Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control

Yuan Tian; Sanyi Tang; Yuan Tian; Sanyi Tang

doi:10.3934/mbe.2021362

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 7318-7343. doi: 10.3934/mbe.2021362

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Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control

Yuan Tian ¹,
Sanyi Tang ^{2
,
,}

1.
School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China
2.
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China

Academic Editor: Yang Kuang

Received: 23 July 2021 Accepted: 24 August 2021 Published: 30 August 2021

Spraying insecticides and releasing natural enemies are two commonly used methods in the integrated pest management strategy. With the rapid development of biotechnology, more and more realistic factors have been considered in the establishment and implementation of the integrated pest management models, such as the limited resources, the mutual restriction between pests and natural enemies, and the monitoring data of agricultural insects. Given these realities, we have proposed a pest-natural enemy integrated management system, which is a nonlinear state-dependent feedback control model. Besides the anti-predator behavior of the pests to the natural enemies is considered, the density dependent killing rate of pests and releasing amount of natural enemies are also introduced into the system. We address the impulsive sets and phase sets of the system in different cases, and the analytic expression of the Poincaré map which is defined in the phase set was investigated. Further we analyze the existence, uniqueness, global stability of order-1 periodic solution. In addition, the existence of periodic solution of order-$ k $ ($ k\geq2 $) is discussed. The theoretical analyses developed here not only show the relationship between the economic threshold and the other key factors related to pest control, but also reveal the complex dynamical behavior induced by the nonlinear impulsive control strategies.
- nonlinear impulsive,
- Poincarémap,
- periodic solution,
- existence and stability,
- integrated pest management
Citation: Yuan Tian, Sanyi Tang. Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7318-7343. doi: 10.3934/mbe.2021362

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Abstract

Spraying insecticides and releasing natural enemies are two commonly used methods in the integrated pest management strategy. With the rapid development of biotechnology, more and more realistic factors have been considered in the establishment and implementation of the integrated pest management models, such as the limited resources, the mutual restriction between pests and natural enemies, and the monitoring data of agricultural insects. Given these realities, we have proposed a pest-natural enemy integrated management system, which is a nonlinear state-dependent feedback control model. Besides the anti-predator behavior of the pests to the natural enemies is considered, the density dependent killing rate of pests and releasing amount of natural enemies are also introduced into the system. We address the impulsive sets and phase sets of the system in different cases, and the analytic expression of the Poincaré map which is defined in the phase set was investigated. Further we analyze the existence, uniqueness, global stability of order-1 periodic solution. In addition, the existence of periodic solution of order-$ k $ ($ k\geq2 $) is discussed. The theoretical analyses developed here not only show the relationship between the economic threshold and the other key factors related to pest control, but also reveal the complex dynamical behavior induced by the nonlinear impulsive control strategies.

