I have a set of bivariate biological data that has a clearly non-linear distribution. I have found that a power curve best fits the data under several metrics (e.g., AIC, residuals versus fits plot, etc.). Specifically, I've noticed that a fractional power curve of $f(x)=x^{b}$ (where b is some fraction) seems to reliably model the data but a quadratic function $f(x)=x^{2}+x$ ends up being heavily influenced by the most extreme values on the x axis and is not robust to subsetting the data (i.e., subsetting the data at random produces a dramatically different curve). Overall the function seems mostly linear but higher values of x produce slightly higher than predicted values of y. I cannot log-transform the data, as the data are already log-transformed (and log-transformation normalizes their distribution) and this non-linearity is present even after log-transformation. When I submitted the research for consideration the reviewers wanted me to provide a theoretical justification as to why a power curve is preferable over a quadratic or exponential model.
I know that a quadratic equation, i.e., $f(x)=x^2+x$, is one where the best-fit line is nonlinear and the slope $dx/x$ changes at a constant rate. I also know that an exponential function, such as $f(x)=n^x$, is used when the slope of the best fit line is expected to change at a constant rate in proportion to x. I also know that exponential functions do not have constant elasticity, unlike polynomial or power curves.
What I'm having more trouble finding is what are the theoretical expectations that would lead one to assume a power function versus a quadratic or exponential one? I've seen a lot of questions talking about the difference between quadratic and exponential models (e.g., this question on Quora), but not so much how either relates to a power model. I.e., I know an equation of $f(x)=x^{1/2}$ would have a constant $dx/x$ of 0.5, but I don't know what theoretical expectations would make somebody consider that a more reasonable approximation of the relationship than quadratic model (or vice versa).