Intro to Logistic Growth Functions

Logistic growth function y= The logistic function is frequently used to model biological populations. Its basic form is. Y = K e β 0 +β 1 t / [1+e β 0 +β 1 t ] where Y is the population size, K is the (asymptotic) maximum potential population, i.e., the carrying capacity of its habitat, β 0 shifts the function left or right, and β 1 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: 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 β 0 and β 1 . So if you know K , you can calculate the LHS log ratio ln[Y/(K-Y)] for any value of Y. 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: 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. E MSY 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).