References

[1]	V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES J. Mar. Sci., 3 (1928), 3–51. doi: 10.1093/icesjms/3.1.3
[2]	A. Lotka, Undamped oscillations derived from the law of mass action, J. Am. Chem. Soc., 42 (1920), 1595–1599. doi: 10.1021/ja01453a010
[3]	C. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 1–60. doi: 10.4039/Ent971-1
[4]	P. J. Wangersky, Lotka-volterra population models, Ann. Rev. Ecol. Syst., 9 (1978), 189–218. doi: 10.1146/annurev.es.09.110178.001201
[5]	B. Liu, Y. J. Zhang, L. S. Chen, Dynamic complexities in a lotka-volterra predator-prey model concerning impulsive control strategy, Int. J. Bifurcat. Chaos, 15 (2005), 517–531. doi: 10.1142/S0218127405012338
[6]	I. Perissi, U. Bardi, T. E. Asmar, A. Lavacchi, Dynamic patterns of overexploitation in fisheries, Ecol. Model., 359 (2017), 285–292. doi: 10.1016/j.ecolmodel.2017.06.009
[7]	A. T. Keong, H. M. Safuan, K. Jacob, Dynamical behaviours of prey-predator fishery model with harvesting affected by toxic substances, Matematika, 34 (2018), 143–151. doi: 10.11113/matematika.v34.n1.1018
[8]	J. López-Gómez, R. Ortega, A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. Differential Equ., 1 (1996), 403–423.
[9]	L. L. Feng, Z. J. Liu, An impulsive periodic predator-prey Lotka-Volterra type dispersal system with mixed functional responses, J. Appl. Math Comput., 45 (2014), 235–257. doi: 10.1007/s12190-013-0721-x
[10]	S. Choo, Global stability in n-dimensional discrete Lotka-Volterra predator-prey models, Adv. Differ. Equ.-NY., 11 (2014), 1–17.
[11]	J. C. Van Lenteren, Integrated pest management in protected crops, in Integrated pest management, Chapman & Hall, (1995), 311–320.
[12]	M. Kogan, Integrated pest management: historical perspectives and contemporary developments, Annu. Rev. Entomol., 43 (1998), 243–270. doi: 10.1146/annurev.ento.43.1.243
[13]	M. Barzman, P. Bàrberi, A. N. E. Birch, P. Boonekamp, S. Dachbrodt-Saaydeh, B. Graf, et al., Eight principles of integrated pest management, Agron. Sustain. Dev., 35 (2015), 1199–1215. doi: 10.1007/s13593-015-0327-9
[14]	M. L. Flint, R. Ven den Bosch, Introduction to integrated pest management, Springer-Verlag, New York, 1981.
[15]	S. Y. Tang, L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Cont. Dyn.-B, 4 (2004), 759–768.
[16]	S. Y. Tang, Y. N. Xiao, L. S. Chen, R. A. Cheke, Integrated pest management models and their dynamical behaviour, B. Math. Biol., 67 (2005), 115–135. doi: 10.1016/j.bulm.2004.06.005
[17]	B. Liu, Y. J. Zhang, L. S. Chen, L. H. Sun, The dynamics of a prey-dependent consumption model concerning integrated pest management, Acta. Math. Sin., 21 (2005), 541–554. doi: 10.1007/s10114-004-0476-2
[18]	X. N. Liu, L. S. Chen, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, Solitons Fract., 16 (2003), 311–320. doi: 10.1016/S0960-0779(02)00408-3
[19]	X. Wang, Y. Tian, S. Y. Tang: A Holling type II pest and natural enemy model with density dependent IPM strategy, Math. Probl. Eng., 2017 (2017), 8683207.
[20]	G. Jiang, Q. Lu, The dynamics of a prey-predator model with impulsive state feedback control, Discrete Cont. Dyn.-B, 6 (2006), 1301–1320.
[21]	S. Y. Tang, B. Tang, A. L. Wang, Y. N. Xiao, Holling II predator-prey impulsive semi-dynamic model with complex Poincare map, Nonlinear Dynam., 81 (2015), 1575–1596. doi: 10.1007/s11071-015-2092-3
[22]	S. Y. Tang, R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257–292. doi: 10.1007/s00285-004-0290-6
[23]	J. Yang, S. Y. Tang, Holling type II predator-prey model with nonlinear pulse as state-dependent feedback control, J. Comput. Appl. Math., 291 (2016), 225–241. doi: 10.1016/j.cam.2015.01.017
