| Citation: |
Xiaoyu Hou, Jingli Fu, Huidong Cheng. SENSITIVITY ANALYSIS OF PESTICIDE DOSE ON PREDATOR-PREY SYSTEM WITH A PREY REFUGE[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 270-293. doi: 10.11948/20210153
|
SENSITIVITY ANALYSIS OF PESTICIDE DOSE ON PREDATOR-PREY SYSTEM WITH A PREY REFUGE
-
Abstract
A type of predator-prey model with refuge and nonlinear pulse feedback control is established. First, we construct the Poincar$ \acute{\rm{e}} $ map of the model and analyse its main properties. Then based on the Poincar$ \acute{\rm{e}} $ map, we explore the existence, uniqueness and global stability of the order-1 periodic solution, and also the existence of order-$ k $($ k \geq $2) periodic solutions of the system. These theoretical analysis gives the relationship between pesticide dosage and spraying cycle and economic threshold, and the relationship between the pesticide dose $ D $ and threshold $ ET $. The results show that choosing the appropriate spraying period and finding the corresponding pesticide dose under the certain economic threshold can control the number of pests and the healthy growth of crops could be controlled. Moreover, we study the influence of refuge on population density. The results show that with the increase of refuge intensity, the population of predators decreases or even becomes extinct. The parameter sensitivity analysis shows that the change of control parameters and pesticide dosage is very sensitive to the critical condition of the stability of the boundary periodic solution.
-
Keywords:
- Impulsive semi-dynamic system /
- refuge /
- Poincaré map /
- periodic solutions
-
-
References
[1] E. Bonotto, LaSalle's theorems in impulsive semidynamical systems, Nonlinear Analysis Theory Methods and Applications, 2009, 71(5-6), 2291-2297. doi: 10.1016/j.na.2009.01.062 [2] N. Bairagi and B. Chakraborty, Complexity in a prey-predator model with prey refuge and diffusion, Ecological Complexity, 2018, 37, 11-23. [3] L. Chen and H. Cheng, Integrated pest control modeling drives the rise of "semi-continuous dynamic system theory", Mathematical Modeling and Its Applications, 2021, 10(01), 1-16. [4] H. Cheng and X. Hou, A predator-prey model with Holling-Tanner functional response, Mathematical Modeling and Its Applications, 2021, 10(02), 32-43. [5] W. Chivers, W. Gladstone, R. Herbert and M. Fuller, Predator-prey systems depend on a prey refuge, Journal of Theoretical Biology, 2014, 360, 271-278. doi: 10.1016/j.jtbi.2014.07.016 [6] L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Applied Mathematics Letters, 2009, 22(9), 1330-1334. doi: 10.1016/j.aml.2009.03.005 [7] M. Huang, J. Li, X. Song and H. Guo, Modeling impulsive injections of insulin: towards artificial pancreas, SIAM Journal on Applied Mathematics, 2012, 72(5), 1524-1548. doi: 10.1137/110860306 [8] L. Ji and C. Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear Analysis Real World Applications, 2010, 11(4), 2285-2295. doi: 10.1016/j.nonrwa.2009.07.003 [9] J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, Journal of Mathematical Biology, 2012, 65(4), 623-652. doi: 10.1007/s00285-011-0474-9 [10] B. Liu, Y. Tian and B. Kang, Dyamics on a Holling Ⅱ predator-prey model with state-dependent impulsive control, International Journal of Biomathematics, 2012, 5(3), 93-110. [11] Z. Li and L. Chen, Dynamical behaviors of a trimolecular response model with impulsive input, Nonlinear Dynamics, 2010, 62(1), 167-176. [12] J. Li and Z. Wu, Mathematical model of intergenerational influence of autocorrelation function of birth rate in China, Mathematical Modeling and Its Applications, 2020, 9(04), 28-36. [13] H. Mainul, M. Sabiar and E. Venturino, Effect of a functional response-dependent prey refuge in a predator-prey model, Ecological Complexity, 2014, 20(12), 284-256. [14] L. Nie, J. Peng, Z. Teng and L. Hu, Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects Mathematical Methods in the Applied Sciences, 2011, 224(2), 544-555. [15] E. Olivares and R. Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, 2003, 166(1-2), 135-146. doi: 10.1016/S0304-3800(03)00131-5 [16] A. Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Applied Mathematics Letters, 2005, 18(7), 729-732. doi: 10.1016/j.aml.2004.05.012 [17] L. Pang, S. Liu, F. Liu, X. Zhang and T. Tian, Mathematical modeling and analysis of tumor-volume variation during radiotherapy-ScienceDirect, Applied Mathematical Modelling, 2021, 89, 1074-1089. doi: 10.1016/j.apm.2020.07.028 [18] X. Qiu and H. Xiao, Qualitative analysis of Holling type Ⅱ predator-prey systems with prey refuges and predator restricts, Nonlinear Analysis Real World Applications, 2013, 14(4), 1896-1906. doi: 10.1016/j.nonrwa.2013.01.001 [19] H. Qi, X. Leng, X. Meng and T. Zhang, Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients, Qualitative Theory of Dynamical Systems, 2018, 18, 347-369. [20] W. Qin, S. Tang and A. Cheke, The Effects of Resource Limitation on a Predator-Prey Model with Control Measures as Nonlinear Pulses, Mathematical Problems in Engineering, 2014, 2014(2), 