A statistical model is linear with respect to a parameter if an estimate of that parameter can be obtained in a single iteration.

That is, the maximum-likelihood estimate of the parameter can be defined analytically by an equation derived from the deterministic model structure and error specification that requires only the observed data for input.

A statistical model is non-linear with respect to a parameter if the maximum-likelihood estimate of that parameter requires, in addition to the deterministic model structure, the error specification and the observed data, the value of at least one other parameter in the model.

Since initially the value of that parameter is usually a guess, parameter estimation requires more than one iteration to sequentially update the parameter values until their values converge to their maximum-likelihood estimates.

Linear statistical models are rarely robust enough to adequately describe biological or ecological systems.

Consequently most realistic biological and ecological models are non-linear in design and require non-linear methodologies for parameter estimation and interpretation of their uncertainty.

Note that simple linear regression is a linear model and gets part of its name from that fact, and not from the straight line defined by the deterministic model.

Likewise, a quadratic linear regression is a linear model despite the curvature of the deterministic model, hence such models are often referred to as curvilinear models.