Our equation greatly simplifies for λ = 1.
In that case, we have the condition that
This mathematical condition is the same as the physical condition of an exact balance between female survival probability and females born per year in the ecosystem.
For instance, for s = 0.9 and b = 0.25 then lα must be .1/.25 = 0.40.
Since l&alpha = so * s1 * s and s = 0.9, then the stability condition occurs whenever so * s1 = 0.44.
This represents and example of the necessary "fine tuning" of a parameter set to achieve stability. This is why stability is rare and certainly difficult to maintain if you ever get there.
As another example, consider the case when all 3 survival probabilities are the same (s= so=s1 = 0.5).
In this case lα = .125 and 1 -s = 0.5. Therefore b would have to be 4 to achieve stability. Clearly this is an unphysical situation.
In this way, all of these various parameters are related at some level and are not completely independent and arbitrary. This is another way of saying that, since the species has survived, there has obviously been some balance achieved, over time, between these parameters.
However, at any given time in any system of mammals (except humans), λ is always less than 1 (sometimes only slightly less than 1), because of the difficulty of balance the parameters to achieve stability. On long timescales, such balance can occur, but at any snapshot in time, there is no balance.
The importance of determining λ is that, once we know its value we can then estimate the characteristic timescale of the population crash as
where N is the number of known individuals in the population (8000 in this case).
As stated earlier, λ is fractional growth rate, so if you find that lambda = 0.98 then that means 2% species reduction per year and the doubling time (or in this case the halving time) of the population is 70/2 = 35 years. After so many halving times, the species is essentially extinct. For instance, using 8000 and .98 produces a population crash time of 450 years, or 450/35 = 13 halving times. Note that 213 = 8192.
So let's work out an example:
So we would have
l α = 0.90 * 0.15 * 0.70 = .095
l α b = .095 *.25 = 0.0236
so,
as our equation to solve. This can not be solved analytically, but only by trial and error (which is why you program this). Note, however, that λ can not be less than .90 else the left hand side of the equation becomes negative. So you only have to guess at λ in the range of 0.90 to 1.
The solution for this case is λ = 0.93 which leads a population crash time of 123 year for a starting N=8000. Note, if N would be 80000 then the crash time is 155 years, so again, N is not the driver, λ is.