In this lab we will use (and build) models to for application to two types of important applications: (1) harvest management, in which we are trying to obtain a harvest yield from a population without depleting the population's ability to replenish itself, and (2) conservation management, in which populations are small, declining, in fragmented habitats, or all three, and the goal is to increase or maintain the population. Many of the principles we will use were already developed in Lab 3 (e.g., density-dependence, stochastic effects) but will be applied to solving specific harvest or conservation problems. Others are covered in the lectures on applications of models to harvest management and conservation management, and in Chapter 11.

HARVEST MODELS

Harvest under density independence

Here we simply extend our previous models of density independent growth to a case where a exponentially-growing population is harvested at a constant rate. We begin with deterministic growth (SE_r=0). Try different harvest rates (h) from hrate=0 to hrate=0.30. What happens when harvest rate exactly equals r ? When it exceeds r? Try setting SE_r to different values (0.01, 0.1, 0.2, etc.).

You should notice some definite trends here, the main one being that harvest (H) increases over time without limit if h < r; is static if h= r; and decreases (approaching zero) if h > r. Cumulative harvest (which is also calculated) likewise increases. This simply by virtue of the fact that H is always proportional to N (H= hN), so if N increases without bounds so will H.

We can explore this a little further by projecting growth out over a long time horizon (e.g., 100 years) and seeing the impacts on harvest and cumulative harvest of different choices of harvest rate. Try this both for deterministic (SE_r=0) and stochastic (SE_r=0.1, 0.2, etc.) conditions. Examine the graphs and start thinking about what this means for harvest management, especially how we set harvest rates to maximize long term harvest. Hang onto those thoughts for the next section on density dependence.

Harvest under density dependence- MSY

Again, we're just extending our previous models of density dependent growth to a case where a population growing according to the logistic model is harvested at a constant rate. Again, begin with deterministic growth (SE_r=0) and try different harvest rates (h) from hrate=0 to hrate=0.20.Try setting SE_r to different values (0.01, 0.1, 0.2, etc.).

You should notice a couple of differences to the previous (density independent) results. First, as long as h < r, the population will grow toward K and equilibrate just below that level. Try starting the population out above K to see that this fact is not dependent on initial conditions. Second, H likewise eventually levels off. However, notice that the harvest that one can take each year and still reach equilibrium also depends on harvest rate, but not in a monotonic fashion; unlike density independent growth future harvest cannot be maximized by keeping harvest rates low. We can explore this a further by projecting growth out over a long time horizon (e.g., 200 years) and seeing the impacts on harvest and cumulative harvest of different choices of harvest rate. The plots of H versus N (now at equilibrium after 200 years) and h reveal two things. First, there exists a optimal population size, somewhere between 200 and 300 in this example, which yield the maximum harvest without depleting the population. Second, there is also an optimal harvest rate, somewhere haround h = 0.05, that likewise provided the greatest harvest yield. Both of these results follow directly from maximum sustained yield, as described in the lecture notes on the topic and in Chapter 11.

Stock-recruitment models

A type of model frequently used in fisheries biology establishes a relationship between the abundance of the spawning stock of fish, and the number of new recruits from reproduction, via a density-dependent function. As an example of such a relationship, take the Ricker model, one form of which is

R = a N exp(-bN)

where R is number of recruits, N is the number of fish in the spawning stock, and a and b are parameters that are related to maximum growth rate and the strength of the density dependent relationship. The Ricker model has also been parameterized as a population dynamics model model, one version of which is

N_t+1 = N_t exp[r_max (1-N_t/ K)].

This functional form has certain advantages over the discrete logistic when N_t is very large, by eliminating the possibility of N_t < 0 (theoretically possible with the discrete logistic. The Beverton-Holt model is another type of stock-recruitment model that is similar to the Ricker model.

Dynamic models of harvest- Anderson model

We return to the example we ended the last lab with, the Anderson (1975) model for mallards. The model has been modified slightly to explicitly incorporate an objective, namely the cumulative, long term harvest of mallards. The model is described in the flowchart below and in a Python program.

