Modelling the effects of delayed mortality and saturated incidence on the population dynamics of infected ash trees

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Abstract

This study centers on infectious disease models that feature a delayed mortality rate and the Holling II type functional response function. Through a theoretical analysis, it is confirmed that the solutions of the model possess positivity and boundedness, and the equilibrium points are obtained with their local and global stability conditions ascertained. The research indicates that, in the absence of time delay, transcritical and Hopf bifurcations emerge in the model under specific parameter conditions, thus furnishing a foundation to understand the dynamics of disease transmission. Subsequently, a time delay is incorporated into the system, and its impact on the Hopf bifurcation is analyzed. Numerical simulations depict the dynamic behaviors of the model at different delay times. For instance, the system can swiftly reach a stable state without delay, and periodic oscillations occur when the delay surpasses the critical value, thus validating the theory and suggesting that the delayed death factor has a significant influence. By comparing with actual data, the model effectively depicts the spread and lethality of the disease in infected ash trees across different regions, and uncovers the population differences and dynamic change patterns. Thus, it offers targeted strategies for disease prevention and control, as well as the ecological management of ash trees, which is relevant to the study on the impacts of delayed mortality and the Holling-II functional response function on the population dynamics of infected ash trees.

Keywords

bifurcation / delayed mortality / Holling type II / stability

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References

[1] H. Wu, C. A. L. Dahlsjo, Y. Malhi, Evaluating the impact of an invasive pathogen on tree population decline: An evidence based modelling approach, For. Ecol. Manage., 566 (2024), 122098. https://doi.org/10.1016/j.foreco.2024.122098

[2] K. Mokni, M. Ch-Chaoui, B. Mondal, U. Ghosh, Rich dynamics of a discrete two dimensional predator-prey modelusing the NSFD scheme, Math. Comput. Simul., 225 (2024), 992-1018. https://doi.org/10.1016/j.matcom.2023.09.024

[3] M. Ch-Chaoui, K. Mokni, A multi-parameter bifurcation analysis of a prey-predator model incorporating the prey Allee effect and predator-induced fear, Nonlinear Dyn., 113 (2025), 18879-18911. https://doi.org/10.1007/s11071-025-11063-w

[4] K. Mokni, H. Mouhsine, M. Ch-Chaoui, Multi-parameter bifurcations in a discrete Ricker-type predator-prey model with prey immigration, Math. Comput. Simul., 233 (2025), 39-59. https://doi.org/10.1016/j.matcom.2025.01.020

[5] H. Mouhsine, K. Mokni, M. Ch-Chaoui, Exploring multi-parameter bifurcations and immigrationdriven dynamics in a discrete-time Bazykin-Berezovskaya prey-predator model, Nonlinear Dyn., 113 (2025), 34101-34131. https://doi.org/10.1007/s11071-025-11818-5

[6] G. Charron, M. Gauthier, V. Aucoin, P. Tanguay, Outbreak of Phytophthora abietivora in a Québec forest nursery: Emergence of a new Phytophthora tree pathogen?, For. Pathol., 55 (2025), e70041. https://doi.org/10.1111/efp.70041

[7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. https://doi.org/10.1137/S0036144500371907

[8] L. Chen, J. Sun, Global stability of an SI epidemic model with feedback controls, App. Math. Lett., 28 (2014), 53-55. https://doi.org/10.1016/j.aml.2013.09.009

[9] L. Wang, D. Zhou, Z. Liu, D. Xu, X. Zhang, Media alert in an SIS epidemic model with logistic growth, J. Biol. Dyn., 11 (2016), 120-137. https://doi.org/10.1080/17513758.2016.1181212

[10] R. M. Anderson, R. M. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367. https://doi.org/10.1038/280361a0

[11] S. Gao, L. Chen, J. J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045. https://doi.org/10.1016/j.vaccine.2006.05.018

[12] T. Zhang, Z. Teng, Pulse vaccination delayed SEIRS epidemic model with saturation incidence, Appl. Math. Modell., 32 (2008), 1403-1416. https://doi.org/10.1016/j.apm.2007.06.005

[13] D. Xiao, S. Ruan, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472. https://doi.org/10.1137/S0036139999361896

[14] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology , Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1686-9

[15] M. Kot, Elements of Mathematical Ecology , Cambridge University Press, Cambridge, 2001. https://doi.org/10.1017/CBO9780511608520

[16] B. Pirayesh, A. Pazirandeh, M. Akbari, Local bifurcation analysis in nuclear reactor dynamics by Sotomayor’s theorem, Ann. Nucl. Energy, 94 (2016), 716-731. https://doi.org/10.1016/j.anucene.2016.04.021

[17] B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation , Cambridge University Press, Cambridge, 1981.

[18] J. Wei, S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. D, 130 (1999), 255-272. https://doi.org/10.1016/S0167-2789(99)00009-3

[19] J. Wei, M.Y. Li, Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear Anal., 60 (2005), 1351-1367. https://doi.org/10.1016/j.na.2003.04.002

[20] Y. Qu, J. Wei, Bifurcation analysis in a predator-prey system with stage-structure and harvesting, J. Franklin Inst., 347 (2010), 1097-1113. https://doi.org/10.1016/j.jfranklin.2010.03.017

[21] N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. https://doi.org/10.1007/s11538-008-9299-0

[22] I. Matisone, R. Matisons, A. Jansons, The Struggle of Ash-Insights from long-term survey in Latvia, Forests, 12 (2021), 340. https://doi.org/10.3390/f12030340

[23] V. Timmermann, N. E. Nagy, A. M. Hietala, I. Borja, H. Solheim, Progression of ash dieback in Norway related to tree age, disease history and regional aspects, Balt. For, 23 (2017), 150-158.

[24] S. Klesse, G. von Arx, M. M. Gossner, C. Hug, A. Rigling, V. Queloz, Amplifying feedback loop between growth and wood anatomical characteristics of Fraxinus excelsior explains size-related susceptibility to ash dieback, Tree Physiol., 41 (2021), 683-696. https://doi.org/10.1093/treephys/tpaa091