A responsivity theory of call timing patterns in bat echolocation

Background The temporal patterning of bat echolocation is extensively characterised, but a general mathematical account linking call timing to acoustic propagation, relative motion, and behaviourally selected information remains incomplete. Results We present the responsivity framework, which represents each call event as a delayed active-sensing cycle comprising an acoustic acquisition interval and a subsequent behavioural interval. Their proportional relation is described by the responsivity coefficient, kᵣ k r , connecting call timing to effective anchor distance, relative velocity, call duration, and the selected echo window. Analytical derivation, call-by-call simulation, internal validation, and comparison with field recordings show how the framework generates hyperbolic call-rate–distance relationships, shifts approach timing with echo-window selection, and places the terminal buzz at the near-target limit of the same timing process. Responsivity and kinematics have separable terminal effects: kᵣ k r constrains terminal update density and attainable call rate, whereas closing velocity determines the time available within the terminal region. A buzz-readiness state precedes the complete expression of buzz, and transient sonar strobe groups emerge under multi-target anchor allocation without an explicit grouping rule. Field recordings show broad correspondence with simulations in the organisation of movement, call timing and call duration, and illustrate conditional evaluation of acquisition-window and anchor assumptions. Conclusions Responsivity theory provides a process-based descriptor of echolocation timing and a diagnostic framework for hypothesis generation and empirical testing. The framework identifies the measurements needed to test how selected acoustic information governs subsequent call timing.