Logistic Growth Functions

The logistic function is frequently used to model biological populations. Its basic form is

Y = K / [1+e-(β01t)]

or equivalently

Y = Keβ01t / [1+eβ01t]

where Y is the population size, K is the (asymptotic) maximum potential population, i.e., the carrying capacity of its habitat, β0 shifts the function left or right, and β1 is the intrinsic maximum rate of net growth of the stock (reproduction minus natural mortality) and t is time.

This figure illustrates a logistic growth function based on the parameters in the yellow-highlighted cells: the cumulative stock is plotted in green; its annual increase is plotted in red, and the relationship between stock growth and stock size is plotted in blue.

You probably think this function looks ugly, but it is mathematically very tractable (easy to use). The stock growth per year turns out to be a simple quadratic function of stock size Y:

ΔY/Δt = β1Y(1 - Y/K)

Maximum growth occurs at Y = K/2, and growth approaches zero as Y approaches K, the maximum stock the habitat can sustain, where reproduction equals natural mortality.

By algebraic manipulation the logistic growth function can written as a linear function of the two parameters β0 and β1.

ln[Y/(K-Y)] = β0 + β1X.

So if you know K, you can calculate the LHS log ratio ln[Y/(K-Y)] for any value of Y < K. If you have empirical data on stock sizes Y(t) at various ages t, then you can use simple linear regression to estimate the other two parameters β0 and β1.

I encourage you to download the simple Excel spreadsheet shown above and tweak the parameters to see how they affect the stock and growth functions.

Many biological populations exhibit growth behavior that approximates logistic growth. When the stock is small relative to environmental carrying capacity, net stock growth is rapid; as the stock approaches the maximum environmental carrying capacity, net stock growth approaches zero. The population stabilizes where reproduction equals natural mortality.

Harvest strategies

When some portion of a biological stock is harvested for human use, it is replenished over time by growth. The maximum sustainable annual harvest or yield (MSY) equals the stock's maximum annual growth. In the figure below, a maximum of 5 units can be harvested and regenerated annually. Any larger harvest would not be sustainable.

In many cases, the growth curve of a stock will not be known until its recovery rates under various harvest levels are observed. If healthy stock recovery requires a minimum critical biomass, and over-harvesting drives the stock below that critical level, then regeneration may be very slow, and require a halt in harvesting. Some species such as the carrier pigeon may simply go extinct after the stock falls below a critical level.

In general, the effort required to harvest a unit of the resource will vary inversely with the stock size: as the stock gets small, the animals get harder to find. The relationship between harvest effort E and sustainable yield Y can often be modeled as a quadratic function:

Y = B0E - B1E2.

Graphically, the Effort-Yield relationship is a reverse of the Stock-Growth relationship, since larger sustained Effort implies smaller stock size. Most sustainable harvest levels less than MSY can be achieved with either of two Effort levels. Obviously the smaller Effort level is the more efficient.

Suppose each unit of the resource sells for $P and each unit of Effort costs $C.
So TR = P*Y(E),   TC = C*E   and   Profit = TR - TC. We can plot these against Effort.

In any single time-period, the profit-maximizing level of harvest Effort occurs at E*, where the slope of TR equals the slope of TC (i.e., the MVP of Effort = MFC). Since the cost of Effort C is positive, this is less than the Effort level corresponding to MSY. EMSY is only efficient if Effort is costless.

The economically optimal harvest strategy depends on the growth behavior of the stock, the economic value of the resource, and harvest costs. In the next few classes we will address management strategies for timber (slow-growing, and clear-cutting is much more efficient than continual harvesting) and fish stocks (fast-growing and harvested continuously).