### Mathematical Biosciences and Engineering

2019, Issue 6: 7963-7981. doi: 10.3934/mbe.2019401
Research article Special Issues

# Optimization of an integrated feedback control for a pest management predator-prey model

• Received: 04 March 2019 Accepted: 28 August 2019 Published: 02 September 2019
• In this paper, a Leslie-Gower predator-prey model with ratio-dependence and state pulse feedback control is established to investigate the effect of spraying chemical pesticides and supplement amount of beneficial insects at the same time. Firstly, the existence, uniqueness and asymptotic stability of the periodic solution are proved by using successor functions method and the analogue of the Poincaré criterion when the equilibria points $E_{*}$ and $E_{0}$ are stable, and the existence of the limit cycle without impulse system are verified when the equilibrium $E_{*}$ is the unstable point. Furthermore, to obtain the minimum cost per period of controlling pests, we propose the optimization problem and calculate the optimal threshold. Finally, the feasibility of our model is proved by numerical simulation of a concrete example.

Citation: Zhenzhen Shi, Huidong Cheng, Yu Liu, Yanhui Wang. Optimization of an integrated feedback control for a pest management predator-prey model[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7963-7981. doi: 10.3934/mbe.2019401

### Related Papers:

• In this paper, a Leslie-Gower predator-prey model with ratio-dependence and state pulse feedback control is established to investigate the effect of spraying chemical pesticides and supplement amount of beneficial insects at the same time. Firstly, the existence, uniqueness and asymptotic stability of the periodic solution are proved by using successor functions method and the analogue of the Poincaré criterion when the equilibria points $E_{*}$ and $E_{0}$ are stable, and the existence of the limit cycle without impulse system are verified when the equilibrium $E_{*}$ is the unstable point. Furthermore, to obtain the minimum cost per period of controlling pests, we propose the optimization problem and calculate the optimal threshold. Finally, the feasibility of our model is proved by numerical simulation of a concrete example.

