# Mathematical Modeling of Host - Pest Interactions in Stage-Structured Populations: A Case of False Codling Moth [Thaumatotibia leucotreta]

• Ochwach O. Jimrise Department of Physical Sciences, Chuka University, P.O. Box 109-60400 Kenya
• Mark O. Okongo Department of Physical Sciences, Chuka University, P.O. Box 109-60400, Kenya
• Moses M. Muraya Department of Plant Sciences, Chuka University, P.O. Box 109-60400, Kenya
Keywords: False codling moth (FCM) (Thaumatotibia lucotreta) is a significant pest due to its potential economic impact on many susceptible fruits in most temperate regions of the world. Efforts to control the codling moth in the past mostly relied on the use of broad spectrum insecticide sprays, which has resulted in the development of insecticide resistance, and the disruption of the control of secondary pests. Understanding the dynamic of this pest is of great in importance in order to effectively employ the most effective control strategies. In this study, a mathematical model of host-false codling moth interactions is developed and qualitatively analysed using stability theory of system of differential equations. The basic offspring number with respect to FCM free equilibrium is obtain using next generation matrix. The condition for local and global asymptotic stability of FCM free and coexistence equilibria are established. The model is analysed numerically and graphically represented to justify the analytical results.

### Abstract

False codling moth (FCM) (Thaumatotibia lucotreta) is a significant pest due to its potential economic impact on many susceptible fruits in most temperate regions of the world. Efforts to control the codling moth in the past mostly relied on the use of broad spectrum insecticide sprays, which has resulted in the development of insecticide resistance, and the disruption of the control of secondary pests. Understanding the dynamic of this pest is of great in importance in order to effectively employ the most effective control strategies. In this study, a mathematical model of host-false codling moth interactions is developed and qualitatively analysed using stability theory of system of differential equations. The basic offspring number with respect to FCM free equilibrium is obtain using next generation matrix. The condition for local and global asymptotic stability of FCM free and coexistence equilibria are established. The model is analysed numerically and graphically represented to justify the analytical results.

