where EXP is Euler’s number, and TIME is a built-in function that takes on the
same value as the current period of the model run. With this specification,
the nominal death rate exponentially declines from 4 to 1 %. This is an example
for our third case mentioned above, and the results it generates are shown in
Fig. 3.4. Now, the mean population size is approximately 27.
We note that at first the population rises in the expected exponential way and
then the effect of the standard deviation takes over, causing an average population
about 26. None of these populations continue to grow. After about t ¼ 2,000, a mean
value is achieved that holds for the rest of the run, in all cases although the first and
third cases clearly show a greater variation.
Can you develop any more scenarios? Should the death rate be allowed to vary
so greatly from period to period? Perhaps the death rate should be a state variable
and thus change more slowly. Can you set up such a model? Should this distribution
be normal? Is it a bit unreal to cut the normal distribution off at 0.01 on the low side
and 1.0 on the high side? How about a normal distribution that has different
standard deviations for the different sides of the mean? What about the idea of
making the standard deviation a nonlinear function of the population size? What
about the possibility of a delay of death rate decreases with no delay of death rate
decreases? Try out some of these modifications of the model, make an educated
guess before you run them, then explain your results.
Now that we have seen some of the effects of randomness on population
dynamics, we will return to deterministic models—models that yield the same
results from run to run. The absence of randomness, however, not necessarily
means that we will be able to make precise forecasts of a system’s behavior once
we know its history and current state. Rather, unforeseen, chaotic events may occur.
This is the topic of the following chapter.
Fig. 3.4
3.1 Risky Population Model
45
same value as the current period of the model run. With this specification,
the nominal death rate exponentially declines from 4 to 1 %. This is an example
for our third case mentioned above, and the results it generates are shown in
Fig. 3.4. Now, the mean population size is approximately 27.
We note that at first the population rises in the expected exponential way and
then the effect of the standard deviation takes over, causing an average population
about 26. None of these populations continue to grow. After about t ¼ 2,000, a mean
value is achieved that holds for the rest of the run, in all cases although the first and
third cases clearly show a greater variation.
Can you develop any more scenarios? Should the death rate be allowed to vary
so greatly from period to period? Perhaps the death rate should be a state variable
and thus change more slowly. Can you set up such a model? Should this distribution
be normal? Is it a bit unreal to cut the normal distribution off at 0.01 on the low side
and 1.0 on the high side? How about a normal distribution that has different
standard deviations for the different sides of the mean? What about the idea of
making the standard deviation a nonlinear function of the population size? What
about the possibility of a delay of death rate decreases with no delay of death rate
decreases? Try out some of these modifications of the model, make an educated
guess before you run them, then explain your results.
Now that we have seen some of the effects of randomness on population
dynamics, we will return to deterministic models—models that yield the same
results from run to run. The absence of randomness, however, not necessarily
means that we will be able to make precise forecasts of a system’s behavior once
we know its history and current state. Rather, unforeseen, chaotic events may occur.
This is the topic of the following chapter.
Fig. 3.4
3.1 Risky Population Model
45