Dynamics in a ratio-dependent Lotka–Volterra competitive-competitive-cooperative system with feedback controls and delays

This study investigates the dynamical behavior of a ratio-dependent Lotka–Volterra competitive-competitive-cooperative system with feedback controls and delays. Compared with previous studies, both ratio-dependent functional responses and time delays are considered. By employing the comparison method, the Lyapunov function method, and useful inequality techniques, some sufficient conditions on the permanence, periodic solution, and global attractivity for the considered system are derived. Finally, a numerical example is also presented to validate the practicability and feasibility of our proposed results.


Introduction
As is well known, competition and mutualism(cooperation) are two important interactions among species. Competition occurs when two species use the same resources or harm each other when seeking resources, whereas mutualism is defined as the living of two species in close association with one another with the benefit of both [1]. Notably, the Lotka-Volterra models were proposed by Lotka [2] and Volterra [3] for the first time, and now they have become the most important means to explain this type of ecological phenomenon. In particular, the Lotka-Volterra competitive model, mutualism (cooperative) model, and predator-prey model characterize competitive, cooperative, and predator-prey interactions between species that are of great interest in the study of dynamical behaviors of systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. However, pure competition as described by the Lotka-Volterra model often results in species exclusion or coexistence with reduced carrying capacity of both species and does not help the coexistence of multiple species, although it is a driving force for natural selection [1]. Hence, when modeling we should consider more interactions between species such as competition, cooperation, and predator-prey [18][19][20][21][22][23][24][25][26].
However, to the best of our knowledge, no study has been conducted to date for dynamics on the three-species ratio-dependent Lotka-Volterra competitive-competitivecooperative system with feedback controls and delays. Therefore, based on the above models, analysis, and reasons, in this study we extend systems (1.2) and (1.3) to the following system: (1.4) The aim of this study is to use the inequality techniques, comparison method and to construct suitable Lyapunov functionals to establish some new and sufficient conditions on the permanence, periodic solution, and global attractivity for system (1.4).
In this paper, the initial conditions for system (1.4) take the following form: For system (1.4) we introduce the following assumptions: For a continuous and bounded function In this paper, we need the following definition and lemmas.

Permanence and periodic solution
In this section, we obtain some new and sufficient conditions for the permanence and periodic solution of system (1.4).
Proof From the first equation of system (1.4), for t ≥ τ , we havė Then, by Lemma 2.1 and the comparison theorem, there exists a constant T 0 > 0 such that By the fourth equation of system (1.4), for t > T 0 , we havė Then, by Lemma 2.2 and the comparison theorem, there exists a constant T 1 > 0 such that From the fifth and sixth equation of system (1.4), for t ≥ τ and i = 2, 3, we havė Then, by Lemma 2.2 and the comparison theorem, there exists a constant T 2 > 0 such that Next, from the second equation of system (1.4), for t > T 2 , we havė Then, by Lemma 2.1 and the comparison theorem, there exists a constant T 3 > 0 such that Similar to the above discussion, from the third equation of system (1.4), there exists a constant T 4 > 0 such that On the other hand, from the third equation of system (1.4), for t ≥ τ , we havė Then, by Lemma 2.1 and the comparison theorem, there exists a constant T 5 > 0 such that Finally, from the fourth equation of system (1.4), for t ≥ τ , we havė Then, by Lemma 2.2 and the comparison theorem, there exists a constant T 6 > 0 such that This completes the proof of Theorem 3.1.

Theorem 3.2 Assume that (H 1 ) holds and
Then, by Lemma 2.2 and the comparison theorem, there exists a constant T M 2 > T M 1 such that Next, from the fifth and sixth equation of system (1.4), we can obtain a sufficiently large positive constant T N 1 such thaṫ Then, by Lemma 2.2 and the comparison theorem, there exists a constant T N 2 > T N 1 such that This completes the proof of Theorem 3.2.
As a direct result of Lemma 2.3, from Theorem 3.1 and Theorem 3.2 we have the following.

One example
In this section one example is given to illustrate the effectiveness of our results obtained in this paper.
Example We consider the following system: By direct calculation, we can get  It is easy to show that system (5.1) satisfies the conditions of Theorem 3.2, Theorem 4.1, Corollary 3.1, and Corollary 4.1. Hence, system (5.1) is permanent, globally attractive and has a globally attractive positive periodic solution.
From Fig. 1 we can see that system (5.1) is permanent and has a globally attractive positive periodic solution.

Conclusion
In this study, we are concerned with system (1.4). First, using the inequality techniques and the comparison method, we obtained a set of conditions that ensure that the system is permanent and at least has a positive periodic solution. Second, using the Lyapunov function method, we derived sufficient conditions on the global attractivity of the system. Finally, we provided a suitable example to illustrate the feasibility of our main results. Because we extended systems (1.2) and (1.3) to system (1.4), we also obtained some sufficient conditions for the permanence, periodic solution, and global attractivity of system (1.4). Hence, system (1.4) and the results obtained in this study can be seen as the supplements and extensions of previously known studies [13,17,20].