### References

[1] Alemneh, H. T., Makinde, O. D., and Theuri, D. M. (2019). Mathematical modelling of msv pathogen interaction with pest invasion on maize plant. Global Journal of Pure and Applied Mathematics, 15(1):55–79.
[2] Anderson, R. and May, R. (1979). Population Biology of Infectious Diseases: Part I. Na- ture, 280:361–7.
[3] Anguelov,M.R.,Dufourd,C.,andDumont,Y.(2016). MathematicalModelforPest-Insect Control using Mating Disruption and Trapping. Applied Mathematical Modelling.
[4] Barclay, H. J. and Haniotakis, G. E. (1991). Combining Pheromone-Baited and Food- Baited Traps for Insect Pest Control: Effects of Developmental Period. Population Ecology, 33(2):269–285.
[5] Barclay, H. J., Steacy, R., Enkerlin, W., and van den Driessche, P. (2016). Modeling Dif- fusive Movement of Sterile Insects Released along Aerial Flight Lines. International Journal of Pest Management, 62(3):228–244.
[6] Barclay, H. J. and Van Den Driessche, P. (1990). A sterile Release Model for Control of a Pest with two Life Stages under Predation. The Rocky Mountain Journal of Mathematics, pages 847–855.
[7] Bhattacharyya, R. and Mukhopadhyay, B. (2014). Mathematical study of a pest control model incorporating sterile insect technique. Natural Resource Modeling, 27(1):61– 79.
[8] Blomefield, T. et al. (1989). Economic Importance of False Codling Moth, Cryptophlebia leucotreta, and Codling Moth, Cydia Pomonella, on Peaches, Nectarines and Plums. Phytophylactica, 21(4):435–436.
[9] Bhunu, C. and Mushayabasa, S. (2011). Modelling the transmission dynamics of pox-like infections. International Journal of Applied Mathematics.
[10] Blomefield, T. et al. (1989). Economic Importance of False Codling Moth, Cryptophlebia leucotreta, and Codling Moth, Cydia Pomonella, on Peaches, Nectarines and Plums. Phytophylactica, 21(4):435–436.
[11] Boardman, L., Grout, T. G., and Terblanche, J. S. (2012). False codling moth thaumatotibia leucotreta (lepidoptera, tortricidae) larvae are chill-susceptible. Insect Science, 19(3):315–328.
[12] Byers, J. A. (2014). Simulation of Mating Disruption and Mass Trapping with Competitive Attraction and Camouflage. Environmental Entomology, 36(6):1328–1338.
[13] Castillo-Chavez, C. and Song, B. (2004). Dynamical models of tuberculosis and their applications. Mathematical Biosciences Engineering, 1(2):361.
[14] Chouinard,D.,Vanoosthuyse,G.,Pelletier,F.,Bellerose,F.,Bourgeois,S.,andDominique, P. (2015). A Phenology Model for Codling Moth Management in Quebec Apple Orchards. Acta Horticulturae, 1068(5):51–56.
[15] Gianni, G., Pasquali, S., Parisi, S., and Winter, S. (2014). Modelling the Potential Distribution of Bemisia Tabaciin Europe in Light of the Climate Change scenario. Pest Management science, 70:1611–1623.
[16] Goh, B. (1976). Global stability in two species interactions. Journal of Mathematical Biology, 3(3):313–318.
[17] Ikemoto, Y., Ishikawa, Y., Miura, T., Asama, H. (2009). A mathematical model for caste differentiation in termite colonies (isoptera) by hormonal and pheromonal regulations. Sociobiology, 54(3), 841.
[18] Hofmeyr, J. H., Hofmeyr, M., Lee, M., Kong, H., and Holtzhausen, M. (1998). Assessment of a Cold Treatment for the Disinfestations of Export Citrus from False Codling Moth, Thaumatotibia leucotreta (Lepidoptera: Tortricidae): a Report to the People’s Republic of China. Citrus Research International http://www. citrusres.
com/sites/default/files/documents/FCMor
[19] FPEAK (2021). Protocols for the Management of the False Codling Moth (Thaumatotibia Leucotreta) in Roses in Kenya. Kenya Flower Council Technical Committee, Kenya Plant Health Inspectorate Service (KEPHIS), Fresh Produce Exporters Association of Kenya (FPEAK), Kenya Agricultural Livestock Research Organization (KALRO) and the Europe-AfricaCaribbean-Pacific Liaison Committee (COLEACP) in the scope of its NExT Kenya programme. https://fpeak.org/wpcontent/uploads/2021/05/FCM-Manual.pdf
[20] La Salle, J. P. (1966). An invariance principle in the theory of stability. Space Flight and Guidance Theory,
[21] Liu, X. and Dai, B. (2018). Threshold dynamics of a delayed predator–prey model with impulse via the basic reproduction number. Advances in Difference Equations, 2018(1):454.
[22] May, R. M. and Anderson, R. M. (1978). Regulation and Stability of Host-Parasite Population Interactions: II. Destabilizing Processes. The Journal of Animal Ecology, pages 249–267.
[23] Mkiga, A., Mohamed, S., du Plessis, H., Khamis, F., and Ekesi, S. (2019). Field and Laboratory Performance of False Codling Moth, Thaumatotibia Leucotreta (Lepidoptera: Troticidae) on Orange and Selected Vegetables. Insects, 10(3):63.
[24] Mondaca, L. L., Da-Costa, N., Protasov, A., Ben-Yehuda, S., Peisahovich, A., Mendel, Z., and Ment, D. (2020). Activity of
metarhizium brunneum and beauveria bassiana against early developmental stages of the false codling moth thaumatotibia leucotreta. Journal of Invertebrate Pathology, 170:107312.
[25] Murray, J. (2002). Mathematical Biology (eds Antman, SS, Marsden, JE, Sirovich, L. Wiggins, S.) 175–256. 126
[26] Okongo, M. (2016). Modeling HIV-AIDS Co-Infections with Malaria and Tuberculosis in the Presence of Antiretroviral Treatment and Counseling. PhD thesis, Kenya: Chuka University.
[27] Potgieter, L. (2013a). A mathematical Model for the Control of Eldana Saccharina Walker using the Sterile Insect Technique. PhD thesis, Stellenbosch: Stellenbosch University.
[28] Savary, S., Teng, P. S., Willocquet, L., and Nutter Jr, F. W. (2006). Quantification and Modeling of Crop Losses: A Review of Purposes. Annu. Rev. Phytopathol., 44:89– 112.
[29] Sergio, R. (2014). The Optimal Release of Sterile Males in Pest Management. All Graduate Plan B and other Reports, 408. 128
[30] Stibick, J., Bloem, S., Carpenter, J., Ellis, S., and Gilligan, T. (2008). New Pest Pesponse Guidelines: False Codling Moth Thaumatotibia leucotreta. Technical report, USDA– APHIS–PPQ–Emergency and Domestic Programs, Riverdale, Maryland. http ….
[31] Ullah, R., Zaman, G., and Islam, S. (2013). Stability analysis of a general sir epidemic model. VFAST Transactions on Mathematics, 1(1):57–61.
[32] Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2):29–48.
[33] Venette, R. C., Davis, E. E., DaCosta, M., Heisler, H., and Larson, M. (2003). Mini Risk Assessment False Codling Moth, Thaumatotibia (Cryptophlebia) leucotreta (Meyrick)(Lepidoptera: Tortricidae). University of Minnesota, Department of Entomology, CAPS PRA, pages 1–30.
[34] Verreynne, S. et al. (2009). Fruit size and crop load prediction for citrus. SA Fruit Journal, 8(5):63–67.
[35] Witzgall, P., Kirsch, P., and Cork, A. (2010). Sex Pheromones and their Impact on Pest Management. Journal of Chemical Ecology, 36(1):80–100.
Published
2021-09-27
How to Cite
Jimrise, O., Okongo, M. O., & Muraya, M. (2021). Mathematical Modeling of Host - Pest Interactions in Stage-Structured Populations: A Case of False Codling Moth [Thaumatotibia leucotreta]. Journal of Progressive Research in Mathematics, 18(4), 1-21. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2087
Section
Articles