99-114. [21] D. Sapna, Effects of prey refuge on a ratio-dependent predator-prey model with stage-structure of prey population, Applied Mathematical Modelling, 2013, 37(6), 4337-4349. doi: 10.1016/j.apm.2012.09.045 [22] K. Sun, T. Zhang and Y. Tian, Theoretical study and control optimization of an integrated pest management predator-prey model with power growth rate, Mathematical Biosciences, 2016, 279, 13-26. doi: 10.1016/j.mbs.2016.06.006 [23] K. Sun, T. Zhang and Y. Tian, Dynamics analysis and control optimization of a pest management predator-prey model with an integrated control strategy, Applied Mathematics and Computation, 2017, 292, 253-271. doi: 10.1016/j.amc.2016.07.046 [24] K. Sun, Y. Tian, L. Chen and A. Kasperski, Nonlinear modelling of a synchronized chemostat with impulsive state feedback control, Mathematical and Computer Modelling, 2010, 52(1), 227-240. [25] Y. Tian, S. Tang and A. Cheke, Nonlinear state-dependent feedback control of a pest-natural enemy system, Nonlinear Dynamics, 2018, 94, 2243-2263. doi: 10.1007/s11071-018-4487-4 [26] Y. Tian, T. Zhang and K. Sun, Dynamics analysis of a pest management prey-predator model by menas of interval state monitoring and control, Nonlinear Analysis: Hybrid Systems, 2017, 23, 122-141. doi: 10.1016/j.nahs.2016.09.002 [27] S. Tang, Y. Xiao, L. Chen and A. Cheke, Integrated pest management models and their dynamical behaviour, Bull Math Biol, 2005, 67(1), 115-135. doi: 10.1016/j.bulm.2004.06.005 [28] S. Tang and A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, Journal of Mathematical Biology, 2005, 50(3), 257. doi: 10.1007/s00285-004-0290-6 [29] S. Tang, J. Liang, Y. Tan and A. Cheke, Threshold conditions for integrated pest management models with pesticides that have residual effects, Journal of Mathematical Biology, 2013, 66(1-2), 1-35. doi: 10.1007/s00285-011-0501-x [30] Y. Tao, W. Xia and X. Song, Effect of prey refuge on a harvested predator-prey model with generalized functional response, Communications in Nonlinear Science and Numerical Simulation, 2011, 16(2), 1052-1059. doi: 10.1016/j.cnsns.2010.05.026 [31] J. Wang, H. Cheng, X. Meng and S. Pradeep, Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse, Advances in Difference Equations, 2017, 2017(1), 252. doi: 10.1186/s13662-017-1300-5 [32] C. Wei and L. Chen, A Leslie-Gower Pest Management Model with Impulsive State Feedback Control, Journal of Biomathematics, 2012, 27(4), 123-134. [33] W. Xie and P. Weng, Existence of a periodic solution for a predator-prey model with patch-diffusion and feedback control, Journal of Experimental Biology, 2013, 216(9), 1561-1569. [34] Y. Xiao, H. Miao, S. Tang and H. Wu, Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models, Advanced Drug Delivery Reviews, 2013, 65(7), 940-953. doi: 10.1016/j.addr.2013.04.005 [35] Y. Xiao, X. Xu and S. Tang, Sliding Mode Control of Outbreaks of Emerging Infectious Diseases, Bulletin of Mathematical Biology, 2012, 74(10), 2043. [36] J. Yang, T. Zhang and S. Yuan, Turing Pattern Induced by Cross-Diffusion in a PredatoršCPrey Model with Pack Predation-Herd Behavior, International Journal of Bifurcation and Chaos, 2020, 30(7), 2050103. doi: 10.1142/S0218127420501035 [37] J. Yang and Y. Tan, Effects of pesticide dose on Holling Ⅱ predator-prey model with feedback control, Journal of Biological Dynamics, 2018, 12(1), 527-550. doi: 10.1080/17513758.2018.1479457 [38] X. Yu, S. Yuan and T. Zhang, Asymptotic properties of stochastic nutrient-plankton food chain models with nutrient recycling, Nonlinear Analysis Hybrid Systems, 2019, 34, 209-225. doi: 10.1016/j.nahs.2019.06.005 [39] X. Yu, S. Yuan and T. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Applied Mathematics and Computation, 2019, 347, 249-264. doi: 10.1016/j.amc.2018.11.005 [40] T. Zhang, W. Ma, X. Meng and T. Zhang, Periodic solution of a prey-predator model with nonlinear statefeedback control, Applied Mathematics and Computation, 2015, 266, 95-107. doi: 10.1016/j.amc.2015.05.016 [41] W. Zhang, S. Zhao, X. Meng and T. Zhang, Evolutionary analysis of adaptive dynamics model under variation of noise environment, Applied Mathematical Modelling, 2020, 84, 222-239. doi: 10.1016/j.apm.2020.03.045 [42] T. Zhang, N. Gao and J. Wang, Dynamic System of Microbial Culture Described by Impulsive Differential Equations, Mathematical Modeling and Its Applications, 2019, 8(01), 1-13. [43] H. Zhang, Y. Cai, S. Fu and W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied Mathematics and Computation, 2019, 356, 328-337. doi: 10.1016/j.amc.2019.03.034 [44] H. Zhang, P. Georgescu and L. Chen, On the impulsive controllability and bifurcation of a predator-pest model of IPM, Biosystems, 2008, 93(3), 151-171. doi: 10.1016/j.biosystems.2008.03.008 -
-
Figures(11)
Export File
Citation
Xiaoyu Hou, Jingli Fu, Huidong Cheng. SENSITIVITY ANALYSIS OF PESTICIDE DOSE ON PREDATOR-PREY SYSTEM WITH A PREY REFUGE[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 270-293. doi: 10.11948/20210153
Format
Content
DownLoad:
- Figure 1. Schematic diagram of the mathematical model.
