I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a function with certain properties.
In biology, there is something called the 'Species-Area Relationship' (SAR) which describes the increase of species numbers with an increase in area investigated. Usually, a power functions is best describing this. Now there is the special case of very small areas where vales are close to zero for a number of area sizes.
One can use piecewise regressions such as
y = f1 (x) = c + (x ≤ T) z1 x + (x > T) [(z1 – z2) T + z2 x], y = number of species x = Area T = Breakpoint z1, z2 = slopes of breakpoint functions on LHS and RHS of T
Sadly, the breakpoint function has an unrealistic "break" which is quite rough. Hence, I am trying to find a smooth version of this breakpoint function that connects the two breakpoint functions on the left and right handside with a smooth transition function.
Trying a logistic function it becomes already quite close. However, it is unrealistically bound between 0 and 1.
Now I am looking for an integral of this logistic function
z = f4’ (x) = z1 + (z2 - z1) (1 /(1 +exp (–k x)))
Then I only had to add a constanct 'c' to lift it beyond 1...and here is where I have to stop since my brain starts hurting.
Could anyone help me out and try to find the integral of the second function? Or give a hint how to achieve a smooth transition in the BP-function?
many thanks already for all grey matter turned into words,
PS: I am not sure if this post would better fit to 'crossvalidated' list on stackexchange?