Abstract
In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.
1. Introduction
In recent few decades, population dynamical models have received considerable attention [Citation13, Citation19, Citation31, Citation43, Citation57]. In the research of population dynamical models, competitive systems play an important role in describing the interaction among the multi-species. A lot of competitive systems have been explored by numerous scholars. For example, Hou [Citation16] analysed the permanence of a competitive Lotka–Volterra system with delays, Balbus [Citation2] addressed the attractivity and stability in a competitive system of PDEs of Kolmogorov type, Shi et al. [Citation39] focused on the extinction of a nonautonomous Lotka–Volterra competitive system with infinite delay and feedback controls, Liu and Wang [Citation32] discussed the asymptotic behaviour of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations, Kulenovi and Nurkanovi [Citation20] made a theoretical discussion on the global behaviour of a two-dimensional competitive system of difference equations with stocking. For more detailed publications on the permanence and global attractivity behaviour of predator–prey models, one can see [Citation4, Citation6, Citation7, Citation9, Citation11, Citation12, Citation15, Citation17, Citation18, Citation21, Citation23–25, Citation28–30, Citation33–36, Citation40, Citation41, Citation44, Citation45, Citation47–52, Citation55, Citation56].
In 1992, Gopalsamy [Citation14] explored the following competitive model on predator species and prey species: (1) (1) where stand for the densities of two competing species at time t, respectively. denote the intrinsic growth rates of predator species and prey species, represent the effects of intra-specific competition, and are the effects of inter-specific competition. Considering that the parameters, in natural world, often vary due to the change of surroundings, we think that it is important to introduce the time-varying parameters into the predator–prey model. Varying parameters will lead to much more different dynamical behaviour in predator–prey models than that of predator–prey models with fixed parameters. Motivated by this idea, we can modify system (Equation1(1) (1) ) as the form (2) (2) where are functions with respect to the time t. Since predator species and prey species live in a real fluctuating environment and many exploitation activities of people might lead to abrupt vary, Tan et al. [Citation42] set up the following impulsive competitive model: (3) (3) where and N is the set of positive integers, all the coefficients are all continuous almost periodic functions which are bounded above and below by positive constants, and are constants and are impulse points with .
Considering that two species are constantly in the competition, and when a species suffers damage from another one by competition, another one could benefit, the duration time of density for species would also play an important role, we modify system (Equation1(1) (1) ) as the following form (4) (4) where stands for the hunting delay. Many scholars [Citation1, Citation3, Citation5, Citation8, Citation10, Citation22, Citation26, Citation27, Citation37, Citation38, Citation53, Citation54] argue that discrete dynamical models which are described by difference equations are often regarded as more suitable tools to depict the dynamical relationship among different species than continuous ones since the species owns non-overlapping generations. Discrete dynamical model is also thought to be a very useful tool to carry out numerical simulations for the continuous ones. In addition, we would like to point out that external force often make the parameters of biological systems vary. For example, the competition and cooperation model of two enterprises is usually interfered by manpower, material resources, financial resources and so on. What we are interested in is how to real the dynamical behaviour of biological systems which are affected by external force. In control language, the external force can be called as control variables. From a mathematical point of view, it is of great interest to investigate the effect of control variables on the dynamics for biological systems. Stimulated by the analysis above, we can modify system (Equation4(4) (4) ) as follows (5) (5) where and denote the densities of two competing species at the generation, respectively, and is the control variable. are bounded nonnegative sequences and are integer-valued sequences. As far as I know, it is first time to probe into system (Equation5(5) (5) ) involving feedback control. We believe that this work on the permanence and global attractivity of the discrete competitive model has significant theoretical meaning and tremendous potential application in preserving population coexistence and maintaining ecological balance.
The chief task of this study is to probe into the permanence and global attractivity of system (Equation5(5) (5) ). In order to set up the key conclusions, the following assumptions are needed: Denote and where is bounded sequence. Let . For system (Equation5(5) (5) ), we give the initial value as follows: (6) (6) It is not difficult to see that solutions of (Equation5(5) (5) ) and (Equation6(6) (6) ) are well defined for all and satisfy We plan the structure of this work as follows. Section 2 presents the necessary definitions and lemmas and states the permanence result of system (Equation5(5) (5) ) Section 3 deals with the existence of periodic solution and global attractivity of periodic solution for system (Equation5(5) (5) ) without delay. Section 4 carries out computer simulation to illustrate the feasibility and effectiveness of our results derived is Sections 2 and 3. A brief conclusion is drawn in Section 5.