[24]	W. Gao, S. Y. Tang, The effects of impulsive releasing methods of natural enemies on pest control and dynamical complexity, Nonlinear Anal.- Hybri., 5 (2011), 540–553. doi: 10.1016/j.nahs.2010.12.001
[25]	T. Y. Wang, L. S. Chen, Nonlinear analysis of a microbial pesticide model with impulsive state feedback control, Nonlinear Dyn., 65 (2011), 1–10. doi: 10.1007/s11071-010-9828-x
[26]	B. Liu, Y. Tian, B. L. Kang, Dynamics on a Holling II predator-prey model with state-dependent impulsive control, Int. J. Biomath, 5 (2012), 1–18.
[27]	S. Y. Tang, W. H. Pang, R. A. Cheke, J. H. Wu, Global dynamics of a state-dependent feedback control system, Adv. Differ. Equ., 2015 (2015), 322. doi: 10.1186/s13662-015-0661-x
[28]	G. Jiang, Q. Lu, L. Qian, Complex dynamics of a Holling type II prey-predator system with state feedback control, Chaos, Solitons Fract., 31 (2007), 448–461. doi: 10.1016/j.chaos.2005.09.077
[29]	W. J. Qin, S. Y. Tang, R. A. Cheke, The effects of resource limitation on a predator-prey model with control measures as nonlinear pulses, Math. Probl. Eng., 2014 (2014), 450935.
[30]	M. J. B. Vreysen, Monitoring sterile and wild insects in area-wide integrated pest management programmes, in Sterile Insect Technique (eds. V.A. Dyck, J. Hendrichs and A.S. Robinson), Springer-Verlag, (2005), 325–361.
[31]	R. L. Nadel, M. J. Wingfield, M. C. Scholes, S. A. Lawson, B. Slippers, The potential for monitoring and control of insect pests in Southern Hemisphere forestry plantations using semiochemicals, Ann. Forest Sci., 69 (2012), 757–767. doi: 10.1007/s13595-012-0200-9
[32]	F. W. Ravlin, Development of monitoring and decision-support systems for integrated pest management of forest defoliators in North America, Forest Ecol. Manag., 39 (1991), 3–13. doi: 10.1016/0378-1127(91)90156-P
[33]	A. R. Ives, A. P. Dobson, Antipredator behavior and the population dynamics of simple predator-prey systems, Am. Nat., 130 (1987), 431–447. doi: 10.1086/284719
[34]	R. Ramos-jiliberto, E. Frodden, A. Aránguiz-Acu${\text{\bar n}}$a, Pre-encounter versus post-encounter inducible defense in predator-prey systems, Ecol. Model., 200 (2007), 99–108. doi: 10.1016/j.ecolmodel.2006.07.023
[35]	Y. Sait${\text{\bar o}}$, Prey kills predator: counter-attack success of a spider mite against its specific phytoseiid predator, Exp. Appl. Acarol., 2 (1986), 47–62. doi: 10.1007/BF01193354
[36]	Y. Choh, M. Ignacio, M. W. Sabelis, A. Janssen, Predator-prey role reversals, juvenile experience and adult antipredator behaviour, Sci. Rep., 2 (2012), 1–6.
[37]	I. U. Khan, S. Y. Tang, The impulsive model with pest density and its change rate dependent feedback control, Discrete Dyn. Nat. Soc., 2020 (2020), 4561241.
[38]	V. A. Kuznetsov, I. A. Makalkin, M. A. Talor, A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295–321. doi: 10.1016/S0092-8240(05)80260-5
[39]	B. Tang, Y. N. Xiao, S. Y. Tang, R. A. Cheke, A feedback control model of comprehensive therapy for treating immunogenic tumours, Int. J. Bifurcat. Chaos, 26 (2016), 1650039. doi: 10.1142/S0218127416500395
[40]	B. Tang, Y. N. Xiao, R. A. Cheke, N. Wang, Piecewise virus-immune dynamic model with HIV-1 RNA-guided therapy, J. Theor. Biol., 377 (2015), 36–46. doi: 10.1016/j.jtbi.2015.03.040
[41]	Q. Li, Y. N. Xiao, Global dynamics of a virus-immune system with virus-guided therapy and saturation growth of virus, Math. Probl. Eng., 2018 (2018), 4710586.
[42]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert W Function, Adv. Comput. Math., 5 (1996), 329–359.

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