Mathematical Model for False Codling Moth Control Using Pheromone Traps
Keywords:False codling moth, Mathematical modeling, stability Analysis, pest, Pheromone Traps, Plant pest model
False codling moth (FCM) is regarded as the most significant indigenous pest. Over 70 host plants are attacked by larval, many of which are horticultural crops with fruit, pods, and berries, such as beans, grapes, citrus, capsicum, avocado, guava, pomegranate, and ornamental plants. They eat macadamia nuts, cotton, tea, and a variety of other wild plants as well. Female moths are drawn to the flower heads as well as other parts of the plant, making this pest especially problematic on roses grown for cut flowers. Therefore, controlling this pest is of importance. For more effective control, pheromone traps are used to capture males attracted to the artificial pheromone. In this study, mathematical model of FCM control using pheromone trap is developed. The model, is based on biological and ecological assumptions, and is governed by an ODE system. The coexistence and pest free equilibria is determined through a theoretical analysis. The theoretical analysis of the model allows for the identification of pheromone threshold values that are practical for field applications. We show that there is a threshold above which the global asymptotic stability of the trivial equilibrium is guaranteed. Finally, we demonstrate the theoretical results through numerical experiments.
 W. C. Agosta, Chemical Communication: the Language of Pheromones. Henry Holt and Company, (1992).
 H. T. Alemneh, O. D. Makinde, D. M. Theuri, Mathematical modelling of msv pathogen interaction with pest invasion on maize plant, Global Journal of Pure and Applied Mathematics, 15(1):55â€“79 (2019).
 R. Anderson, R. May, Population Biology of Infectious Diseases: Part I. Nature, 280:361â€“7.(1979).
 M.R.Anguelov, C.Dufourd, Y.Dumont, (2016), Mathematical Model for Pest-Insect Control using Mating Disruption and Trapping. Applied Mathematical Modelling. (2016).
 H. J. Barclay, G. E. Haniotakis, . Combining Pheromone-Baited and Food- Baited Traps for Insect Pest Control: Effects of Developmental Period. Population Ecology, 33(2):269â€“285 (1991).
 H. J. Barclay, R. Steacy, W. Enkerlin, P. van den Driessche, Modeling Diffusive Movement of Sterile Insects Released along Aerial Flight Lines. International Journal of Pest Management, 62(3):228â€“244.(2016).
 H. J. Barclay, P. Van Den Driessche, A sterile Release Model for Control of a Pest with two Life Stages under Predation. The Rocky Mountain Journal of Mathematics, pages 847â€“855.(1990).
 H.Bestmann, J. Erler, O. Vostrowsky, Determination of Diel Periodicity of Sex Pheromone Release in Three Species of Lepidoptera by â€˜Closed-LoopStrippingâ€™. Experientia, 44(9) (1988):797â€“799
 Bhattacharyya, R. and Mukhopadhyay, B. (2014). Mathematical study of a pest control model incorporating sterile insect technique. Natural Resource Modeling, 27(1):61â€“ 79.
 T.Blomefield, Economic Importance of False Codling Moth, Cryptophlebia leucotreta, and Codling Moth, Cydia Pomonella, on Peaches, Nectarines and Plums. Phytophylactica, 21(4):435â€“436.(1989).
 C.Bhunu, S. Mushayabasa, Modelling the transmission dynamics of pox-like infections. International Journal of Applied Mathematics.(2011).
 L. Boardman, T.G. Grout, J.S. Terblanche, . False codling moth thaumatotibia leucotreta (lepidoptera, tortricidae) larvae are chill-susceptible. Insect Science, 19(3):315â€“328.(2012)
 J. H. Borden, Use of Semiochemicals to Manage Coniferous Tree Pests in Western Canada. In Behaviour Modifying Chemicals for Insect Management: Application of Pheromones and other Attractants, (1990).
 E. Boissard, P. Degond, S. Motsch, (2011). Trail Formation Based on Directed Pheromone Deposition. Hal.
 J. A.Byers, Simulation of Mating Disruption and Mass Trapping with Competitive Attraction and Camouflage. Environmental Entomology, 36(6):1328â€“1338. (2014).
