if by MPC you mean Model Predictive Control, then that is "merely" an advanced process control framework and NOT a modeling method.

You would typically use a MPC if you have a multivariate control problem and/or if the model equations are not invertible and such hinder their direct use in a control framework. For instance, if you would want to study how you can maintain a population at a fixed level (called set-point) and you could for instance interfere with the system by reducing the number of predators or prey (i.e. variables that you can manipulate) then an MPC in combination with a model might be a good option.

Typically a linear model is used with MPC.

Thank you. @Moritz