Simulation study of territory size distributions in subterranean termites

In this study, on the basis of empirical data, we have simulated the foraging tunnel patterns of two subterranean termites, Coptotermes formosanus Shiraki and Reticulitermes flavipes (Kollar), using a two-dimensional model. We have defined a territory as a convex polygon containing a tunnel pattern and explored the effects of competition among termite territory colonies on the territory size distribution in the steady state that was attained after a sufficient simulation time. In the model, territorial competition was characterized by a blocking probability Pblock that quantitatively describes the ease with which a tunnel stops its advancement when it meets another tunnel; higher Pblock values imply easier termination. In the beginning of the simulation run, N=10, 20,…,100 territory seeds, representing the founding pair, were randomly distributed on a square area. When the territory density was less (N=20), the differences in the territory size distributions for different Pblock values were small because the territories had sufficient space to grow without strong competitions. Further, when the territory density was higher (N>20), the territory sizes increased in accordance with the combinational effect of Pblock and N. In order to understand these effects better, we introduced an interference coefficient γ. We mathematically derived γ as a function of Pblock and N: γ(N,Pblock)=a(N)Pblock/(Pblock+b(N)). a(N) and b(N) are functions of N/(N+c) and d/(N+c), respectively, and c and d are constants characterizing territorial competition. The γ function is applicable to characterize the territoriality of various species and increases with both the Pblock values and N; higher γ values imply higher limitations of the network growth. We used the γ function, fitted the simulation results, and determined the c and d values. In addition, we have briefly discussed the predictability of the present model by comparing it with our previous lattice model that had been used to explain the territory size distributions of mangrove termites on the Atlantic coast of Panama.