Abstract
A nonlinear discrete switching host-parasitoid model with Holling type II functional response function, in which the switch is guided by an economic threshold (ET), is proposed. Thus, if the weighted density of two generations of the host population increases and exceeds the ET, then integrated pest management (IPM) measures are enacted, i.e., biological and chemical measures are implemented together, assuming that the chemical immediately precedes the biological inputs to avoid pesticide-induced deaths of the natural enemies. First, the existence and local stability of the equilibria of two subsystems were studied, and the existence and coexistence of several types of equilibria of a nonlinear switching system were analysed. Next, the nonlinear switching system was investigated by numerical simulation, showing that the system exhibits quite complex dynamic behaviour. A two-dimensional bifurcation diagram revealed the existence and coexistence regions of different types of equilibria including regular and virtual equilibria. Moreover, period-adding bifurcations in two-dimensional parameter spaces were found. One-dimensional bifurcation diagrams revealed that the system has periodic, quasiperiodic, and chaotic solutions, Neimark–Sacker bifurcation, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the initial densities of hosts and parasitoids associated with host outbreaks and their biological implications are discussed.
Highlights
Pest control is important in subjects such as agriculture, fisheries, and ecology [1]
In order to control pests more effectively and to consider the impact of control measures on the environment, the integrated pest management (IPM) strategy was proposed [3,4,5,6], which requires the combination of chemical control and biological control. ere are some important concepts in IPM, including the economic injury level (EIL) [7,8,9], economic threshold (ET) [8,9,10], and threshold policy control (TPC) [11,12,13]. e purpose of IPM is not to eliminate pests completely, but to control them at levels below the EIL. us, the classical method is to maintain the density of a pest population below the EIL by spraying pesticides and releasing natural enemies and/or other measures such as cultural control once the density of the pest population reaches the ET
Erefore, in order to design the best control policies to prevent host outbreaks, one of the possible ways is to choose a desirable switching curve such that all equilibria of subsystem SG1 become regular and all equilibria of subsystem SG2 become virtual. us, the parameters a and ET should be carefully chosen such that the interior equilibria of the two subsystems SG1 and SG2 are in region III-1
Summary
Pest control is important in subjects such as agriculture, fisheries, and ecology [1]. Yang et al [14] proposed and analysed a Holling type II host-parasitoid model with IPM intervention as a pulse control strategy, including models with fixed pulses and unfixed pulses. In order to apply such a threshold policy control (TPC) to IPM measures, a switching system (or Filippov system) is proposed and studied. Such systems, based on the ordinary differential equation (ODE) model for the predator-prey interaction, have been investigated in recent publications (see [12, 13, 16,17,18] for relevant references).
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