A mathematical analysis of mode share and travel distance in active mobility with respect to facility density

Active mobility (AM), such as walking and cycling, is gaining attention as a sustainable transport mode in urban areas.It offers low environmental impact, efficient land use, and health benefits, though its use is constrained by travel distance.In compact cities where urban functions are concentrated, AM becomes a practical travel option.This study considers a setting where uniformly distributed demand visits the nearest facility based on Manhattan distance.The probability of choosing AM is modeled using a Gaussian decay function of distance.We examine how facility density influences both the number of AM users and the average distance for all travelers.For randomly distributed facilities, we derive simple analytical expressions for these quantities.We further show that the facility density maximizing the average travel distance equals the Gaussian decay parameter.This implies the existence of an optimal facility density that balances urban spatial scale with human mobility characteristics.