2. Permanence
In this section, we firstly introduce the definition of permanence and several lemmas which is needed to set up the key results.
Definition 2.1
We say that system (Equation5(5) (5) ) is permanent provided that ∃ two positive constants and m such that for each positive solution of system (Equation5(5) (5) ) satisfies Give the single species discrete model as follows: (7) (7) where and are strictly positive sequences of real numbers defined for and . Similarly to the proofs of Propositions 1 and 3 in [Citation3], one can obtain the following result.
Lemma 2.1
Every solution of system (Equation7(7) (7) ) with initial value satisfies where
Give the first order difference equation (8) (8) where and are positive constants. Following Theorem 6.2 of Wang and Wang [Citation46, page 125], one has the following result.
Lemma 2.2
[Citation46]
Assume that , for any initial value , there exists a unique solution of (Equation8(8) (8) ) which can be expressed as follows: where . Thus, for any solution of system (Equation8(8) (8) ), .
Lemma 2.3
[Citation46]
Let . For any fixed n, is a nondecreasing function with respect to r, and for , the following inequalities hold: If , then for all .
Proposition 2.1
For system (Equation5(5) (5) ), assume that (H1) holds, then where
Proof.
Let be any positive solution of system (Equation5(5) (5) ) with the initial condition . It follows from system (Equation5(5) (5) ) that (9) (9) Set , then (Equation9(9) (9) ) is equivalent to (10) (10) By virtue of (Equation10(10) (10) ), one has which leads to (11) (11) Then (12) (12) By means of (Equation12(12) (12) ) and system (Equation5(5) (5) ), one gets (13) (13) It follows from (Equation13(13) (13) ) and Lemma 2.1 that (14) (14) ∀ , it follows (Equation14(14) (14) ) that ∃ such that ∀ (15) (15) In view of the third and fourth equations of the system (Equation5(5) (5) ), one can easily get (16) (16) Then (17) (17) Applying Lemmas 2.2 and 2.3, we have (18) (18) Setting , we get (19) (19) which completes the proof.
Theorem 2.1
Let and be defined by (Equation14(14) (14) ) and (Equation19(19) (19) ), respectively. Assume that (H1) and (H2) hold, then system (Equation5(5) (5) ) is permanent.
Proof.
By applying Proposition 2.1, we are easy to see that to end the proof of Theorem 2.1, it is enough to show that under the conditions of Theorem 2.1, In view of Proposition 2.1, ∀ , ∃ for all , (20) (20) It follows from system (Equation5(5) (5) ) and (Equation20(20) (20) ) that ∀ , (21) (21) Set , then (Equation21(21) (21) ) is equivalent to (22) (22) In view of (Equation22(22) (22) ), we obtain (23) (23) Then (24) (24) Thus (25) (25) Applying (Equation25(25) (25) ) and (Equation5(5) (5) ), one has (26) (26) In view of Lemmas 2.1 and 2.3, we get (27) (27) where (28) (28) Setting in (Equation28(28) (28) ), we have (29) (29) where (30) (30) Without loss of generality, we assume that . For any positive constant ε small enough, it follows from (Equation29(29) (29) ) that there exists enough large which satisfies (31) (31) ∀ . By means of (Equation5(5) (5) ) and (Equation31(31) (31) ), one gets (32) (32) Hence (33) (33) By Lemmas 2.1 and 2.2, we get (34) (34) Setting in the above inequality leads to (35) (35) which completes the proof.
3. Existence and stability of periodic solution of system (5) without delay
In this section, we are to discuss the stability of system (Equation5(5) (5) ) with , that is, we discuss the following system (36) (36) Throughout this section we always assume that are all bounded negative periodic sequences with a common periodic ω and satisfy (37) (37) Also it is assumed that the initial values of (Equation36(36) (36) ) are given by (38) (38) Applying the similar approach, under some conditions, we can obtain the permanence of system (Equation36(36) (36) ). We still let and be the upper bound of and , and and be the lower bound of and .
Theorem 3.1
In addition to (Equation37(37) (37) ), suppose that (H1) and (H2) hold, then system (Equation36(36) (36) ) owns a periodic solution .
