One form of model that is commonly used in population dynamics is the discrete dynamical model, which has the general form:

P_n+1 = F(P_n)

for some function, F(P). This function is called the updating function, which we will study in more detail later. The model states that the population at the next time step is some function of the population at the current time. If the updating function is linear, we obtain a Malthusian growth model. One of the most popular forms is a quadratic or logistic model, where the linear part gives Malthusian growth and the quadratic part is negative due to crowding effects on the population. There are numerous other types of updating functions that ecologists use for F(P) when they are modeling. This lab will study four different updating functions: Logistic, Beverton-Holt, Ricker's, and Hassell's. These will all be fit to an experimental study of beetles grown on a restricted diet.

A. C. Crombie [1] studied Oryzaephilus surinamensis , the saw-tooth grain beetle, with an almost constant nutrient supply (maintained 10 g. of cracked wheat weekly). The data below show the adult population of Oryzaephilus from Crombie's study (with some minor modifications to fill in uncollected data, provide an initial shift of one week, and give slightly different versions for different students).

a. The discrete logistic growth model for the adult population P_n can be written

where the constants r and m must be determined from the data.

Plot P_n+1 vs. P_n, which you can do by entering the adult population data from times 2-30 for P_n+1 and times 0-28 for P_n. (Be sure that P_n is on the horizontal axis.) To find the appropriate constants use Excel's trendline with its polynomial fit of order 2 and with the intercept set to 0 (under options). In your Lab, write the equation of the model, which fits the data best. Clearly, write the best values of the constants, r and m. Also, find the sum of square errors between the data and this model.

Find the equilibria for this model. Note that equilibria are found by solving

P_e = F(P_e)

Write the derivative of the updating function, then find the value of the derivative at all equilibria. An equilibrium is stable if the value of the derivative at the equilibrium is less than 1 and unstable if the value is greater than 1. Write the stability for each of the equilibria.

b. Another important model used for population dynamics is the Beverton-Holt model, which is given by

where the constants a and b must be determined from the data. Find the least squares best fit of this updating function to the data. and give the least sum of squares error. Clearly, write the best values of the constants, a and b. (You can make an initial guess of a = 3 and b = 200.)

Find the equilibria for this model. Write the derivative of the updating function, then find the value of the derivative at all equilibria. Write the stability for each of the equilibria.

c. Another model used for population dynamics is Ricker's model, which is more often used for fish populations and is given by

where the constants a and b must be determined from the data. Find the least squares best fit of this updating function to the data. and give the least sum of squares error. Clearly, write the best values of the constants, a and b. (You can make an initial guess of a = 2.5 and b = 500.)

Find the equilibria for this model. Write the derivative of the updating function, then find the value of the derivative at all equilibria. Write the stability for each of the equilibria.

d. Another model used for population dynamics is Hassell's model, which is often used for insect populations and is given by

where the constants a and b must be determined from the data. We will assume that c = 1.8 is a value known for this particular species. Find the least squares best fit of this updating function to the data. and give the least sum of squares error. Clearly, write the best values of the constants, a and b. (You can make an initial guess of a = 2.5 and b = 600.)

Find the equilibria for this model. Write the derivative of the updating function, then find the value of the derivative at all equilibria. Write the stability for each of the equilibria.

e. Graph all four updating functions 1. logistic growth, f(P) 2. Beverton-Holt function, B(P) 3. Ricker's function, R(P) 4. Hassell's function, H(P). Include the original data in your graph and add the identity map,

P_n+1 = P_n .

(All of these functions are to be on a single graph and labeled properly. Take your domain to be approximately twice the value of the largest equilibrium and take the range to be about 1.5 times the largest equilibrium.) Use your techniques from this course to find all intercepts, vertical and/or horizontal asymptotes, critical points, maxima and minima, and points of inflection for all four updating functions. Write clearly all of this information about each of these functions, including stating when one doesn't apply. (You only need to be concerned about when these occur for P_n greater than or equal to zero, which is where real populations exist.)

f. Discuss the similarities and differences that you observe between four models. Compare the models to the experimental data. Which model appears to fit the data best? Compare the values and stability of the equilibria for each of the models. How does this match the data at large times? Find the equilibria on the graph and relate this to the identity map.

g. The discrete population models are given by the equations:

where the functions are given above for each model and the best fitting parameters have been found. In this part of the problem, you simulate each of the models with the discrete dynamical models, and use Excel's Solver to best fit the initial value, p₀. As an initial guess, start with your initial population as (p₀ = 4) starting at t = 0 and simulate the growth for 30 weeks. Use Excel's Solver to find the best possible p₀ value that minimizes the square error between the populations in the data and the popualtions given by each of the models. (You are performing 4 different simulations for the 4 models.) Give this best p₀ values for each model and list the populations at times t = 4, 8, 14, and 30. Which model best matches the actual initial starting population? Give the percent error between the models and the data at these times. Also, include the value of the least sum of square errors between your simulations and the data. Which model gives the least sum of square errors? Graph the data and the simulations for all 4 models.

h. Write a short discussion that compares and contrasts the four models. Which model do you believe is better and why?