Dynamical behaviour of a Lotka–Volterra competitive-competitive–cooperative model with feedback controls and time delays

The aim of this paper is to investigate the dynamical behaviour of a class of three species Lotka–Volterra competitive-competitive–cooperative models with feedback controls and time delays. By developing a new analysis technique, we obtain some sufficient conditions that ensure these models have the dynamical property of permanence. We also give some sufficient conditions that guarantee the global attractivity of positive solutions for this system by constructing a new suitable Lyapunov function. Finally, we give some numerical simulations to illustrate our results in this paper.


Introduction
The modelling and analysis of the dynamics of biological populations by means of differential equations are of the primary concern in population growth problems. A well-known and extensively studied class of models in population dynamics is the Lotka-Volterra system which models certain types of interactions among various species. In the real world, the growth rate of a natural species will not often respond immediately to changes in its own species or that of an interacting species, but will rather do so after a time lag. Time delays are introduced to make the model respond better to impersonal law (see, [1][2][3][4][5][6][7][8][9][10][11]).
Lu et al. in [2] proposed and studied the following Lotka-Volterra system with discrete delays with initial conditions x i (t) = φ i (t) ≥ 0, t ∈ [−τ 0 , 0]; φ i (0) > 0, (i = 1, 2) CONTACT Changyou Wang wangchangyou417@163.com; Linrui Li linrui020213@163.com; Qiuyan Zhang zqy1607@cuit.edu.cn; Rui Li liruimath@qq.com where r i , a i , a ij and τ ij are constants with a i > 0, a ij ≥ 0(i, j = 1, 2) and τ 0 = max{τ ij : i, j = 1, 2}, φ ij is continuous on [−τ 0 , 0]. They show that delays can change the permanence for Lotka-Volterra cooperative systems. For certain delays with the same length, the delayed system has a similar property to the corresponding system without delays in the sense of permanence, but for a general delay case, the delays may destroy the permanence for the system. In 2010, Nakata and Muroya considered the following nonautonomous Lotka-Volterra cooperative systems with time delays (see, [3]) where x i (t)(i = 1, 2) denote the density of i-species at time t, τ is a positive constant and r i (t), a l ij (t) (1 ≤ i, j ≤ 2; 0 ≤ l ≤ 2) are continuous, bounded and strictly positive functions as t ∈ [−τ , +∞). They obtained some sufficient conditions for the permanence of the system (2). Xu and Zu [4] investigated the following two-species delayed competitive model with stage structure and harvesting By using the differential inequality theory, some new sufficient conditions which ensure the permanence of the system are established. In [5], the authors considered the following competitor-competitor-mutualist Lotka-Volterra systems with discrete time delays ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ẋ And some sufficient conditions which guarantee the boundedness, permanence and global attraction for system (4) were obtained. In 2011, Xu et al. [6] studied the dynamical behaviours for the following Lokta-Volterra predator-prey model with two delays Its linear stability and Hopf bifurcation are investigated by analysing the associated characteristic transcendental equation. Some explicit formulate for determining the stability and the direction of the Hopf bifurcation periodic solutions are obtained by using normal form theory and centre manifold theory.
One can find that 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. The practical interest in ecology is the question of whether or not an ecosystem can withstand those disturbances which persist for a finite period of time. In the control systems, we regard the disturbance functions as control variables. These are of significance in the control of ecology balance. One of the methods to research 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 harvesting procedure. In fact, during the last decade, the qualitative behaviour of the population dynamics with feedback control has been studied extensively. In 2009, Nie et al. [12] considered the following non-autonomous predator-prey Lotka-Volterra system with feedback controls where x 1 (t) is the prey population density and x 2 (t) is the predator population density, b 1 (t) and a 11 (t) are the intrinsic growth rate and density-dependent coefficient of the prey, respectively; b 2 (t) and a 22 (t) are the intrinsic growth rate and densitydependent coefficient of the predator, respectively; a 12 (t) is the capturing rate of the predator and a 21 (t) is the rate of conversion of nutrient into the reproduction of the predator; u i (t)(i = 1, 2) are control variables. They studied whether or not the feedback controls have an influence on the permanence of a positive solution of the general nonautonomous predator-prey Lotka-Volterra-type systems and establish the general criteria on the permanence of system (6), which is independent of some feedback controls. In addition, by constructing an appropriate Lyapunov function, some sufficient conditions are obtained for the global stability of any positive solution to system (6). In [13], Yang, Wang and Chen proposed and studied the following cooperation system with feedback controls where b i , a ij , α i , η i , a i , i, j = 1, 2 are positive constants. x i (t), (i = 1, 2) are the densities of the species at time t, u i (t), (i = 1, 2) denote feedback controls. They showed that if system (7) has a positive equilibrium, then feedback controls can only influence the position of the positive equilibrium, and have no influence on the stability. In 2018, Wang et al. [14] considered the following three-species Lokta-Volterra predator-prey system with By using a comparison theorem and constructing a suitable Lyapunov function as well as developing some new analysis techniques, the authors established a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the global attractivity of positive solution for the predator-prey system (8). Furthermore, some conditions for the existence, uniqueness and stability of a positive periodic solution for the corresponding periodic system were obtained by using the fixed point theory and some new analysis method. More work on feedback controls can be found in (cf. [15][16][17][18][19][20][21] and the references cited therein). As is known to all, the Lotka-Volterra system with time delay and feedback control can respond better to impersonal law. In recent years, more and more attention has been paid to some ecosystem models with both feedback control and time delay (see, [22][23][24][25][26][27]). In 1993, Gopalsamy et al. [22] studied a class of autonomous single-species ecosystem with feedback control and time delay where u(t) denotes an indirect control variable, τ , a 2 , a, b, c, r ∈ (0, ∞) and a 1 ∈ [0, ∞). Some sufficient conditions were obtained for the global asymptotic stability of the positive equilibrium for the system (9). In order to show that whether the feedback control variables play an essential role on the persistent property of Lotka-Volterra cooperative systems or not, Xu and Chen [26] established and studied the following system with time delay and feedback control They obtained some new sufficient conditions which ensured the system to be permanent, and showed that feedback control variables had no influence on the permanence of the system. In 2017, Xu and Li [27] considered the following competition and cooperation model of two enterprises with multiple delays and feedback controls (11) Some sufficient conditions that guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution for the system (11) were obtained by constructing a suitable Lyapunov functional and using the comparison theorem of differential equations.
However, as far as we know, no work has been done until now for the three-species Lotka-Volterra system with feedback control and time delay. Motivated by the above work, we propose and investigate the following three species Lotka-Volterra competitive-cooperative model with feedback controls and time delays where x i (t), i = 1, 2, 3 stands for the densities of the species at time t, and u i (t), i = 1, 2, 3 are the indirect control variables. The given coefficients a 12 (t), a 13 (t), a 21 (t), a 23 (t), a 31 (t), , a l 33 (t), (l = 1, 2) are continuous, bounded and strictly positive functions on [0, ∞). r i (t), (i = 1, 2, 3) denote the intrinsic growth rate of the i-th species at time t. Especially, a l ii (t), (i = 1, 2, 3, l = 1, 2) denote the internal competitive coefficient of the three species at time t. a 12 Due to the biological interpretation of the system (8), it is reasonable to consider only positive solution of the system (8), in other words, take admissible initial conditions Obviously, the solutions of system (12) with the initial values (13) are positive for all t ≥ 0.
Comparing the systems (4) and (12), one could see that we introduce the control variables u i (t) (i = 1, 2, 3) so as to implement a feedback control mechanism. Our main purpose in this paper is to establish some sufficient conditions which ensure the system to be permanence and global attractivity by constructing a new appropriate Lyapunov function and developing a new analysis technique. This paper is organized as follows: In Section 2, we provide the conditions for the permanence to system (12). In Section 3, by constructing a nonnegative Lyapunov function, we shall derive sufficient conditions for the global attractivity of positive solution for the Lotka-Volterra Competitive-Competitive-Cooperative model (12). Some numerical simulations to the system are given in Section 4.

