Dynamic analysis of wild and sterile mosquito release model with Poincaré map

Yufei Wang; Huidong Cheng; Qingjian Li; Yufei Wang; Huidong Cheng; Qingjian Li

doi:10.3934/mbe.2019385

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7688-7706. doi: 10.3934/mbe.2019385

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Dynamic analysis of wild and sterile mosquito release model with Poincaré map

1.
College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
2.
College of Foreign Languages, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

Received: 08 May 2019 Accepted: 13 August 2019 Published: 21 August 2019

In recent years, the application of impulsive semi-dynamic system in state-dependent feed-back control has attracted extensive attention, but most models only discuss their special cases without delving into their complex dynamics. Therefore, we establish the wild and sterile mosquito system with integrated mosquito control, and use the Poincaré map method to conduct a comprehensive analysis of the model dynamics. First, the main properties of Poincaré map such as monotonicity, continuous differentiability, extremum and fixed point are discussed. Second, we prove the existence and stability of boundary periodic solution and study the influence of its parameters on the system. Then the ex-istence and global stability of the order-1 periodic solution and the existence condition of the order-k (k > 1) periodic solution are analyzed. Finally, our conclusion is verified by numerical analysis. The results show that the population density of wild mosquitoes can be controlled below the threshold by integrated mosquito control.
- impulsive semi-dynamic system,
- integrated mosquito control,
- Poincaré map,
- periodic solution
Citation: Yufei Wang, Huidong Cheng, Qingjian Li. Dynamic analysis of wild and sterile mosquito release model with Poincaré map[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7688-7706. doi: 10.3934/mbe.2019385

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Abstract

In recent years, the application of impulsive semi-dynamic system in state-dependent feed-back control has attracted extensive attention, but most models only discuss their special cases without delving into their complex dynamics. Therefore, we establish the wild and sterile mosquito system with integrated mosquito control, and use the Poincaré map method to conduct a comprehensive analysis of the model dynamics. First, the main properties of Poincaré map such as monotonicity, continuous differentiability, extremum and fixed point are discussed. Second, we prove the existence and stability of boundary periodic solution and study the influence of its parameters on the system. Then the ex-istence and global stability of the order-1 periodic solution and the existence condition of the order-k (k > 1) periodic solution are analyzed. Finally, our conclusion is verified by numerical analysis. The results show that the population density of wild mosquitoes can be controlled below the threshold by integrated mosquito control.

