Duality in Phase Space and Complex Dynamics of an Integrated Pest Management Network Model

Abstract

Fragmented habitat patches between which plants and animals can disperse can be modeled as networks with varying degrees of connectivity. A predator–prey model with network structures is proposed for integrated pest management (IPM) with impulsive control actions. The model was analyzed using numerical methods to investigate how factors such as the impulsive period, the releasing constant of natural enemies and the mode of connections between the patches affect pest outbreak patterns and the success or failure of pest control. The concept of the cluster as defined by Holland and Hastings is used to describe variations in results ranging from global synchrony when all patches have identical fluctuations to n-cluster solutions with all patches having different dynamics. Heterogeneity in the initial densities of either pest or natural enemy generally resulted in a variety of cluster oscillations. Surprisingly, if n > 1, the clusters fall into two groups one with low amplitude fluctuations and the other with high amplitude fluctuations (i.e. duality in phase space), implying that control actions radically alter the system's characteristics by inducing duality and more complex dynamics. When the impulsive period is small enough, i.e. the control strategy is undertaken frequently, the pest can be eradicated. As the period increases, the pest's dynamics shift from a steady state to become chaotic with periodic windows and more multicluster oscillations arise for heterogenous initial density distributions. Period-doubling bifurcation and periodic halving cascades occur as the releasing constant of the natural enemy increases. For the same ecological system with five differently connected networks, as the randomness of the connectedness increases, the transient duration becomes smaller and the probability of multicluster oscillations appearing becomes higher.