Effective condition of insecticidal rotation

Several strategies have been used in insecticide resistance management (IRM) to prevent the evolution of resistance, but the spatial aspects of insecticide application are crucially important among these strategies. Here, we consider a structured environment that consists of on-farm and off-farm fields where crops are planted periodically in on-farm fields during cultivation periods. We define the basic reproduction rate (R_0) of resistance as the expected number of offspring of a resistant individual divided by that of a susceptible individual under the condition that the proportion of resistance is extremely small; it is measured as the quantity per cycle of the cultivation period. We calculate log_e (R_0 ) using realistic dose-survival curves under a given fitness cost of resistance genes. The evolution of resistance occurs if and only if the log_e (R_0 ) value is larger than 0. Then, we propose a procedure for calculating the optimal design of rotational spraying that prevents the evolution of resistance, that is, the evolutionary stable strategy (ESS) for farmers, satisfying the mortality required for managing the abundance of insects. We consider the following controllable factors in calculating the optimal design: the dose of insecticide, the number of sprays, the number of different types of insecticides, and potentially, the size of on-farm fields.