As simple as they are, the population models discussed previously have several important applications to natural resource management, two of which I'll discuss here.
Sustainable yield harvest
The essence of "sustainability" is the ability to extract a gain (harvest or other offtake) from a resource without depleting it, thus assuring maintenance of the resource for the future. In the context of animal populations that are harvested (game animals, commercial and recreational fisheries), this boils down to "how much harvest can we take every year and still maintain the stock."
Mathematically, under the deterministic logistic growth model this by definition the population grows each year by the equation
N[t+1]<-N[t]+ rmaxN[t](1 - N[t]/K)
If we take a harvest H[t] from the population each year we have
N[t+1]<-N[t]+ rmaxN[t](1 - N[t]/K) - H[t]
By re-arranging things slightly we see that in order to maintain a stationary population N[t+1]<-N[t] , H[t] must be exactly
H[t]<-rmaxN[t](1 - N[t]/K)
Anything more, and population growth is negative; less than that and the population grows but we have "lost" some harvest opportunity. The idea of maximum sustainable yield (MSY) comes from this principle, after recognizing that there many possible yields, all sustainable, that differ according to the value N[t] takes between 0 and K. In fact, yield is 0 at either N=0 or N=K, and increases up to a unique maximum value at an optimal N, N=N.msy, which gives N.msy= K/2 providing msy=r*K/4. The attached script file illustrates the yield curve (plot of yield vs. N) for specific values of r and K (which you can change), and derives N.msy and msy for these values.
Note that, while useful, the msy model makes several important assumptions or simplifications, including:
In practice, therefore, MSY, while useful conceptually, must be used with caution or we risk overharvesting the stock (something that has commonly occurred, for example, with oceanic fisheries).
Forestry applications
Although the above description is framed in terms of wildlife or fish populations, the same principles apply to forestry and other renewable resources. In fact, the basic mathematical models of population growth are similar or even identical to wildlife population models. Typical forest growth models follow the a logistic-type relationship, with rapid initial growth and an upper limit, although the response is typically in terms of basal area, tree height, volume, or similar attributes. The rate of growth expressed in relationship to time or plotted against basal area etc. provides the typical yield model. Maximum yield is generally obtained by examining the peak of the yield curve (although discounting factors may apply), providing an optimal age at which maximum return can be obtained (e.g., rotation age). We will discuss forestry application more in the next section.
Population viability analysis
In conservation biology, typically we are focused on smaller populations, where spatial isolation and stochastic factors create a real risk of extinction (local or global) on near -term time scales. For these problems, we almost always use stochastic simulation models to project population growth and persistence (e.g., over 100 years).
For example, we might assume the population is at an initial abundance of 10 and has a mean growth of r =0.05, but that annual growth varies stochastically based on a normal (0.05, 0.10) distribution. We would then stochastically simulate the population over 100 years, keeping track of whether or not the population perists to year 100 or goes extinct (falls to 0 or below some critical threshold for persistence). Because these are stochastic simulations, each trajectory is actually a single sample (persists or not), so it is important to repeat the simulations numerous (1,000 X or more) times. The proportion of simulated populations that persist is then taken as an estimate of persistence probability (or its complement, extinction risk). The attached code runs multiple stochastic simulations using a user-defined function and keeps track of proportion that persist and its sd. Note that the function is based on a simple (but stochastic) exponential model; it would be straightforward to modify as needed to allow density dependence under the logistic model.