Aerobic vs anaerobic respiration comparison
I think the reason as to why glucose concentration is faster in the aerobic case than the anaerobic one, is perfectly explained by Chris. To summarize:- 


  The energy requirement of the organism in both the condition remains more or less
  constant. Since aerobic respiration generates more energy per glucose,(38ATP)
  it takes more time for the same concentration of the glucose to be
  depleted, than it takes in the case of anaerobic respiration where
  less energy is released.(2ATP)  


The non-linearity in case of anaerobic respiration can probably be explained thus:-
In the aerobic case, glucose is oxidised to water and carbon dioxide, both of which do not influence the respiratory process. The first step of glycolysis, the phosphorylation of glucose has a very negative free energy change, i.e. is practically irreversible. Therefore a dip in glucose concentration will not affect the rate of energy release and therefore the rate of energy released will stay constant resulting in a linear graph. But in the anaerobic case, the final product is un-metabolised ethyl alcohol. The conversion of pyruvic acid formed as a result of glycolysis to ethyl alcohol is not so strongly irreversible(smaller free energy change), and at high product concentrations, becomes considerably reversible. This means that the ethyl alcohol produced will affect the rate of glucose metabolism, and will retard it by negative feedback (Le chatelier's principle). This will lower the speed of glucose utilisation at low concentrations of glucose, i.e. high concentrations of ethyl alcohol. This reduce the value of slope at lower concentrations of glucose in the graph, and hence produces the non-linearity.