Asymptotic Dynamics of a Larch Budmoth Interaction with Harvesting and Thinning Operatio

Abstract

The larch budmoth (Zeiraphera diniana) poses a significant threat to larch-dominated forests, particularly in the cooler regions of the northern hemisphere, where periodic outbreaks of this pest are well-documented. These outbreaks can substantially damage larch trees, affecting ecological balance and economic stability. The pest control measures, such as chemical spraying, biological agents, and cultural practices, are optimal strategies for mitigating larch budmoth damage while minimizing environmental impacts. We propose a discrete-time mathematical model by incorporating pest control measures, explicitly harvesting and thinning operations, to manage larch budmoth infestations. This is a novel modelling framework that integrates these interventions and evaluates their effectiveness using the Beverton-Holt equation, which helps in understanding how the model exhibits a monotonically increasing reproduction rate. We discuss the biological context of plant quality of larch trees and budmoth population density. The existence of the model's fixed point has been observed through numerical simulations due to the model's complexity. We then study the parametric conditions necessary for the local stability of the equilibria. The trivial fixed point exhibits saddle point characteristics, while the stability of the positive fixed point is discussed based on the Jury conditions. Moreover, the existence of period-doubling bifurcation using the center manifold theorem based on the residue rate of the pesticide is provided. Enhanced accuracy is achieved in this model due to the consideration of residual rates and the timing of harvesting operations. When applying pest control, the most efficient pest control agents sometimes cause the most extreme pest outbreaks, called paradoxical emergence. Our work observes that the conditions for paradox emergence are never met. Specifically, the paradoxical increase never occurs in this scenario: The harvesting or thinning process ultimately results in an equilibrium population size smaller than before the harvesting or thinning was applied. We perform numerical simulations to validate theoretical findings. The proposed model suggests that the larch budmoth population increases with residual rates while plant quality decreases under the same conditions, and the larch budmoth population decreases with mortality rates while plant quality increases under the same conditions.