An ant navigation model based on Weber's law

We analyze an ant navigation model based on Weber's law, where the ants move across a pheromone landscape sensing the area using two anten-nae. The key parameter of the model is the angle 2β representing the span of the ant's sensing area. We show that when β < π/2 ants are able to follow (straight) pheromone trails proving that for initial conditions close to the trail, there exists a Lyapunov function that ensures ant trajectories converge on and follow the pheromone trail, with these solutions being locally asymptotically stable. Furthermore, we indicate that the features of the ant trajectories such as convergence speed or oscillation wave length are controlled by the angle β. For β > π/2, we present numerical evidence that indicates that ants are unable to follow pheromone trails. We also assess our model by comparing it to previous experimental results, showing that the solutions' behavior falls into biologically meaningful ranges. Our work provides solid mathematical support for experimental studies where it was found that ant perception follows a Weber's law, by proving that such models lead to the desired robust and stable trail following.