 [1] H. Guo and X. Song, An impulsive predator-prey system with modified Leslie-Gower and Holling type Ⅱ schemes, Chaos Solitons Fract., 36 (2008), 1320–1331. [2] R. Kooij, J. Arus and A. Embid, Limit cycles in the Holling-Tanner model, Publicacions Matematiques, 41 (1997), 149–167. [3] E. Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, Siam J. Appl. Math., 59 (1999), 1867–1878. [4] 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. [5] Y. Li, Y. Li, Y. Liu, et al., Stability analysis and control optimization of a prey-predator model with linear feedback control, Discrete Dyn. Nature Soc., 2018 (2018), 4945728. [6] C. Li and S. Tang, Analyzing a generalized pest-natural enemy model with nonlinear impulsive control, Open Math., 16 (2018), 1390–1411. [7] J. Gu, Y. Zhang and H. Dong, Dynamic behaviors of interaction solutions of (3+1)-dimensional shallow water wave equation, Comput. Math. Appl., 76 (2018), 1408–1419. [8] Z. Shi, J. Wang, Q. Li, et al., Control optimization and homoclinic bifurcation of a prey-predator model with ratio-dependent, Adv. Differ. Equ., 2019 (2019), 2. [9] Y. Tian, S. Tang and R. A. Cheke, Nonlinear state-dependent feedback control of a pest-natural enemy system, Nonlinear Dyn., 94 (2018), 2243–2263. [10] 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. [11] T. Zhang, Y. Song, T. Zhang, et al, A stage-structured predator-prey SI model with disease in the prey and impulsive effects, Math. Model. Anal., 18 (2013), 505–528. [12] 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. [13] Z. Jiang, X. Bi, T. Zhang, et al., Global hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient, Math. Biosci. Eng., 16 (2019), 3807–3829. [14] 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), 367. [15] K. Liu, T. Zhang and L. Chen, State-dependent pulse vaccination and therapeutic strategy in an SI epidemic model with nonlinear incidence rate, Comput. Math. Method. M., 2019 (2019), article ID 3859815. [16] F. Wang, B. Chen, Y. Sun, et al., Finite time control of switched stochastic nonlinear systems, Fuzzy Set. Syst., 35 (2019), 140–152. [17] S. Tang, X. Tan, J. Yang, et al., Periodic solution bifurcation and spiking dynamics of impacting predator-prey dynamical model, Int. J. Bifurcat. Chaos, 28 (2018), 1850147. [18] G. Pang and L. Chen, Periodic solution of the system with impulsive state feedback control, Nonlinear Dyn., 78 (2014), 743–753. [19] Z. Zhao, L. Pang and X. Song, Optimal control of phytoplankton fish model with the impulsive feedback control, Nonlinear Dyn., 88 (2017), 2003–2011. [20] Y. Gong and J. Huang, Bogdanov-takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin-E, 30 (2014), 239–244. [21] T. Zhang, X. Meng and Y. Song, The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments, Nonlinear Dyn., 64 (2011), 1–12. [22] J. Jiao, L. Chen, J. Nieto, et al., Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey, Appl. Math. Mech., 195 (2008), 316–325. [23] Y. Pei, C. Li and L. Chen, Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay, Math. Comput. Simulat., 79 (2009), 2994–3008. [24] J. Jiao, S. Cai and L. Li, Dynamics of a periodic switched predator-prey system with impulsive harvesting and hibernation of prey population, J. Franklin I., 353 (2016), 3818–3834. [25] K. Sun, T. Zhang and Y. Tian, Dynamics analysis and control optimization of a pest management predator-prey model with an integrated control strategy, Appl. Math. Comput., 292 (2017), 253–271. [26] S. Tang and L. Chen, Global attractivity in a food-limited population model with impulsive effects, J. Math. Anal. Appl., 292 (2004), 211–221. [27] R. N. Guedes, G. Smagghe, J. D. Stark, et al., Pesticide-induced stress in arthropod pests for optimized integrated pest management programs, Ann. Rev. Entomol., 61 (2015), 43. [28] P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245. [29] F. Liu, Q. Xue and K. Yabuta, Boundedness and continuity of maximal singular integrals and maximal functions on triebel-lizorkin spaces, Sci. China-Math., Mathematics, Doi: 10.1007/s11425-017-9416-5. [30] 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. [31] W. Lv and W. Fang, 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. [32] Z. Zhao, Z. Li and L. Chen, Existence and global stability of periodic solution for impulsive predatorprey model with diffusion and distributed delay, J. Appl. Math. Comput., 33 (2010), 389–410. [33] P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234. [34] 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. [35] 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), 41. [36] T. Liu and H. Dong, The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach Mathematics, 7 (2019), 170. [37] Q. Wang, Y. Zhang, Z. Wang, et al., Periodicity and attractivity of a ratio-dependent Leslie system with impulses, J. Math. Anal. Appl., 376 (2011), 212–220. [38] S. Tang, B. Tang, A. Wang,et al., Holling Ⅱ predator-prey impulsive semi-dynamic model with complex Poincaré map, Nonlinear Dyn., 81 (2015), 1575–1596. [39] 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. [40] M. Han, L. Zhang, Y. Wang, et al., The effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations, Nonlinear Anal-Real., 47 (2019), 236–250. [41] F. Liu, Z. Fu and S. T. Jhang, Boundedness and continuity of marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces, Front. Math. China, 14 (2019), 95–122. [42] F. Liu, Rough maximal functions supported by subvarieties on Triebel-Lizorkin spaces Revista de la Real Academia de Ciencias Exactas, F´ ısicas y Naturales. Serie A. Matemáticas, 112 (2018), 593–614. [43] X. Meng, F. Li and S. Gao, Global analysis and numerical simulations of a novel stochastic ecoepidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701–726. [44] R. P. Gupta and M. Banerjee, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339–366. [45] Z. Zhao, L. Yang and L. Chen, Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type Ⅱ schemes J. Appl. Math. Comput., 35 (2011), 119–134. [46] C. Wei and L. Chen, A Leslie-Gower pest manangement model with impulsive state feedback control, J. Biomath., 27 (2012), 621–628. [47] J. D. Flores and E. González-Olivares, A modified Leslie-Gower predator-prey model with ratiodependent functional response and alternative food for the predator, Math. Methods Appl. Sci., 40 (2017), 2313-2328. [48] Z. Liang and H. Pan, Qualitative analysis of a ratio-dependent Holling-Tanner model, Math. Anal. Appl., 334 (2007), 954–964. [49] K. Sun, T. Zhang and Y. Tian, Theoretical study and control optimization of an integrated pest management predator-prey model with power growth rate, Math. Biosci., 279 (2012), 13–26. [50] C. Wei, J. Liu and L. Chen, Homoclinic bifurcation of a ratio-dependent predator-prey system with impulsive harvesting, Nonlinear Dyn., (2017), 1–12. [51] Z. Liang, X. Zeng, G. Pang, et al., Periodic solution of a Leslie predator-prey system with ratio-dependent and state impulsive feedback control Nonlinear Dyn., (2017), 1–15. [52] Y. Tian, K. Sun and L. Chen, Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system, Int. J. Biomath., 7 (2014), 1450018. [53] C. J. Edholm, B. Tenhumberg, C. Guiver, et al., Management of invasive insect species using optimal control theory, Ecol. Model., 381 (2018), 36–45.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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