globally attractive. By employing the software package MATLAB7.1, we can solve the numerical solutions of the equations (7) which are shown in Figures 1, Figures 2, Figures 3 and Figures 4.
Figures 1 shows that the permanence of the systems (7) with the initial conditions . It is not difficult to find that () and () as . From Figures 2, Figures 3 and Figures 4, one can find that for any two solutionsand of the system (7) with the different initial conditions, which show that the predator-prey model (7) is globally attractive.
.
Fig. 2. Dynamic behavior of the first componentof the solutions to system (7) with the different initial conditions.
Fig. 3. Dynamic behavior of the first componentof the solutions to system (7) with the different initial conditions.
Fig. 4. Dynamic behavior of the first componentof the solutions to system (7) with the different initial conditions.
4. Conclusion
This paper presents the use of Lyapunov stability theorem for systems of nonlinear differential equations. This method is a powerful tool for solving nonlinear differential equations in mathematical physics, chemistry and engineering etc. The technique constructing a suitable Lyapunov function provides a new efficient method to handle the nonlinear structure.
We have dealt with the problem of positive solution for a Lotka-Volterra predator-prey system with ratio-dependent functional responses and feedback controls. The general sufficient conditions have been obtained to ensure the global attractive of positive solution for the predator-prey model. Moreover, some numerical simulations to the equation are given to illustrate our results. In particular, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of the Lotka-Volterra predator-prey system.
Remark: Obviously, the models (3) is the extension of model (4). Adding delay term to the proposed model (3) is our next research work.
Acknowledgments
This work is supported Science Fund for Distinguished Young Scholars (cstc2014jc yjjq40004) of China, the National Nature Science Fund (Project nos.11372366 and 61503053) of China, the Science and Technology Project of Chongqing Municipal Education Committee (Grants no. kj1400423) of China, and the excellent talents project of colleges and universities in Chongqing of China.
References