Proof.
Let . It is easy to see that Ω is an invariant set of system (Equation36(36) (36) ). Then we can define a mapping F on Ω by (39) (39) for . Clearly, F depends continuously on , then F is continuous and maps a compact set Ω into itself. So, F owns a fixed point . Thus one can know that system (Equation36(36) (36) ) owns the solution which passes through . This ends the proof of Theorem 3.1.
Next, we are to deal with the globally stability property of the periodic solution derived in Theorem 3.1.
Theorem 3.2
In addition to the conditions of Theorem 3.1, suppose that the following condition (H3) holds, then the ω periodic solution obtained in Theorem 3.1 is globally attractive.
Proof.
Assume that is any positive solution of system (Equation36(36) (36) ). Let (40) (40) To finish the proof, we only need to prove that (41) (41) Notice that (42) (42) where . In a similar approach, one gets (43) (43) where . Also, one has (44) (44) (45) (45) By (H3), we can choose a such that (46) (46) In view of Proposition 2.1 and Theorem 2.1, ∃ which satisfies (47) (47) It follows from (Equation42(42) (42) )m and (Equation43(43) (43) ) that (48) (48) (49) (49) Also, for , one has (50) (50) (51) (51) Let , then . It follows from (Equation48(48) (48) )–(Equation51(51) (51) ) that (52) (52) for . Then we get (53) (53) Thus (54) (54) This completes the proof.
Remark 3.1
In [Citation14, Citation42], the authors dealt with the continuous or pulsing competitive model without time delays and feedback controls. In this paper, we consider the practical situation and introduce time delays and feedback controls. Based on this viewpoint, the acquired outcomes of our work are new and replenish the outcomes of [Citation14, Citation42].
Remark 3.2
We do not investigate the existence of periodic solution and global attractivity of system (Equation5(5) (5) ) since the introduction of delay leads to the difficulties in analysis methods. We leave this aspect for future work.
4. Examples
In this section, we will execute computer simulations via Matlab software to confirm the rationality of our derived key conclusions.
Example 4.1
Consider the following system (55) (55) where . Then . We can verify that all the hypotheses in Theorem 2.1 are true. Thus one can easily know that system (Equation55(55) (55) ) is permanent. The computer simulation figures are clearly presented in Figures .
Example 4.2
Consider the following system (56) (56) where . Then . We can verify that all the hypotheses in Theorem 3.1 are true. Thus one can easily know that system (Equation56(56) (56) ) owns globally attractive periodic solution. The computer simulation figures are clearly presented in Figures .
5. Conclusions
In this current work, we propose a new discrete predator–prey competitive model involving delays and feedback controls. By virtue of the difference inequality knowledge, a novel sufficient condition guaranteeing the permanence of the system is set up. We find that under a suitable parameter conditions, two species will keep a state of coexistence. The study reveals that feedback control effect and time delays play a vital role in remaining the co-existence of two species In addition, we also obtain the sufficient conditions which ensure the existence and stability of unique globally attractive periodic solution of the system without time delays. The derived results own significant theoretical guiding value in keeping a balance of biological population. Meanwhile, our results are new and supplement the existed results in [Citation14, Citation42].
Data availability statement
No data were used to support this study.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
References
- R.P. Agarwal, Difference equations and inequalities: theory, method and applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, 2nd ed., Marcel Dekker, New York, NY, USA, 2000.
- J. Balbus, Attractivity and stability in the competitive systems of PDEs of Kolmogorov type, Appl. Math. Comput. 237 (2014), pp. 69–76.
- F.D. Chen, Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems, Appl. Math. Comput. 182 (2006), pp. 3–12.
- F.D. Chen, Permanence of a discrete n-species food-chain system with time delays, Appl. Math. Comput. 185 (2007), pp. 719–726.
- F.D. Chen, Permanence for the discrete mutualism model with time delays, Math. Comput. Model. 47 (2008), pp. 431–435.
- F.D. Chen, Permanence in a discrete Lotka-Volterra competition model with deviating arguments, Nonlinear Anal. Real World Appl. 9 (2008), pp. 2150–2155.
- F.D. Chen, Y.M. Chen, Z. Li, and L.J. Chen, Note on the persistence and stability property of a commensalism model with Michaelis-Menten harvesting and Holling type II commensalistic benefit, Appl. Math. Lett. 134 (2022), Article ID 108381.
