Abstract
The organisms need energy to survive and reproduce. So, efficient foraging way becomes important to the predator. Ecologists observe that organisms will learn to be optimal and develop the optimal foraging theory. The majority of classical optimal foraging theories focus on deterministic models. In our study, we discuss the stochastic model to imitate the realistic environment. We consider the realistic problems, what information can we extract from data and does the predator provide with learning ability? In fact, we do not know what foraging rule adopted by the predator. All we have are data by observing or experiment.In this article, we try to simultaneously estimate the relative environmental and foraging parameters by sing backward dynamic programming as a pseudo tool whatever foraging rule adopted by the predator. We have elaborate results in different environmental assumptions and foraging styles. Also, we extend our discussions to more practical situations, for examples, the patch sizes may be different and the variety of the environment is larger. We not only give an optimal foraging rule, but also give an executable form to calculate. In addition, we also apply this dynamic tool to bio-statistics, finding a new agent by using dynamic optimal sample size, to sample the environment as the predator enters an unfamiliar environment, and so on.