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Your fitted curve will look like this

You need to include information about the sigma information of each Gaussian. When you say percentage coming from each population, I think you are asking about the total counts (integral) in each Gaussian. For a Gaussian in your form, the integral is:

You have the arguments to np.trapz reversed. It should be

You can define the function to take a variable number of arguments using def func(x, *args). The *args variable then contains something like (a1, b1, a2, b2, a3, ...). It would be possible to loop over those and sum the Gaussians but I'm showing a vectorized solution instead.

Since curve_fit can no longer determine the number of parameters from this function, you can provide an initial guess that determines the amount of Gaussians you want to fit. Each Gaussian requires two parameters, so [1, 1]*n produces a parameter vector of the correct length.

This is how a simple solution with least_squares would look like:

It looks like you are simply trying to use linear regression to find the best values of c_10 and c_01 to best fit your λ and P data.

So I suggest using LinearRegression from scikit-learn.

I copied your λ and P data into a csv file named 'curvefitdata.csv'.

This is really not a scipy or a Python question, and is perhaps better suited for the sister site https://datascience.stackexchange.com/

However here is my proposed solution for coming up with good initial values. It does rely on the knowledge of the lognormal distribution as in for example https://en.wikipedia.org/wiki/Log-normal_distribution

The curve you are trying to fit is a shifted (by b) and scaled (by a) log-normal probability density. First we want to estimate these. As the log-normal density takes values in $[0,+infty)$, the initial estimate for parameter b coiuld be set to the lowest value of the intensity. Also, the integral of the log-normal density should be 1 for definition, hence a should be set to the integral of intensity shifted by b. After that we have a (true) lognormal density in intensity and we can estimate m and s from the mean of the distribution and its variance (in fact we just use the second moment). This logic is implemented in the function get_initial_values below. I also modify your func to make sure t is always strictly positive as otherwise it messes log, leads to division by zero, etc. Also it had a type n vs m. Here is the full code

Try this fit, which works. The original function will not really fit the data. For large x it converges to a. That's okay. One has a drop to smaller x but only if b is positive. Moreover, one is lacking a parameter that is scaling the drop-rate. So with a<0 and b<0 one gets the right shape, but is converging to a negative number. i.e. an offset is missing. Hence the use of y = a + b / ( x + c )

I am not into Matlab, so I don't know, which additional work the Matlab fit does to estimate starting values for a non-linear fit. I can say, though, that curve_fit does non at all, i.e. all values are assumed to be on the order of 1. The easiest way, would have been to rescale the x axis to the range [0, 2 pi]. Hence, the problem of the OP is, once again, wrong starting values. Rescaling requires, however, the knowledge that the main wave to be fitted is approximately the width of the data set. Moreover, we need to assume that all other fit parameters are also of the order 1. Luckily, this is the case, so this would have worked: