Abstract
Song and Xiang (2006) developed an impulsive differential equations model for a two-prey one-predator model with stage structure for the predator. They demonstrate the conditions on the impulsive period for which a globally asymptotically stable pest-eradication periodic solution exists, as well as conditions on the impulsive period for which the prey species is permanently maintained under an economically acceptable threshold. We extend their model by including stage structure for both predator and prey as well as by adding stochastic elements in the birth rate of the prey. As in Song and Xiang (2006), we find the conditions under which a globally asymptotically stable pest eradication periodic solution exists. In addition, we numerically show the relationship between the stochastically varying birth rate of the prey and the necessary efficacy of the pesticide for which the probability of eradication of the prey species is above 90%. This is significant because the model recognizes varying environmental and climatic conditions which affect the resources needed for pest eradication.
Highlights
It is well-known that a variety of pest species pose a serious health risk to humans and pets, as well as causing great damage to property and crops
A number of recent articles have mathematically modeled a variety of integrated pest management (IPM) approaches using impulsive differential equations, taking into account, for example, stage structure in the predator species and periodically varying environmental conditions (Song and Xiang, 2006)
Here we seek a value of E, the pesticide potency or application effectiveness, under randomly varying birth rates in an attempt to maximize the eradication probability
Summary
It is well-known that a variety of pest species pose a serious health risk to humans and pets, as well as causing great damage to property and crops. As is more realistic in most ecosystems, we consider a random birth rate following a prior distribution with a mean that replaces the fixed birth rate of the previous models considered in Zhang et al (2003, 2005, 2007); Tang et al (2005); Song and Xiang (2006) This approach generalizes the model to accommodate random fluctuations, not just periodic fluctuations, in the birth rate due to environmental and climatic factors. The stochastic birth rate component in the proposed model accommodates factors such as shortened day length and lower temperatures, which may induce varying levels of egg production (Paulson et al, 2009) It recognizes that a fixed birth rate really represents an “average” birth rate, which may produce misleading results as to the resources necessary to ensure a high probability of pest eradication. We present numerical results showing the relationship between the birth rate parameter b and value of E, the pesticide potency or application effectiveness
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
