Insects breathe using one or a combination of three gas exchange patterns; continuous, cyclic and discontinuous, which vary in their rates of exchange of oxygen, carbon dioxide and water. In general, there is a trade-off between lowering gas exchange using discontinuous exchange that limits water loss at the cost of lower metabolic rate. These patterns and hypotheses for the evolution of discontinuous exchange have been examined for relatively large insects (>20 mg) over relatively short periods (<4 h), but smaller insects and longer time periods have yet to be examined. We measured gas exchange patterns and metabolic rates for adults of a small insect pest of grain, the red flour beetle, Tribolium castaneum (Coleoptera: Tenebrionidae), using flow-through respirometry in dry air for 48 h. All adults survived the desiccating measurement period; initially they used continuous gas exchange, then after 24 h switched to cyclic gas exchange with a 27% decrease in metabolic rate, and then after 48 h switched to discontinuous gas exchange with increased interburst duration and further decrease in metabolic rate. The successful use of the Qubit, a lower cost and so more common gas analyser, to measure respiration in the very small T. castaneum, may prompt more flow-through respirometry studies of small insects. Running such studies over long durations may help to better understand the evolution of respiration physiology and thus suggest new methods of pest management.

We consider a fractional equation involving the left and right Riemann–Liouville fractional integrals and with Sturm–Liouville boundary-value conditions. We establish the variational structure of the problem and, by using critical-point theory, the existence of an unbounded sequence of solutions is obtained.

Let $L:\Bbb R^N\times\Bbb R^N\rightarrow\Bbb R$ be a Borelian function andconsider the following problems $$\inf\left\{F(y)=\int_a^bL(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\} \qquad\quad\! (P)$$ 
 $$ \inf\left\{F^{**}(y)=\int_a^bL^{**}(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N), y(a)=A,\,y(b)=B\right\}\cdot \quad\;\ \! (P^{**})$$ We give a sufficient condition, weaker then superlinearity, under which $\inf F=\inf F^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.