We have a nonlinear regression model with $m$ parameters ($\alpha_1,\alpha_2, ..., \alpha_m$) and $n$ regressors $(X_1, .... X_n)$, predicting an outcome Y:
$Y = f(\alpha_1,\alpha_2, ..., \alpha_m; X_1, .... X_n) + error$
We define a second model in which one of the parameters (e.g $\alpha_1$) is a linear function of a new regressor ($X_{n+1}$)
$Y = f(\alpha_1 = a + b*X_{n+1} ,\alpha_2, ..., \alpha_m; X_1, .... X_n) + error$
So the second model is similar to the first model except that the value of $\alpha_1$ for the $i$th data point is linearly related to $x_{i,n+1}$. The second model has one more parameter than the first model ($(a, b, \alpha_2, ..., \alpha_m)$ compared to $(\alpha_1,\alpha_2, ..., \alpha_m)$)
We fit both models to data and obtain the maximum likelihoods. What's the best way to check if the second model is significantly better than the first one? Can we use likelihood ratio test?