Permanence
In order to establish a permanence result for the system (12), we need some preparations. Firstly, we introduce the following notations and definitions. Given a function g(t) defined on [t 0 + ∞), we set As a direct corollary of Lemma 2.1 of Chen [1], we have.

Lemma 2.3 (see [3], Lemma 2.3):
Assume that for y(t) > 0, it holds thaṫ If the system (14) holds, then, there exists a positive constant m y > 0 such that For the system (12), let Then the system (12) is permanent.
Proof. By the fifth equation of system (8), we havė Moreover, similar to the above discussion of the fifth equation of system (12), from the sixth of system (12), we have Next, suppose that lim sup and From system (12), one has From (18), (19), we can obtain Thus, by the assumption of the Theorem 2.1 and (20), it holds that Moreover, by (19) and the assumption of the Theorem 2.1, it follows that By integrating both sides of (21) from t k − τ to t k further, we have Therefore However, it leads to a contradiction with (17). Thus, we have Moreover, similar to the above discussion, we can also obtain that According to the first equation of system (21) and (22), it follows thaṫ where x * 1 is the unique positive solution of equation x 1 [r m 1 − (a 1l 11 + a 2l 11 )x 1 ] + a m 13 P 1 = 0.
Similar to the above discussion of the first equation of system (12), from (23) and the second equation of system (12), we obtaiṅ where x * 2 is the unique positive solution of the following equation From the third equation of system (12), we obtaiṅ By Lemma 2.2, it holds that where x * 3 is the unique positive solution of the following equation From the fourth equation of system (12), one haṡ Thus, according to Lemma 2.1, it follows that On the contrary, from the first equation of system (12), we havė By Lemma 2.3, one easily verifies that By the same way, from the second and third equations of system (12), we deducė Thus, by Lemma 2.3, we have According to the fourth equation of system (12), we havė Similarly, from the fifth and sixth equations of system (12), it follows thaṫ Moreover, by Lemma 2.1, it follows that and From (15), (16), and (24)-(33), this completes the proof of Theorem 2.1.

