Catastrophe control of aphid populations model

Lichun Zhao; Jingna Liu; Bing Liu; Yuan Li; Huiyan Zhao; Lichun Zhao; Jingna Liu; Bing Liu; Yuan Li; Huiyan Zhao

doi:10.3934/mbe.2022336

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 7122-7137. doi: 10.3934/mbe.2022336

Previous Article Next Article

Research article Special Issues

Catastrophe control of aphid populations model

1.
School of Mathematics and Information Science, Anshan Normal University, Ashan 114007, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China
3.
State Key Laboratory of Crop Stress Biology in Arid Areas, School of Plant Protection, Northwest A and F University, Yangling 712100, China

Received: 08 February 2022 Revised: 28 March 2022 Accepted: 18 April 2022 Published: 13 May 2022

Considering the effect of the natural enemy on aphid populations, the corresponding model with delay is built. The model is analyzed using the qualitative theory of differential equations and catastrophe theory etc. For the outbreak phenomenon of aphid populations, the corresponding management model is proposed and the catastrophe controller is designed to keep the system in a virtuous cycle by means of the qualitative theory of impulsive differential equations. In the mean time, some simulations are carried to prove the results. The paper not only provides a new method for catastrophe control but also expands the application fields of catastrophe control.
- catastrophe theory,
- aphid populations,
- catastrophe control,
- delay effect,
- the qualitative theory of impulsive differential equations
Citation: Lichun Zhao, Jingna Liu, Bing Liu, Yuan Li, Huiyan Zhao. Catastrophe control of aphid populations model[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 7122-7137. doi: 10.3934/mbe.2022336

Related Papers:

Abstract

Considering the effect of the natural enemy on aphid populations, the corresponding model with delay is built. The model is analyzed using the qualitative theory of differential equations and catastrophe theory etc. For the outbreak phenomenon of aphid populations, the corresponding management model is proposed and the catastrophe controller is designed to keep the system in a virtuous cycle by means of the qualitative theory of impulsive differential equations. In the mean time, some simulations are carried to prove the results. The paper not only provides a new method for catastrophe control but also expands the application fields of catastrophe control.