 R. Carde, Using pheromones to disrupt mating of moth pests. Â´ Perspectives in ecological theory and integrated pest management. Cambridge University Press, Cambridge, (2007). pages 122â€“169.
 R. T. Carde, A. K. Minks, Control of Moth Pests by Mating Disruption: Successes and Constraints. Â´ Annual Review of Entomology, 40(1), (1995):559â€“585.
 R. T. Carde, Principles of Mating Disruption. Behavior-Modifying Chemicals for Pest Management: Applications of Pheromones and Other Attractants. Marcel Dekker, New York,(1990), pages 47â€“71.
 C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering, 1(2):361.(2004).
 D. Chouinard,G.Vanoosthuyse,F.Pelletier, F.Bellerose, S.Bourgeois, P. Dominique, A Phenology Model for Codling Moth Management in Quebec Apple Orchards. Acta Horticulturae, 1068(5):51â€“56.(2015).
 G. Gianni, S. Pasquali, S. Parisi, S. Winter, Modelling the Potential Distribution of Bemisia Tabaciin Europe in Light of the Climate Change scenario. Pest Management science, 70:1611â€“1623.(2014).
 T. M. Gilligan, M. E. Epstein, K. M. Hoffman, Discovery of false codling moth, Thaumatotibia leucotreta (Meyrick), in California (Lepidoptera: Tortricidae). Proceedings of the Entomological Society of Washington, 113(4),(2011), 426-435.
 B.Goh, Global stability in two species interactions. Journal of Mathematical Biology, 3(3):313â€“318.(1976).
 P.Howse, J. Stevens, O.T. Jones, Insect Pheromones and their use in Pest Management. Springer Science and Business Media (2013).
 . H.Hofmeyr, M. Hofmeyr, M. Lee, H. Kong, M. Holtzhausen, Assessment of a Cold Treatment for the Disinfestations of Export Citrus from False Codling Moth, Thaumatotibia leucotreta (Lepidoptera: Tortricidae): Report to the Peopleâ€™s Republic of China. Citrus Research International, http://www. citrusres. com/sites/default/files/documents/FCM% 20cold% 20disinfestation% 20study% 20f or% 20Korea, 201998.(1998).
 FPEAK, 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 and Livestock Research Organization (KALRO) and the Europe-Africa-Caribbean-Pacific Liaison Committee (COLEACP) in the scope of its NExT Kenya programme, (2021). https://fpeak.org/wp-content/uploads/2021/05/FCM-Manual.pdf
 Y.Ikemoto, Y. Ishikawa, T. Miura, H. Asama, A mathematical model for caste differentiation in termite colonies (isoptera) by hormonal and pheromonal regulations. Sociobiology, 54(3), 841. (2009).
 D. Kalajdzievska, M.Y. Li, Modeling the Effects of Carriers on the Transmissions Dynamics of Infectious Diseases. PhD thesis, University of
 U. T. Koch, W. Luder, U.Andrick, R. T. Staten, R. T. Card Â¨ e, Measurement by electroantennogram of airborne pheromone in cotton treated for Â´ mating disruption of Pectinophora gossypiella following removal of pheromone dispensers. Entomologia experimentalis et applicata, 130(1), 1-9. (2009).
 J. P. La Salle, An invariance principle in the theory of stability. Space Flight and Guidance Theory,(1966).
 X.Liu, B. and Dai, Threshold dynamics of a delayed predatorâ€“prey model with impulse via the basic reproduction number. Advances in Difference Equations, 2018(1):454.(2018).
 A. Korobeinikov, G. C. Wake, Lyapunov functions and global stability for sir, sirs, and sis epidemiological models. Applied Mathematics Letters, 15(8), (2002):955â€“960.
 R. M.May, R. M. Anderson, Regulation and Stability of Host-Parasite Population Interactions: II. Destabilizing Processes. The Journal of Animal Ecology, (1978) pages 249â€“267.
 M.Mkiga, S. Mohamed, H. du Plessis, F.Khamis, S. Ekesi, Field and Laboratory Performance of False Codling Moth, Thaumatotibia Leucotreta (Lepidoptera: Troticidae) on Orange and Selected Vegetables. Insects, 10(3),(2019):63.