- X. Chen and F.D. Chen, Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control, Appl. Math. Comput. 181 (2006), pp. 1446–1454.
- F.D. Chen and M.S. You, Permanence for an integrodifferential model of mutualism, Appl. Math. Comput. 186 (2007), pp. 30–34.
- Y.M. Chen and Z. Zhou, Stable periodic of a discrete periodic Lotka-Volterra competition system, J. Math. Anal. Appl. 277 (2003), pp. 358–366.
- J. Dhar and K.S. Jatav, Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories, Ecol. Complex. 16 (2013), pp. 59–67.
- X.H. Ding and C. Liu, Existence of positive periodic solution for ratio-dependent N-species difference system, Appl. Math. Model. 33 (2009), pp. 2748–2756.
- N.H. Du and N.H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differ. Equ. 250 (2011), pp. 386–409.
- K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
- X.Y. Guan and F.D. Chen, Dynamical analysis of a two species amensalism model with Beddington-DeAngelis functional response and Allee effect on the second species, Nonlinear Anal. Real World Appl. 48 (2019), pp. 71–93.
- Z.Y. Hou, On permanence of all subsystems of competitive Lotka-Volterra systems with delays, Nonlinear Anal. Real World Appl. 11 (2010), pp. 4285–4301.
- Z.Y. Hou, On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates, Nonlinear Anal. Real World Appl. 14 (2013), pp. 960–975.
- H.X. Hu, Z.D. Teng, and H.J. Jiang, On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal. Real World Appl. 10 (2009), pp. 1803–1815.
- H.F. Huo, Permanence and global attractivity of delay diffusive prey-predator systems with the michaelis-menten functional response, Comput. Math. Appl. 49 (2005), pp. 407–416.
- M.R.S. Kulenović and M. Nurkanović, Global behavior of a two-dimensional competitive system of difference equations with stocking, Math. Comput. Model. 55 (2012), pp. 1998–2011.
- X.D. Li and J.D. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE. Trans. Automat. Contr. 62 (2017), pp. 3618–3625.
- Y.K. Li and L.H. Lu, Positive periodic solutions of discrete n-species food-chain systems, Appl. Math. Comput. 167 (2005), pp. 324–344.
- X.D. Li and S.J. Song, Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays, IEEE Trans. Neural Netw. Learn. Syst. 24 (2013), pp. 868–877.
- X.D. Li and S.J. Song, Stabilization of delay systems: delay-dependent impulsive control, IEEE. Trans. Automat. Contr. 62 (2017), pp. 406–411.
- X.D. Li and J.H. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica 64 (2016), pp. 63–69.
- X.P. Li and W.S. Yang, Permanence of a discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls, Discrete. Dyn. Nat. Soc. 2008 (2008), Article ID 149267, 8 pp.
- X.P. Li and W.S. Yang, Permanence of a discrete model of mutualism with infinite deviating arguments, Discrete. Dyn. Nat. Soc. 2010 (2010), Article ID 931798, 7 pp.
- Z.C. Li, Q.L. Zhao, and D. Ling, Chaos in a discrete population model, Discrete. Dyn. Nat. Soc. 2012 (2012), Ariticle ID 482459, 14 pp.
- X.Y. Liao, S.F. Zhou, and Y.M. Chen, Permanence and global stability in a discrete n-species competition system with feedback controls, Nonlinear Anal. Real World Appl. 9 (2008), pp. 1661–1671.
- S.Q. Liu and L.S. Chen, Necessary-sufficient conditions for permanence and extinction in lotka-volterra system with distributed delay, Appl. Math. Lett. 16 (2003), pp. 911–917.
- M. Liu and K. Wang, A note on a delay Lotka-Volterra competitive system with random perturbations, Appl. Math. Lett. 26 (2013), pp. 589–594.
- M. Liu and K. Wang, Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations, Math. Comput. Model. 57 (2013), pp. 909–925.
- Z.J. Liu, S.M. Zhong, and X.Y. Liu, Permanence and periodic solutions for an impulsive reaction-diffusion food-chain system with ratio-dependent functional response, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), pp. 173–188.
- C. Lu and L.J. Zhang, Permanence and global attractivity of a discrete semi-ratio dependent predator-prey system with Holling II type functional response, J. Appl. Math. Comput. 33 (2010), pp. 125–135.
