Stable territory formation in ecology and its potential generality in pattern formations

• We propose a new stochastic individual-based scheme to predator–prey systems.
• Our scheme can produce stable Turing patterns.
• Our scheme can work as a logical bridge with the reaction–diffusion theory.

Stable territory formation is frequently observed in ecology. Until now, only the reaction–diffusion scheme has successfully produced stable patterns in the predator–prey system. However, it is a density-based scheme and in principle it cannot be used to derive a comprehensive understanding from a mean-field scheme. The application of our new stochastic individual-based scheme to predator–prey systems successfully produced stable patterns such as net, stripe, and lattice patterns for the first time. This study clearly shows that non-interacting time is an important factor in stable pattern formation. Specifically, of high importance is the existence of finite time to build the appetites of predators. In some cases, extreme increases of the appetites of predators lead to chaotic changes of the population, which are similar to the locust outbreak in Africa.