 L. L. Mondaca, N. Da-Costa, A. Protasov, S. Ben-Yehuda, A. Peisahovich, Z Mendel, D. Ment, . Activity of metarhizium brunneum and beauveria bassiana against early developmental stages of the false codling moth thaumatotibia leucotreta. Journal of Invertebrate Pathology, 170, (2020):107312.
 S.Moore, Moths and Butterflies: False Codling Moth. Citrus Research International IPM Production Guidelines, 3 (2012)(Part 9.4).
 D. K. Mueller, (1995). Pheromones Chapter 25 in Moreland, D. (Ed.) Handbook of Pest Control, 8th Edition. Mallis Handbook and Technical Training Company, (1995).
 J.Murray, J. Mathematical Biology (eds Antman, SS, Marsden, JE, Sirovich, L. and Wiggins, S.) , (2002).175â€“256. 126
 M.Okongo, Modeling HIV-AIDS Co-Infections with Malaria and Tuberculosis in the Presence of Antiretroviral Treatment and Counseling. PhD thesis, Kenya: Chuka University, (2016).
 S. Pasquali,C. Soresina, G.Gilioli, G. The effects of fecundity, mortality and distribution of the initial condition in phenological models. Ecological modelling, 402, (2019):45â€“58.
 R. Peshin, A. K. Dhawan, Integrated Pest Management: Volume 1: Innovation Development Process, volume 1. Springer Science and Business Media , (2009).
 L. Potgieter, A mathematical Model for the Control of Eldana Saccharina Walker using the Sterile Insect Technique. PhD thesis, Stellenbosch: Stellenbosch University ,(2013a).
 S.Savary, P.S. Teng, L. Willocquet, F. W. Nutter Jr, Quantification and Modeling of Crop Losses: A Review of Purposes. Annu. Rev. Phytopathol, 44, (2006).:89â€“ 112.
 R.Sergio, The Optimal Release of Sterile Males in Pest Management. All Graduate Plan B and other Reports, 408. 128,(2014).
 J.Stibick, S. Bloem, J.Carpenter, S. Ellis, T. Gilligan, New Pest Pesponse Guidelines: False Codling Moth Thaumatotibia leucotreta. Technical report, USDAâ€“ APHISâ€“PPQâ€“Emergency and Domestic Programs, Riverdale, Maryland, (2008).
 Z. Tazerouni, A. Talebi, M. Rezaei, Functional response of parasitoids: Its impact on biological control. Parasitoids: Biology, Behavior, and Ecology. Nova Science Publishers Inc., New York , (2019)..
 M.B. Thomas and A.J. Willis, Biocontrol â€“ risky but necessary, Trends Ecol. Evol. 13 (1998), pp. 325â€“329.
 R.Ullah, G. Zaman, S. Islam, Stability analysis of a general sir epidemic model. VFAST Transactions on Mathematics, 1(1)(2013):57â€“61.
 M.Rafikov, E. de Holanda Limeira, Mathematical modelling of the biological pest control of the sugarcane borer. International Journal of Computer Mathematics, 89(3),(2012), 390-401.
 P.Van den Driessche, J.Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2)(2002):29â€“48.
 Z. Varga, Applications of mathematical systems theory in population biology, Period. Math. Hungar. 56 (1) (2008), pp. 157â€“168.
 R. C. Venette, E. E. Davis, M. DaCosta, H. Heisler, M. Larson, Mini Risk Assessment False Codling Moth, Thaumatotibia (Cryptophlebia) leucotreta (Meyrick)(Lepidoptera: Tortricidae). University of Minnesota, Department of Entomology, CAPS PRA,(2003). pages 1â€“30.
 S.Verreynne, Fruit size and crop load prediction for citrus. SA Fruit Journal, 8(5),(2009) :63â€“67.
 B. P. Wilson, L. K. Alan, F. C. Eduardo, Modeling Codling Moth (Lepidoptera: Tortricidae) Phenology and Predicting Egg Hatch in Apple Orchards of the Maule Region, Chile. Chilean Journal of Agricultural Research, (2015).
 P.Witzgall, P. Kirsch, A. Cork, . Sex Pheromones and their Impact on Pest Management. Journal of Chemical Ecology, 36(1),(2010):80â€“100.
 C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence. Nonlinear Analysis: Real World Applications, 11(4),(2010), 3106-3109.