Spatial Pattern in a Chemical Reaction System: Prey and Predator in the Position-Fixed Limit

The spatial pattern in an ecosystem composed of prey and predator is studied. Under the assumption that the position of each individual is fixed, we are able to obtain a stationary pattern. Depending on environmental conditions, both species form specific patterns. In particular, when the death rate of the predator is large or small, the prey then tends to clump, while predators are evenly distributed. Moreover, we observe a paradox that even if the death rate of the predator becomes low, the number of predator decreases; predators cannot easily catch prey because of their clumping behaviour. Such a paradox is never explained by mean-field approximation.