- D. Mukherjee, Permanence and global attractivity for facultative mutualism system with delay, Math. Methods Appl. Sci. 26 (2003), pp. 1–9.
- Y. Muroya, Permanence and global stability in a Lotka-Volterra predator-prey system with delays, Appl. Math. Lett. 16 (2003), pp. 1245–1250.
- Y. Muroya, Persistence and global stability in Lotka-Volterra delay differential systems, Appl. Math. Lett. 17 (2004), pp. 795–800.
- Y. Muroya, Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems, Tokyo J. Math. 28 (2005), pp. 189–200.
- C.L. Shi, Z. Li, and F.D. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Nonlinear Anal. Real World Appl. 13 (2012), pp. 2214–2226.
- Y.G. Sun and S.H. Saker, Positive periodic solutions of discrete three-level food-chain model of Holling type II, Appl. Math. Comput. 180 (2006), pp. 353–365.
- Y. Takeuchi, Global Dynamical Properties of Lotla-Volterra Systems, World Scientific, River Edge, NJ, USA, 1996.
- R.H. Tan, W.F. Liu, Q.L. Wang, and Z.J. Liu, Uniformly asymptotic stability of almost periodic solution for a competitive system with impulsive perturbations, Adv. Differ. Equ. 2014 (2014), 11 pp.
- X.H. Tang, D.M. Cao, and X.F. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay, J. Differ. Equ. 228 (2006), pp. 580–610.
- Z.D. Teng, Y. Zhang, and S.J. Gao, Permanence criteria for general delayed discrete nonautonomous -species Kolmogorov systems and its applications, Comput. Math. Appl. 59 (2010), pp. 812–828.
- X.H. Wang and C.Y. Huang, Permanence of a stage-structured predator-prey system with impulsive stocking prey and harvesting predator, Appl. Math. Comput. 235 (2014), pp. 32–42.
- L. Wang and M.Q. Wang, Ordinary Difference Equation, Xinjiang University Press, China, 1991.
- H. Xiang, K.M. Yang, and B.Y. Wang, Existence and global stability of periodic solution for delayed discrete high-order hopfifled-type neural networks, Discrete. Dyn. Nat. Soc. 2005 (2005), pp. 281–297.
- X.S. Xiong and Z.Q. Zhang, Periodic solutions of a discrete two-species competitive model with stage structure, Math. Comput. Model. 48 (2008), pp. 333–343.
- J.B. Xu, Z.D. Teng, and H.J. Jiang, Permanence and global attractivity for discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls, Period. Math. Hung. 63 (2011), pp. 19–45.
- C.J. Xu, W. Zhang, C. Aouiti, Z.X. Liu, M.X. Liao, and P.L. Li, Further investigation on bifurcation and their control of fractional-order BAM neural networks involving four neurons and multiple delays, Math. Methods. Appl. Sci. 26(3) (2023), pp. 3091–3114. https://doi.org/10.1002/mma.7581.
- C.J. Xu, M.X. Liao, P.L. Li, and S. Yuan, Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks, Chaos Solitons Fractals 142 (2021), Article ID 110535.
- C.J. Xu, Z.X. Liu, M.X. Liao, and L.Y. Yao, Theoretical analysis and computer simulations of a fractional order bank data model incorporating two unequal time delays, Expert. Syst. Appl. 199 (2022), Article ID 116859.
- X.T. Yang, Uniform persistence and periodic solutions for a discrete predator-prey system with delays, J. Math. Anal. Appl. 316 (2006), pp. 161–177.
- W.P. Zhang, D.M. Zhu, and P. Bi, Multiple periodic positive solutions of a delayed discrete predator-prey system with type IV functional responses, Appl. Math. Lett. 20 (2007), pp. 1031–1038.
- R.Y. Zhang, Z.C. Wang, Y.M. Chen, and J.H. Wu, Periodic solutions of a single species discrete population model with periodic harvest/stock, Comput. Math. Appl. 39 (2009), pp. 77–90.
- J.D. Zhao and J.F. Jiang, Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system, J. Math. Anal. Appl. 299 (2004), pp. 663–675.
- J.D. Zhao, Z.C. Zhang, and J. Ju, Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka-Volterra system, Appl. Math. Comput. 230 (2014), pp. 587–596.