Spiracular fluttering decouples oxygen uptake and water loss: a stochastic PDE model of respiratory water loss in insects.

In insect respiration, oxygen from the air diffuses through a branching system of air-filled tubes to the cells of the body and carbon dioxide produced in cellular respiration diffuses out. The tracheal system has a very large surface area, so water loss is a potential threat and the question of how insects regulate oxygen uptake and water loss has been an important issue in insect physiology for the past century. The tracheal system starts at spiracles on the surface of the body that insects can open and close, and three phases are observed experimentally, open or closed for relatively long periods of time and opening and closing rapidly, which is called fluttering. In previous work we have shown that during this flutter phase, no matter how small the percentage of time that the spiracles are open, the insect can absorb almost as much oxygen as if the spiracle were always open, if the insect flutters fast enough. This left open the question of water loss during the flutter phase, which is the question addressed in this paper. We formulate a stochastic diffusion-convection model for the concentration of water vapor in the tracheae. Mathematical analysis of the model yields an explicit formula for water loss as a function of six non-dimensional parameters and we use experimental data from various insects to show that, for parameters in the physiological ranges, water loss during the flutter phase is approximately proportional to the percentage of time open. This means that the insect can solve the oxygen uptake versus water loss problem by choosing to have their spiracles open a small percentage of time during the flutter phase and fluttering rapidly.