Basically the population starts off each year with ducks and ponds counted in spring; ducks are produced according to a production function (Yn) that depends on number of breeding ducks (Nn, in miillions) and number of ponds (Pn, in millions). Surviving adults and juveniles constitute a fall population (Fn). A harvest decision (Hn) is applied to this population, adding to the cumulative objective function; the harvest decision can either be constant (same rate every year) or a time-specific policy can be applied. Ducks then survive to the next breeding season depending on compensatory or additive mortality assumptions. Pond numbers next spring are a function of pond numbers this spring, and rainfall (Rn), a random variable. Try running this model with a variety of assumptions about starting conditions (ducks, ponds), harvest rates, and survival models. The 'additive'(AMH) model is selected via the statement "model=AMH" at the beginning of the program. Comment this statement and uncomment the "model = CMH" statement to change this assumption to compensatory.

CONSERVATION MODELS

Time to extinction and "MVP"

As discussed in lecture, all populations face the risk of eventual extinction. The issue for conservation is what is the likelihood that extinction will occur before a relatively short time horizon, and whether management actions can alter that likelihood. A theoretical relationship developed by MacArthur (1972) and other (see Chapter 11) posits a relationship between expected time to extinction (T_E) , maximum abundance (N_max ; e.g., , K carrying capacity), and average per-capita birth (b) and death (d) rates as

E(T_E) ~ 1/(b N_max ) (b/ d) ^Nmax.

This approximation is encoded in a Python program. Use this program with various values for b, d, and N_max to approximate expected extinction time.

Simulating extinction and quasiextinction

It is well known that the above approximation, which predicts a rapid increase in extinction time as N_max increases, fails to account for other sources of variability that would tend to prevent such a rapid increase. There are also practical limitations, such as the fact that for small populations N_max will be unknown or meaningless; it makes more sense to ask: given today's population and demographic rates, what is the likelihood of extinction over the time horizon? An alternative to analytical models of extinction probability is the use of stochastic simulation models. We have a Python program for the simulation of persistence that projects a population from a specified initial abundance (N) given an assumed form and parameter values for growth. The program assumes logistic growth, but K can be made very large relative to N, effectively producing density-independence for many years. The program simulates over 100 years; this can be raised or lowered by changing the parameter "nyears".

Each time you run the above program you will get a different populations trajectory (or realization of a stochastic process), even if all the parameter values and initial conditions are the same. Some of these "populations" will go extinct over you time horizon; most probably will not. This mimics the situation where a given population either persists or does not to, say 100 years. We would like to estimate the probability that any randomly chosen population with these characteristics will persist (or conversely, go extinct). One way to do that is to simulate under these conditions a large number of population realizations, keep track of which persist and which don't, and use these frequencies to compute an empirical probability. That is, if we simulate n times and x of those time persistence occurs, then our estimate of persistence is simply

p^ = x/n

and of persistence

e^ = 1 - p^ = 1 - x/n

The next Python program performs estimation of persistence (extinction) probabilities by simulating populations' persistence to nyears and performing nreps such simulations and calculating x/n. This is done in the program for a range of initial abundance values, so that you can examine the impacts of initial N on population persistence.

Source-sink models

As discussed in lecture and elaborated in more detail in Chapter 11, populations are not uniformly distributed in space, and dynamic models may need to take this into account in order to accurately describe growth. As special case of these models postulates that one population (the "source") has positive net growth rates ( > 1), and upon reaching carrying capacity exports excess individuals to a second population having net deficit growth ( < 1). The Python matrix model of source-sink dynamics will perform the projections for a source-sink model, given user-defined values for in the source and sink, carrying capacity in the source, and initial abundance in each population (be sure that these values are >1 and <1 for the source and sink respectively). You should be able to convince yourself that the formula for equilibrium abundance in each population works when the source-sink assumptions hold.

Assignment 4. Due by beginning of next lab. Send a 1-2 page write up accompanied by any programs or other methods used to derive solutions to instructor via email.