THE INFLUENCE OF SPECIES' NUMBER AND THE DENSITY OF VACANT SITES ON THE DEFENSIVE ALLIANCE

The multi-species ecological system with cyclic dominance and vacant sites is studied. The theme is the forming problem of species' defensive alliances in the corresponding systems. György Szabó and Gustavo Arial Sznaider have discussed the forming problem of species' defensive alliance in Phys. Rev. E69, 031911 (2004). Now, this problem is discussed in depth. The following conclusions are obtained: Firstly, the necessary and sufficient condition of forming defensive alliances is that the number of species is even. Secondly, in the condition where the number of species is an even number, for any density of vacant sites, there is three forms of defensive alliance as follow:

• Deadlock — the habitats of specie's individuals in each alliance interweave together.

• Relative preponderance — the different of the number of species' individuals between the two defensive alliances keeps steadily.

• Absolute preponderance — the different of the number of species' individuals between the two defensive alliances enlarges continuously.

This paper discusses the relations between the number of species and the forming of the defensive alliances, and shows the corresponding three statistical distributions of discrete density of vacant sites when there are four species.