The discrete logistic growth
model proves valuable in the study of simple organisms. However, sexual
organisms and organisms living in a more complex environment often do
not follow logistic growth very well. Scientists have discovered that
certain animals require a minimum number of animals in a colony before
they reproduce successfully. If the population is too small, then they
have difficulty finding mates or they are incapable of defending the
colony from competitors or predators. This is called the Allee effect.
This problem studies a population of birds that forms colonies to
improve defenses against raptor predators and enhance mating
possibilities. Suppose that the relatively new colony is studied over
10 years, and the annual counts (in thousands) are given in the table
below.
| n (yr) | Pn | n (yr) | Pn |
| 0 | 5.1 | 6 | 10.3 |
| 1 | 5.5 | 7 | 12.4 |
| 2 | 5.9 | 8 | 14.7 |
| 3 | 6.5 | 9 | 16.0 |
| 4 | 7.5 | 10 | 16.2 |
| 5 | 8.7 |
In this problem, we study three models to simulate the data above. In
each of the three models, we find and graph the updating function for
the discrete dynamical model, then we simulate the model with a time
series, adjusting the initial population to best fit the data in the
table.
a. We begin this problem fitting the discrete logistic growth model to
the population data above. If we assume that the population is closed
(meaning that immigration and emigration are ignored), then the
updating function must pass through the origin. Under this assumption,
the
logistic growth model has the following simple quadratic form:
Pn+1 = F(Pn)
= rPn - mPn2
where the constants r and m must be determined from the
data.
Begin by finding the logistic updating function. This is accomplished
by plotting Pn+1 vs.Pn. (Let the data for the
populations from year 0 to 9 represent Pn in one column, then in the
next column enter the populations from year 1 to 10, representing Pn+1.) Use Excel's trendline to fit a polynomial of
order 2 with its intercept set to
zero.
The updating function is used
to simulate the model and compare to the
time series data. Begin with a reasonable guess for the initial
population, P0, then use Excel's solver to
minimize the sum of square errors
between the simulated model and the population data. Give the
model population prediction at n = 5 and
n = 9
and
find the percent error at each of these times from the actual data
given.
b. Recall that equilibria are found by solving
Pe = F(Pe)
where F(P)
is the updating function. Find the equilibria for the
logistic growth model given above.
Write the derivative of the updating function F '(P). Find the value of the
derivative at all equilibria and determine the
local behavior of the solution near each of those equilibria. State
whether
each equilibrium is Stable or Unstable and if it is Monotonic or
Oscillatory.
c. Bird populations are generally not closed systems. The birds can readily fly to other colonies, so the model could need to account for immigration or emigration. The next model that we consider is a minor extension to the discrete logistic model that includes emigration. Under this assumption, the logistic growth model has the full quadratic form:
Pn+1 = G(Pn)
= sPn - kPn2 -
m
where the constants s, k, and m
must be determined from
the data.
Find the new logistic growth with emigration updating function much as
you did in Part a. However, simply use Excel's trendline to fit a
polynomial of
order 2 (without setting the intercept to zero). This updating
function is used to simulate the model with emigration.
You compare the new simulation to the time series data. Again start
with a reasonable guess for the initial population, P0, then use Excel's
solver to minimize the sum of square errors between the simulated model
and
the population data. Give the model population prediction at n = 5 and
n = 9 and
find the percent error at each of these times from the actual data
given.
d. Now the equilibria are found by solving
Pe = G(Pe)
where G(P) is the updating function. Find the equilibria
for the
logistic growth model with emigration given above.
Write the derivative of the updating function G '(P). Find
the value of the
derivative at all equilibria and determine the
local behavior of the solution near each of those equilibria. State
whether
each equilibrium is Stable or Unstable and if it is Monotonic or
Oscillatory.
e. An alternative population model includes the phenomenon called the Allee effect. This model is again
based on a closed
system, so the updating function passes through the origin. However,
the model satisfies a cubic equation, so has three equilibria. This
discrete
population model has the following form:
Pn+1 = G(Pn) = aPn - bPn2 - cPn3
where the constants a, b, and c must be determined from the
data. Find the allee effect model updating function much as you
did
previously. Use Excel's trendline to fit a polynomial of order 3 that
must pass
through the origin.
This updating function is used to simulate the allee effect model. You
compare the new simulation to the time series data. Again start with a
reasonable guess for the initial population, P0, then use Excel's
solver to minimize the sum of square errors between the simulated model
and
the population data.
Give the model population prediction at n = 5 and
n = 9 and
find the percent error at each of these times from the actual data
given.
f. Now the three equilibria are found by solving
Pe =
A(Pe)
where A(P) is the updating function for the model with
the allee
effect. Find the equilibria for this model.
Write the derivative of the updating function A '(P). Find
the value of the
derivative at all equilibria and determine the
local behavior of the solution near each of those equilibria. State
whether
each equilibrium is Stable or Unstable and if it is Monotonic or
Oscillatory.
g. Graph all three updating functions 1. Logistic growth, F(P) 2. Logistic growth with emigration, G(P) 3. Cubic growth (Allee), A(P). Include the original data in your graph and
add the identity map,
Pn+1 = Pn
All of these functions are to be on a
single graph, labeled properly,
and extending to the origin. Take your domain and range to be about 1.5
times the largest equilibrium. Briefly discuss the relationship between
the
equilibria computed above and the identity map shown in your graph.
Discuss the
similarities and differences that you observe between the three
updating functions.
Describe how well each of the updating functions fit the data. (Don't
forget to
relate your sum of square errors.) Which updating function appears to
fit the
data best? Do you expect these models to be valid for large values
of Pn
based on the updating functions? Explain your answer.
h. After finding the updating functions, you simulated the models,
finding the best values of P0 for each of the models. Present a single graph
of the time series formed from the simulation of all three models,
including the
time series data given in the table above. Which model best matches the
actual initial starting population? Which model best fits the data?
Explain.
Write a short discussion that compares and contrasts the three models.
Which model do you believe is better and why? Give strengths
and weaknesses of each
model. Give a brief biological description of what your best models
imply
about this gregarious species of bird.