Student Assignment
- In what circumstances might a population be expected grow in a geometric fashion?
- Suppose you are interested in how the population size of the annual herb Polygonum douglasii growing on a mountainside in Colorado changes over time. You are pretty sure that this population is closed, meaning that there are no immigrants (individuals entering the population) or emigrants (individuals leaving the population) as seeds. In this case the only processes that affect the growth of this population are births and deaths. If this population is growing at a rate of 20% per year; write the discrete time dynamical system for the population.
- Let N be the world population (in billions) at time t, in years since 1975. Then N0 is the world population in 1975; therefore N0= 4 billion. The rate of growth is actually the k value in the formula for exponential growth given above; for this problem, k=0.021.
- What is the exponential growth model for this system?
- By which year will the population reach 7.5 billion?
- What will be the population after 10 years? re started at 1000 bacteria. 7 hours later there were 5000 bacteria.
- A bacterial culture started at 1000 bacteria. 7 hours later there were 5000 bacteria.
1. What is the rate of growth of the system?
2. At what time will the population reach 10000?
- Exponential decay and exponential growth are quite important models in Physics, biology, etc. If relatively small populations (of humans, animals, bacteria, etc.) are left undisturbed, they often grow according to Malthus's law. This law states that the time rate of growth is proportional to the population y(t) present. Model this by a differential equation. Show that the solution is y (t) = y0ekt. Determine y0 and k from two columns of the table, which contains data for the United States. Calculate values for 1860, 1890, ....1980 from the formula. Compare with the observed values. Use the simulator to analyse the data.
|
t
|
0
|
30
|
60
|
90
|
120
|
150
|
180
|
|
Year
|
1800
|
1830
|
1860
|
1890
|
1920
|
1950
|
1980
|
|
Population (millions)
|
5.3
|
13
|
31
|
63
|
106
|
150
|
230
|
(Hint: Continuous growth model)
- If in culture of yeast the rate of growth y'(t) is proportional to the amount y(t ) present at time t, and if y(t) doubles in 1 day, how much can be expected after 3 days at the same rate of the growth? After 1 week
(Hint: Continuous growth model)
