Modelling a two-dimensional random alarm process.

Abstract

This paper deals with the mathematical modelling of two-dimensional alarm processes randomly spreading, amplifying and switching off within limited distributions of particles (individuals). It has been stimulated by recent studies on the enemy alarm behavior upon disturbance in Australian bull-dog ants (Myrmecia). The alarm within a random distribution of a limited number of resting particles in a finite two-dimensional region starts with the excitation, i.e. stochastic movement of a single particle. The excitation or alarm is spread over the distribution by excitation transfer, which occurs if the distance between the moving and a resting particle is below a fixed value. The mathematical model proceeds in three steps: (a) modelling of the stochastic movement of a single excited particle; (b) quantitative description of the area scanned by a single particle; (c) simulation of the whole many-particle process, i.e. amplification and switching off of the alarm. The essential parameters characterizing the single particles' motion are the particle velocity nu, and the turning frequency beta for the statistically independent changes in the direction of movement. Further parameters of the model, which determine the spread of the alarm, are the excitation period T, the capture radius Rc, the particle density rho and the extent of the distribution. The sensitivity of the process to variations of these parameters has been studied by averaging over a great number of stochastic simulations. The results show that the parameters as realistically estimated for the case of the bull-dog ants (nu = 10 cm/s, P = 2/s, T = 5s, Rc = 10 cm, 10 particles within a circular region of radius Rp = 50 cm) represent a possible set which on the average leads to a successful spread of the alarm.