Starting with the thinking of Malthus the Belgium mathematician Pierre Francois Verhulst developed the Verhulst Equation:
which was generalized and simplified by the American Robert May, who renamed it to the Logistic Equation:
x² = λx(1-x)
Where:
x is the starting population x² is new population after ... λ as the rate of increase of that population, including, though abstracted, the resources consumed to sustain the population and its increase
and (1-x) makes it a proportion, a percentage - if you will, of the population and the resources needed to sustain that population.
The advantage to the generalization is it limits the quantification of λ to less than 4 making it easier to crank the machinery around.
As Dr. May ran the numbers things went linear until λ reached 3 and all of sudden the system developed two stable points. At 3.5 the system exhibited four stable points. At 3.56 the system developed eight stable points. As the value increased the stable points went to sixteen, then thirty-two, and kept on increasing until at 3.569946 the system went gonzo-weird (r is λ in this graph):
and there weren't any stable points. The relation of x to x² broke down and became unpredictable or, put another way, there was no relation between a population, its rate of increase, and the size of the next generation of that population, although there is a strong tendency for the population to decrease.
As was stated above, the resources consumed by the population is included in λ. Thus, restriction of supply, to a generation, of resources increases λ.
In a finite world - like our little globe - the various Peaks: Oil, Water, Agricultural productivity, fish stocks, Ability-to-Borrow, Willingness-to-Lend, & etc, contribute to an increase in resource utilization and, thus, increase λ.
Even a dramatic die-off does not necessarily imply a decrease in λ. In fact, it is possible for the population to suffer a catastrophic loss of numbers and for λ to increase. Scenario: resource wars go nuclear, mega-death follows but the oil fields, say, are turned into radioactive 'no-go' areas removing them from the resource inventory.
Ok. The 'sound-bite' here is:
We, whoever 'we' are, is to affect a decrease in λ both on a global and local level and scale.
Starting with the thinking of Malthus the Belgium mathematician Pierre Francois Verhulst developed the Verhulst Equation: which was generalized and simplified by the American Robert May, who renamed it to the Logistic Equation: x² = λx(1-x)Where: x is the starting population x² is new population after ... λ as the rate of increase of that population, including, though abstracted, the resources consumed to sustain the population and its increase.
x is the starting population x² is new population after ... λ as the rate of increase of that population, including, though abstracted, the resources consumed to sustain the population and its increase.
Write the Velhust equation as
dP = r P (1 - P/K) dt
Where K is where the resources needed to sustain the population come in, in the form of an "equilibrium population size"; r is the vegetative growth rate when resources are plentiful (therefore r is resource-independent); and dt is the time step involved in discretization of the differential equation. This means
P' - P = r P (1 - P/K) dt
P' = P (1 + r - rP/K) dt
P' = (1 + r)dt P {1 - rP/[(1 + r)K]} dt
x' = (1 + r)dt x (1 - x)
λ = (1 + r)dt
Put this way, the λ = 3 bifurcation point indicates the point at which the discretization is too coarse to faithfully represent the Velhust equation.
Of course, in reality the logic of the model goes the other way: the logistic equation comes first and the Velhust equation is a continuum approximation to it, valid only if λ < 3.
But, in any case, the point about λ stands: the "resources" are implicit in the units of x, and λ has to do with the vegetative growth rate when the population is small and resources are not a constraint and so is resource-independent. It'd be nice if the battle were only against the right wingers, not half of the left on top of that — François in Paris
