Dear Jordan

I don't know your software, but the process is first you need to test if the intercepts are significantly different or not. If there was a significant difference, the models are different.

But if the intercepts were not significantly different, you need to test the slope coefficients.

Regards,

Hormoz

Looking at your data, I'd recommend using a simple linear regression model rather than a non-linear.

In Prism 9, the simple linear model has its own dialog window, which includes an option to test if the two lines are significantly different (second check box from the top). This will generate a second tab in the results ("Are the lines different?") that performs the analyses that Hormoz Sohrabi mentioned (slope and intercept comparisons).

Avoid a non-linear model in case the diagnostics shows you that a linear model meets the regression assumptions. Your "curves" looks linear, so y_i = b_0 x_i b_1 +e_i (= including the factor variable for the different groups) is one candidate of a linear model, and y_i = b_0 + b_1 x_1 + b_1 x^2_i b_2 + e_i could be an alternative linear model to check.

From the curves shown, I'm in doubt that a non-linear model was fitted, but just the points are connected.