A Logistics Model Group Activity 8 STEM Project Week #11 Consider fencing off several thousand acres of land and placing 1000 rabbits on the land. Initially the rabbits would grow at a constant percent rate. The table below gives the data that the farmer gathered. Population P(t) Time (t) in months 1000 1270 1618 2312 4230 0 2 4 7 12 Plot the data on the grid below. Be sure to label the x and y axis and label the window. Looking at the data, what kind of a function (linear, quadratic, cubic, exponential, or a log equation) do you think would model this data the best? Explain. Make sure to use phrases such as rate of increase or rate of decrease in your explanation. You might see that the growth of the population could be modeled by an exponential equation. In the last activity we looked at representing an exponential equation with both a base b and a base e.
Now, create an exponential regression for the data. What is your regression equation? Your regression should be of the form P(t) = P(0)*b t, where P(0) is the initial population. How well do you think it models the data? Explain. Do you think this model will work for small t values? Explain. Do you think this model will work for large t values? Explain. Using the model, what will the population of the rabbits be in 10 years? Does this seem reasonable? Explain why or why not. In the other exponential growth models (in our other activities), we created a regression equation that had a base of e. Originally in this activity, we have found that the equation P(t) =1000e rt, where P(t) represents the population at time t, and r represents the percent growth rate of the population, will model rt this situation nicely. Using the model, P( t) 1000e, find a value for r and rewrite the model. What is your value for r? How did you find it? What would be the exponential equation that represents this data with a base of e? Explain how you found this. Hint: To find the model that represents this situation with a base of e, we need to change the base from b to e: this is done by finding the value of r in the equation e r = b. If you did not figure out how to solve for e above, here are the steps: e r = b lne r = ln b r = lnb Once you have found r, substitute r into the equation P(t) =1000e rt. Does this model fit the data better or worse than the original exponential model? Explain. Make sure to include in your discussion the issue of when t gets VERY large.
A Logistic Model Hopefully in your discussion you talked about the exponential model NOT modeling the data very well as t gets large. It doesn t model the data very well because it would be impossible for the population to get infinitely large. We would have limitations on the rabbit population such as food supplies, illnesses, and other animals preying on them. We would have a carrying capacity for our population or a maximum population due to outside factors. Another mathematical model called the logistic model can be used to model this type of population behavior for a much longer period of time. The general logistic equation is: c P t 1 bt ae where b and c are both positive values. The value for c represents the carrying capacity of the land. For our problem, let the carrying capacity for the rabbits on the acreage we fenced off be 50,000. The value for b represents the rate of growth (in our above model r represented our rate of growth.) The value of the initial population does not appear directly in the logistic equation. Since the initial b*0 population is when t = 0 and e 1for any value of b, we have P 0 ( ) = c 1+ a. In our initial problem, the initial population is P(0) = 1000, and the carrying capacity is c = 50,000. Under these conditions, what value does a have? c Now, to finish modeling the equation, we need to use the equation P t, substitute the value 1 bt ae that we found for a above and find the value for b. Using the set of data at the beginning of the lesson, use one of the points and find b. [Remember not to use the first data point (0, 1000).] What value did you find for b? What is the model for the original rabbit population question?
Graph the model in your calculator. Sketch the graph on the grid below. Make sure to label your window. Now graph the line y= 50,000 on the same set of axis. What does the line y = 50,000 represent? rt On the same graph, graph your original equation P t 1000e. Notice that your original equation and the logistic equation are very close early on. Due to complexity, we will use the exponential model for short term growth models and the logistic model when we are concerned about approaching the carrying capacity. How does the logistic model behave when we approach the line y = 50,000? Describe what you would expect to happen as a population approached the carrying capacity. Does the logistic model accurately portray the possible events as the model gets close the carrying capacity (the line y = 50,000)?
Hopefully, you realize that the exponential model does well in the short term and logistics model does well in the short and middle range area, but will be less accurate near the carrying capacity. Thus, there are better models when we approach the carrying capacity. These models require additional math concepts and will not be discussed here. Changing the value of b Let s use the same model that we graphed above but now, let s change the value for b so we can see what changing the growth rate does to the graph. We are going to keep the values for a and c the same. Change the value of b to 0.075. Rewrite the model and graph the rabbit population model on your calculator. Also graph the y = 50,000 line on your calculator. Sketch both equations below making sure to label the window. Now, compare the two graphs above. How are the graphs the same? Explain. How are the graphs different? Explain. How does changing the value of b effect the population?
Exercises c 1 ae, rewrite the model letting the initial population be 500 and the carrying capacity be 100,000. What will be your values for a, b and c? What will the new model be? Graph the new model and the line y = 100,000 on the same axis. How does our new model differ from the model that we found in the activity above? Explain. How is it similar? Explain. 1. Using the logistic model P t bt 2. Consider fencing off several thousand acres of land and placing 5000 rabbits on the land. The table below gives the data that the farmer gathered. Population P(t) Time (t) in months 5000 7283 10,490 17,583 36,717 0 2 4 7 12 Find a logistic model that will fit the data. Assume that the carrying capacity is 100,000.