Undesirable consequences of neglecting nonlinearity: response to comments by Liefting et al. (2013) on Rocha &...

In response to our previous study, Liefting et al. argue, in defense of their work on latitudinal variation of developmental-rate reaction norms (RNs), that (1) developmental rate (the reciprocal of development time: rate = time(-1) ) is a more biologically relevant variable than development time; (2) the linear RN model is a valid approximation; and (3) three experimental points suffice to estimate RN parameters. Here, we reply to their comments. First, we give evidence that the complexity of actual development challenges the appealing simplicity of developmental rate. Using the same analysis as Liefting et al. to test their hypothesis with development time, instead of rate, reveals a pattern that is the opposite of their conclusion. Second, we show that a quadratic model is consistent with the whole development-time RNs and explains this contradiction. Third, with the quadratic model, we introduce two parameters to study plasticity: the RN shape (the quadratic coefficient) and RN local plasticity (the derivative of the RN function). The first showed a statistically significant correlation with latitude; and the second showed a continuous variation pattern where all localized patterns can be found (positive, negative, or nonsignificant correlations with latitude) but certainly cannot be generalized.