References

[1]	K. Ciesielski, On stability in impulsive dynamical systems, B. Pol. Acad. Sci. Math., 52 (2010), 81–91.
[2]	Z. Shi, Y. Li and H. Cheng, Dynamic analysis of a pest management smith model with impulsive state feedback control and continuous delay, Mathematics , 7 (2019), 591.
[3]	T. Zhang, X. Liu, X. Meng, et al., Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl., 75 (2018), 4490–4504.
[4]	Y. Li, H. Cheng, J. Wang, et al., Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy, Adv. Differ. Equ., 2018 (2018), 32.
[5]	J. Wang, H. Cheng, H. Liu, et al., Periodic solution and control optimization of a prey-predator model with two types of harvesting, Adv. Differ. Equ., 2018 (2018), 1687–1847.
[6]	N. Gao, Y. Song, X. Wang, et al., Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates, Adv. Differ. Equ., 2019 (2019), 41.
[7]	Y. Li, H. Cheng and Y. Wang, A Lycaon pictus impulsive state feedback control model with Allee effect and continuous time delay, Adv. Differ. Equ., 2018 (2018), 1687–1847.
[8]	L. Zhang, Y. Wang, M. Khalique, et al., Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Comput., 8 (2018).
[9]	Z. Shi, J. Wang, Q. Li, et al., Control optimization and homoclinic bifurcation of a preyCpredator model with ratio-dependent, Adv. Differ. Equ., 7 (2019), 591.
[10]	K. Liu, T. Zhang and L. Chen, State-dependent pulse caccination and therapeutic strategy SI epi-demic model with nonlinear incidence rate, Comput. Math. Method. M., 2019 (2019), 10.
[11]	W. Lv and F. Wang, Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks, Adv. Differ. Equ., 2017 (2017), 374.
[12]	H. Liu and H. Cheng, Dynamic analysis of a prey-predator model with state-dependent control strategy and square root response function, Adv. Differ. Equ., 2018 (2018), 63.
[13]	F. Wang, B. Chen, Y. Sun, et al., Finite time control of switched stochastic nonlinear systems, Fuzzy Set. Syst., 365 (2019), 140–152.
[14]	X. Meng, F. Li and S. Gao, Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701–726.
[15]	Y. Ren, M. Tao, H. Dong, et al., Analytical research of ( 3 + 1 )-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach, Adv. Differ. Equ., 9 (2019), 13.
[16]	J. Wang, H. Cheng, X. Meng, et al., Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse, Adv. Differ. Equ., 2017 (2017), 252.
[17]	Y. Li, Y. Li, Y. Liu, et al., Stability analysis and control optimization of a prey-predator model with linear feedback control, Discrete Dyn. Nat. Soc., 2018 (2018), 12.
[18]	G. Liu, X. Wang and X. Meng, Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity, 2017 (2017), 115.
[19]	F. Wang and X. Zhang, Adaptive finite time control of nonlinear systems under time-varying actu-ator failures, IEEE T. Syst., (2019).
[20]	J. Li, L. Cai and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol. Dynam., 11 (2016), 1.
[21]	W.Mylene, L.Georges, M.Knud, etal., Comparativegenomics: Insecticideresistanceinmosquito vectors, Nature, 423 (2003), 136–7.
[22]	L. Zhang and M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Cont. Dyn. S, 11 (2017), 777–790.
[23]	F, Liu, Continuity and approximate differentiability of multisublinear fractional maximal func-tions, Math. Inequal. Appl., 21 (2018), 25–40.
[24]	P. Stiling, Biological control by natural enemies, Ecology, 73 (1992), 1520–1520.
[25]	A. C. Redfield, The biological control of chemical factors in the environment, Sci Prog, 11 (1960), 150–170.
[26]	Y. Li and X. Liu, H-index for nonlinear stochastic systems with state and input dependent noises, Int. J. Fuzzy Syst., 20 (2018), 759–768.
[27]	F. Zhu, X. Meng and T. Zhang, Optimal harvesting of a competitive n-species stochastic model with delayed diffusions, Math. Biosci. Eng., 16 (2019), 1554–1574.
[28]	L. Marshall, J. W. Miles and A. Press, Integrated mosquito control methodologies, (1983).
[29]	H. J. Barclay, Models for the sterile insect release method with the concurrent release of pesticides, Ecol. Model., 11 (1980), 167–177.
[30]	Y. Li and X. Meng, Dynamics of an impulsive stochastic nonautonomous chemostat model with two different growth rates in a polluted environment, Discrete Dyn. Nat. Soc., 2019 (2019), 15.
[31]	Y. Liu, H. Dong and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 2 (2018), 1–17.
[32]	W. Klassen, C. F. Curtis, W. Klassen, et al., Sterile Insect Technique, (1989).
[33]	M. Q. Benedict and A. S. Robinson, The first releases of transgenic mosquitoes: an argument for the sterile insect technique, Trends Parasitol., 19 (2003), 349–355.
[34]	F. Liu, Q. Xue and K. Yabuta, Rough maximal singular integral and maximal operators supported by subvarieties on Triebel-Lizorkin spaces, Nonlinear Anal., 171 (2018), 41–72.
[35]	T. Feng, Z. Qiu, X Meng, et al., Analysis of a stochastic HIV-1 infection model with degenerate diffusion, Appl. Math. Comput., 348 (2019), 437–455.
[36]	L. Alphey, M. Benedict, R. Bellini, et al., Sterile-insect methods for control of mosquito-borne diseases: an analysis, Vector Borne Zoonot. Dis, 10 (2010), 295–311.
[37]	Y. Li, W. Zhang and X.Liu, H-index for discrete-time stochastic systems with Markovian jump and multiplicative noise, Automatica, 90 (2018), 286–293.
[38]	L. Zhang, Y. Wang, M. Khalique, et al., Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Comput., 8 (2018), 1938–1958.
[39]	J. Li and Z. Yuan, Modelling releases of sterile mosquitoes with different strategies, J. Biol. Dyn., 9 (2015), 1–14.
[40]	H. Barclay and M. Mackauer, The sterile insect release method for pest control: a density-dependent model, Environ. Entomol., 9 (1980), 810–817.
[41]	F. Wang, B. Chen, Y. Sun, et al., Finite time control of switched stochastic nonlinear systems, Fuzzy Set. Syst., (2018).
[42]	H. Qi, X. Leng, X. Meng, et al., Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients, Qual. Theory Dyn. Syst., 18 (2019).
[43]	M. Huang, X. Song and J. Li, Modelling and analysis of impulsive releases of sterile mosquitoes, J. Biol. Dynam., 11 (2017), 1147.
[44]	J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment, B. Math. Biol., 58 (1996), 425–447.
[45]	S. Tang, B. Tang, A. Wang, et al., Holling II predator–prey impulsive semi-dynamic model with complex Poincaré map, Nonlinear Dynam., 81 (2015), 1575–1596.
[46]	D. R. J. Chillingworth, An Introduction to Chaotic Dynamical Systems by Robert L. Devaney, 1986.

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