References

[1]	R. S. Zahler, H. J. Sussmann, Claims and ccomplishments of Applied Catastrophe, Nature, 269 (1977), 59–763. https://doi.org/10.1038/269759a0 doi: 10.1038/269759a0
[2]	D. Chillingworth, Catastrophe theory: selected papers, 1972–C1977: Edited by E.C. Zeeman Addison-Wesley Inc. London, 1978. 675 pp: £ 15.50 hardback; £ 8.50 paperback, Appl. Math. Modell., 2 (1978), 221–222. https://doi.org/10.1016/0307-904x(78)90013-6 doi: 10.1016/0307-904x(78)90013-6
[3]	C. C. Chang, S. H. Sheu, Y. L. Chen, Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, Appl. Math. Modell., 37 (2013), 308–317. https://doi.org/10.1016/j.apm.2012.02.031 doi: 10.1016/j.apm.2012.02.031
[4]	M. J. Bazin, P. T. Saunders, Determination of critical variables in a microbial predator-prey system by catastrophe theory, Nature, 274 (1978), 52–54. https://doi.org/10.1038/275052a0 doi: 10.1038/275052a0
[5]	R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 71–77. https://doi.org/10.1038/269471a0 doi: 10.1038/269471a0
[6]	J. Casti, Catastrophes, control and the inevitability of spruce budworm outbreaks, Ecol. Modell., 14 (1982), 293–300. https://doi.org/10.1016/0304-3800(82)90024-2 doi: 10.1016/0304-3800(82)90024-2
[7]	C. Ouimet, P. Legender, Practical aspects of modelling ecological phenomena using the cusp catastrophe, Oikos, 42 (1988), 265–287. https://doi.org/10.1016/0304-3800(88)90061-0 doi: 10.1016/0304-3800(88)90061-0
[8]	H. Y. Zhao, S. Z. Wang, Y. C. Dong, Apply catastrophe theory to study control strategy of aphid ecosystem, Chin. Sci. Bull., 34 (1989), 1745–1749. https://doi.org/CNKI:SUN:KXTB.0.1989-22-018
[9]	H. Y. Zhao, Study on the cusp catastrophe model, sudden change region and controlling target during strategy of wheat aphid control, Syst. Eng., 6 (1991), 30–35. https://doi.org/10.1038/s41558-020-0835-8 doi: 10.1038/s41558-020-0835-8
[10]	X. D. Zhao, H. Y. Zhao, G. Z. Liu, L. F. Zheng, Prey under nature enemy model parameter grey estimattion, J. Northwest A F Univ., 33 (2005), 65–68. https://doi.org/10.13207/j.cnki.jnwafu.2005.04.015 doi: 10.13207/j.cnki.jnwafu.2005.04.015
[11]	X. L. Wei, H. Y. Zhao, G. Z. Liu, Y. H. Wu, Analysis of pest population dynamics model using swallowtail catastrophe theory, Acta Ecol. Sin., 29 (2009), 5478–5484. https://doi.org/10.3321/j.issn:1000-0933.2009.10.036 doi: 10.3321/j.issn:1000-0933.2009.10.036
[12]	M. K. D. K. Piyaratnea, H. Y. Zhao, Q. X. Meng, APHIDSim: A population dynamics model for wheat aphids based on swallowtail catastrophe theory, Ecol. Modell., 253 (2013), 9–16. https://doi.org/10.1016/j.ecolmodel.2012.12.032 doi: 10.1016/j.ecolmodel.2012.12.032
[13]	Z. Li, H. Y. Zhao, G. Z. Liu, J. J. liu, Population dynamics of insect pests of butterfly catastrophe model and analysis, J. Northwest A F Univ., 40 (2012), 1–6. https://doi.org/10.13207/j.cnki.jnwafu.2012.09.019 doi: 10.13207/j.cnki.jnwafu.2012.09.019
[14]	Y. li, Catastrophe Theory and Its Application in Ecological Regulation of Pests, Ph.D thesis, Northwest Agriculture and Forestry University, 2020. https://doi.org/10.27409/d.cnki.gxbnu.2020.001421
[15]	M. K. D. K. Piyaratnea, The Catastrophe Region Identification, Parameter Estimation and Software Development on Swallowtail Catastrophe Model of the AphidPopulation Dynamics and Its Application, Ph.D thesis, Northwest Agriculture and Forestry University, 2015. https://doi.org/10.27409/d.cnki.gxbnu.2015.001421
[16]	Z. S. Ma, E. J. Bechinski, Life tables and demographic statistics of Russian wheat aphid (Hemiptera: Aphididae) reared at different temperatures and on different host plant growth stages, Eur. J. Entomol., 106 (2009), 205–210. https://doi.org/10.14411/eje.2009.026 doi: 10.14411/eje.2009.026
[17]	Z. S. Ma, E. J. Bechinski, Developmental and phennological modelling of Russian wheat aphit, Ann. Entomol. Soc. Am., 101 (2008), 351–361. https://doi.org/10.1603/0013-8746(2008)101[351:DAPMOR]2.0.CO;2 doi: 10.1603/0013-8746(2008)101[351:DAPMOR]2.0.CO;2
[18]	Z. S. Ma, E. J. Bechinski, A survival-analysis-based simulation model for Russian wheat aphid population dynamics, Ecol. Modell., 216 (2008), 323–332. https://doi.org/10.1016/j.ecolmodel.2008.04.011 doi: 10.1016/j.ecolmodel.2008.04.011