Globally attractive
In this section, we shall prove that the system (12) is globally attractive. To get the sufficient conditions for globally attractive of system (12), we give firstly the following definition and Lemma.
, v 3 (t)) are any two different positive solutions of the system (12). Then from Theorem 2.1, there exist positive constants M i , N i , m i , n i , i = 1, 2, 3 and T, such that Calculating the upper right derivative of V 11 (t) along the solution of system (12), we obtain Next, we define that Then, from (34) and (35), we have Define Let According to (36) and (37), calculating the upper right derivative of V 1 (t), we have Similarly, we define V 21 (t) = |ln x 2 (t) − ln y 2 (t)|, then one obtain On the other hand, define From (40), (41), we have Furthermore, define Let From (42) and (43), we can get the upper right derivative of V 2 (t) Similarly, we define Then, it follows that Let Then, we have Take Moreover, we take Then, we have Take and Moreover, we give a Lyapunov function as follows from (39), (45), (51)-(54), we can obtain for all t ≥ t + τ . In view of the conditions of Theorem 3.1, there exist a constant α > 0 and T * ≥ T + τ such that for all t ≥ T * , it holds that Integrating from T * to t on both sides of (55) and by (56), we have Therefore, V(t) is bounded on [T * , +∞), and we have By (58), we have From the uniformity permanence of the system (12), is uniformity continuous on [T * , +∞). By Lemma 3.1, we can obtain This completes the proof of Theorem 3.1.

Numerical simulation
In this section, we give some numerical simulations to support our theoretical analysis. As an example, we consider the following Lotka-Volterra competitivecompetitive-cooperative model with feedback controls and time delays and choose the periodic function as the coefficients of the model   It is easy to show that the system (60) satisfies the conditions of Theorem 3.1. It follows from Theorem 3.1 that the Lotka-Volterra competitive-competitive-cooperative model (60) is permanent and globally attractive. By employing the software package MATLAB 7.1, we can solve the numerical solutions of the system (60) which are shown in Figures 1-3. Figure 1 shows that the permanence of the systems (60) with time delay τ = 0.075 and the  From Figure 2, it is not difficult to find that the system (60) is globally attractive. Figure 3 shows the dynamical behaviour of the systems (60).

Conclusion
This paper presents the use of Lyapunov stability theorem for system 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 an appropriate Lyapunov function provides a new efficient method to handle the nonlinear structure with time delay and feedback control.
We have dealt with the problem of positive solution for a class of three-species Lotka-Volterra competitive-competitive-cooperative with feedback controls and time delays. By developing some new analysis techniques and constructing a new suitable Lyapunov function, we obtain some sufficient conditions which ensure the system to be permanent and globally attractive. Our results show that feedback control variables and time delay terms have influence both the persistent property and global attractive of system (12). Moreover, some numerical simulations to the system (60) are given to illustrate our results obtained in this paper. In particular, the sufficient conditions that we obtained are very simple and practical, which provide flexibility for the application and analysis of the Lotka-Volterra models with feedback controls and time delays.
Remark: The main contribution and innovation of this paper are as follows: (1) The control variables are introduced to the known model (4) to implement a feedback control mechanism, and the new model can better describe the interactions among multi-species. Obviously, system (4) is the special case of the new system (12). To the best of the author's knowledge, this is the first time such a system is proposed. (2) To study the new model, we obtain some new methods and skills (such as the new structure method of the Lyapunov function, the applications of delay differential inequalities) that can also be used to research other related models with multi-delays and feedback controls. Because of the complexity of the new system, the structure of the Lyapunov function is completed step by step to overcome the difficulties brought about by the multiple time delays and feedback controls, please see pages 10-17. (3) In this paper, the research contents are richer than the related references. We study not only the permanence and global attractivity of the new system but also some numerical simulations to the new system (12) are given to illustrate our results obtained in this paper. (4) The sufficient conditions obtained herein are new, general, and easily verifiable, which provide flexibility for the application and analysis of three-species multi-delays Lotka-Volterra predator-prey model with feedback controls.

Disclosure statement
No potential conflict of interest was reported by the authors.