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This randomness may be due to varying environmental Glafenine and climatic conditions. The stochastic parameter selection mimics the implicit variability of these natural conditions. Thanks to the flexibility introduced with this approach, the resulting model allows for more realistic and accurate prediction of the optimal pesticide level, as explained below, needed for total eradication or economically feasible control of the pests. Just as Song and Xiang (2006) incorporates seasonal variation through the intrinsic growth rate parameter of the prey, the authors in Akman et al. (2014) and Akman et al. (2013) incorporate random variation through parameters in the birth pulses for the prey. Specifically, values for these birth rate parameters are chosen randomly from various probability distributions. KEY CONCEPT 4 Stochasticity Stochasticity allows greater flexibility in modeling by incorporating random variation over selected parameters. By further incorporating stochastic parameters in the birth-pulse function, the proposed mixture model accommodates not only a broad range of birth rates but also different population growth behavior of the pest species. We achieve this by introducing a probabilistic see more mixture of different birth-pulse functions with stochastic weights that optimize the overall efficacy of the model. In statistical modeling theory, it is well-known that the mixture modeling approach produces flexible models with good statistical and probabilistic properties. This is especially useful when X and Y are two competing random variables where unobserved heterogeneity that exists in the data collection process results in lack of fitness. Specifically, if 0 Lenvatinib solubility dmso fY(x) are the probability density functions (pdf) of the random variables (r.v.) X and Y, respectively, then the pdf of the r.v. Z produced by the mixture between X and Y is given by fZ(x)??=??(1?��)fX(x)+��fY(x),x>0. (2) Our model that includes a set of external effects such as environmental and climatic conditions of a more random nature takes its inspiration from the approach given by Equation (2). We use two Beverton-Holt type birth-pulse functions, B1(t)=b1x2(t)q1+x1(t)+x2(t) and B2(t)=b1x2(t)q2+x1(t)+x2(t), which are widely used population density dependent birth-pulse functions Tang and Chen (2002). Additionally, we introduce further stochasticity by considering b1 and b2 as random variables. The impulsive effect for the immature prey in the model used in the stochastic version of System (1) in Akman et al. (2014) is defined by X1(t+)=(x1(t)+��B1(t)+(1?��)B2(t))(1?E), (3) for t = nT, where 0