[19]	Z. S. Ma, E. J. Bechinski, An approach to the nonlinear dynamics of Russian wheat aphid population growth with the cusp catastrophe model, Entomol. Res., 39 (2009), 175–181. https://doi.org/10.1111/j.1748-5967.2009.00216.x doi: 10.1111/j.1748-5967.2009.00216.x
[20]	L. C. Zhao, J. N. Liu, J. Liu, The geometrical analysis of insect pest population model with cusp catastrophe, Math. Pract. Theory, 47 (2017), 273–279.
[21]	Z. Zheng, Optimal and Optimization Control of the Fold Ecosystem with Impulsive Effects, M.S. thesis, Liaoning Normal University, 2014. https://doi.org/10.7666/d.Y2613542
[22]	Y. Li, The Qualitative Analysis of Wheat Aphids Ecosystem Model Based on Catastrophe Theory, M.S. thesis, University of Science and Technology Liaoning, 2015.
[23]	J. Liu, L. C. Zhao, J. N. Liu, Optimization impulsive control of insect pest population model with cusp catastrophe, J. Biomath., 430 (2015), 113–120.
[24]	L. C. Zhao, J. N. Liu, M. Zhang, B. Liu, Analysis and control of a delayed population model with an allee effect, Int. J. Biomath., 2022 (2022), 2250025. https://doi.org/10.1142/S1793524522500255(2022 doi: 10.1142/S1793524522500255(2022
[25]	R. Sun, Research on Catastrophe Control Technique and Its Application, Ph.D thesis, University of Science and Technology Liaoning, 2002. https://doi.org/CNKI:CDMD:1.2003.032576
[26]	X. F. Wang, Research on Catastrophe Control Method and Its Application in Ship Motion, Ph.D thesis, Harbin Enginnering University, 2009. https://doi.org/10.7666/d.y1655662
[27]	R. Sun, X. B. Wang, H. W. Mo, Catastrophe analysis in coupled pitch-roll ship motion, Appl. Math. Comput., 30 (2009), 527–530. https://doi.org/10.3969/j.issn.1006-7043.2009.05.011 doi: 10.3969/j.issn.1006-7043.2009.05.011
[28]	M. Xiao, Z. K. Shi, The control method for catastrophe of out-of-water model of underground mine, Acta Autom. Sin., 38 (2012), 1610–1617. https://doi.org/10.3724/SP.J.1004.2012.01609 doi: 10.3724/SP.J.1004.2012.01609
[29]	Q. H. Ding, Nonlinear Ship Rolling Analysis Based on Catastrophe Theory, M.S. thesis, Harbin Enginnering University, 2009. https://doi.org/10.7666/d.y1488988
[30]	X. H. Zhao, Control and A pplication based on Catastrophe Theory, Harbin Institute of Technology Press, 2013.
[31]	X. H. Zhao, Y. Sun, Z. K. Qi, Catastrophe characteristics and control of pitching supercavitating vehicles at fixed depths, Ocean Eng., 112 (2016), 185–194. https://doi.org/10.1016/j.oceaneng.2015.12.021 doi: 10.1016/j.oceaneng.2015.12.021
[32]	Y. G. Huang, Research on Traffic Congestion Mechenism and Traffic Control Method For Urban Road, Ph.D thesis, South China University of Technology, 2015.
[33]	H. J. Liang, Mutation flow control model simulation analysis based on the large hybrid network, Bull. Sci. Technol., 4 (2015), 205–207. https://doi.org/10.13774/j.cnki.kjtb.2015.04.069 doi: 10.13774/j.cnki.kjtb.2015.04.069
[34]	N. Macdonzld, Time delay in prey-predator models, Math. Biosci., 28 (1976), 321–330. https://doi.org/10.1016/0025-5564(76)90130-9 doi: 10.1016/0025-5564(76)90130-9
[35]	Q. Xiong, X. L. Li, F. Guo, Population dynamic of aphids and predatory natural enemies in the seedling stage of bupleurum Chinese Dc, Acta Agric. Boreali-Occident. Sin., 14 (2005), 78–80.
[36]	D. J. Luo, L. B. Teng, Qualitative Theory of Dynamical Systems, World Scientific, 1993. https://doi.org/10.1142/1914
[37]	L. C. Zhao, L. S. Chen, Q. L. Zhang, The geometrical analysis of a predator-prey model with two state impulses, Math. Biosci., 3 (2012), 67–75. https://doi.org/10.1016/j.mbs.2012.03.011 doi: 10.1016/j.mbs.2012.03.011
[38]	G. R. Jian, Q. S. Lu, Impulsive state feedback control of a predator-prey model, Math. Biosci., 200 (2007), 193–207. https://doi.org/10.1016/j.cam.2005.12.013 doi: 10.1016/j.cam.2005.12.013
[39]	J. H. Zhi, Y. F. Chen, Computation of invariant curves and identifying the type of critical point, Math. Biosci., 31 (2018), 1698–1708. https://doi.org/CNKI:SUN:XTYW.0.2018-06-018
[40]	R. Andrea, V. Marco, F. Alessandro, S. Roberto, dynamical criticality: Overview and open questions, Math. Biosci., 31 (2018), 647–663. https://doi.org/10.1007/s11424-017-6117-5 doi: 10.1007/s11424-017-6117-5

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)