-
Figure 2. The domains of the phase and pulse sets in three cases,
$ A = 1 $ ,$ B = 0.8 $ ,$ C = 0.2 $ . (a)$ k_{1} = 0.7,k_{2} = 0.3,D = 0.5,ET = 0.2 $ ,$ \tau = 0.45 $ .(b)$ k_{1} = 1.5,k_{2} = 0.3,D = 0.5,ET = 0.3 $ ,$ \tau = 0.45 $ .(c)$ k_{1} = 1.5,k_{2} = 0.3,D = 0.5,ET = 0.4 $ ,$ \tau = 0.45 $ .(d)$ k_{1} = 1.5,k_{2} = 0.3,D = 0.5,ET = 0.4 $ ,$ \tau = 0.45 $ . -
Figure 3. The Poincar
$ \acute{\mathit{\text{e}}} $ map$ \varphi $ related to the impulsive point series$ y_k^ + $ in case Ⅰ. The parameters fixed as$ A = 1 $ ,$ B = 0.8 $ ,$ C = 0.2 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ ET = 0.25 $ . (a)$ \tau = 0.15 $ ; (b)$ \tau = 0.45 $ . -
Figure 4. The Poincar
$ \acute{\mathit{\text{e}}} $ map$ \varphi $ related to the impulsive point series$ y_{k}^{+} $ in case Ⅱ. The parameters fixed as:$ A = 0.8 $ ,$ B = 0.8 $ ,$ C = 0.2 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ ET = 0.25 $ . (a)$ \tau = 0.2 $ ; (b)$ \tau = 0.45 $ . -
Figure 5. The density of prey
$ x_{e} $ and predator$ y_{e} $ in system (2.4)as the refuge$ \beta $ changes. Here$ r = 0.2 $ ,$ a = 1 $ ,$ b = 0.8 $ ,$ p = 0.6 $ ,$ q = 0.6 $ , and$ c = 0.45 $ . -
Figure 6. State pulse feedback control strategy of system (2.6),
$ A = 1 $ ,$ B = 0.8 $ ,$ C = 0.2 $ ,$ ET = 0.24 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ \tau = 0.45 $ . -
Figure 7. The order-2 periodic solution of system (2.6).
$ A = 0.8 $ ,$ B = 0.98 $ ,$ C = 0.174 $ ,$ ET = 0.25 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ \tau = 0.45 $ . -
Figure 8. Periodic solutions in the case of Fig.4a and Fig.4b, respectively.
$ A = 0.8 $ ,$ B = 0.8 $ ,$ C = 0.2 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ ET = 0.25 $ . (a)$ \tau = 0.2 $ ; (b)$ \tau = 0.45 $ . -
Figure 9. The path curve of system (2.6) starting from different initial points.
$ A = 1 $ ,$ B = 0.8 $ ,$ C = 0.2 $ ,$ ET = 0.25 $ ,$ k_{1} = 0.7 $ ,$ k_{2} = 0.3 $ ,$ D = 0.5 $ ,$ \tau = 0.45 $ . -
Figure 10. The effects of key parameters on the dose
$ D $ of chemical control, and the parameters are fixed as: (a)$ r = 0.4 $ ,$ k_{1} = 0.4 $ ,$ T = 6 $ ,$ K = 92 $ ; (b)$ r = 1.4 $ ,$ k_{1} = 0.4 $ ,$ ET = 35 $ ,$ K = 92 $ . -
Figure 11. The effects of control parameters on the threshold condition
$ R_{0} $ . (a)$ k_{1} = 0.5 $ ,$ k_{2} = 0.2 $ ,$ K = 92 $ ,$ b = 0.3 $ ,$ p = 0.2 $ ,$ \beta = 0.2 $ ,$ a = 0.3 $ : (a)$ ET = 35 $ and$ c = 0.6 $ ; (b)$ D = 0.5 $ and$ c = 0.3 $ .