Species Accumulation Curve is a graph recording the cumulative number of species of living things found in a particular environment as a function of the cumulative effort expended searching for them. The curve will necessarily be increasing, and will normally be negatively accelerated (that is, its rate of increase will slow down). Further, it converges after a while. An example curve available here
I firstly want to find the equation of the curve that will fit this curve in the best way. Next, with that equation, I want to be able to fit any other curve to observe whether that curve resembles species accumulation curve or not.
Any help will be appreciated.
In a naive model where you search by continually picking random individuals and checking whether they are from a species you haven't seen yet, the curve ought to behave roughly like $$ f(t) = S(1-e^{-t/S}) $$ where $S$ is the total number of species there are to find and the parameter $t$ is the number of samples.
This assumes that each time you pick an individual, the species you get is uniformly distributed over all possible species, which is not really a realistic assumption.
Depending on your needs, it may work for you to fit your observed curve with a sum of a small number of curves of the above form, e.g. $$ f(t) = a_1(1-e^{-b_1 t}) + a_2(1-e^{-b_2 t}) + a_3(1-e^{-b_3 t}) $$