The effect of a non-uniform turning kernel on ant trail morphology

An ordinary differential equation model is constructed for the formation of pheromone trails by ants on a pre-determined network. At each junction of the trails the probability that an ant will turn through any particular angle is given by a turning kernel. We prove analytically using analogies with thermodynamics that turning behaviour determines trail morphology when the turning kernel is steep. We conjecture that this is also true in general for non-uniform turning kernels and present numerical simulations as evidence. Using this conjecture we show the existence of three types of collective foraging: individuals exploring without the use of a trail network, and two distinct types of trail networks; one that consists of low pheromone concentration trails that bend, branch and dissipate and one that consists of high pheromone concentration, straight, unbranched trails. We show that the form of the pheromone response function is crucial in determining the existence and stability of the steady states corresponding to these three foraging strategies, and examine the bifurcations between different trail morphologies as a function of turning kernel steepness for a particular response function.