Modeling A Population Suppose that rabbits reproduce at a rate proportional to their population (uninhibited growth). If initially there are 24 rabbits and 6 months from now (! = 1 2) there are 48 rabbits, 1. a. Write the IVP that describes the rabbit population. b. Find the solution to the IVP. c. How many rabbits will there be in 18 months (! = 1.5) ? 2. A more detailed model reflects that the rabbits will begin to compete for resources – food, shelter from preditors, etc. and that competition will slow the growth as the population increases. For a given area or site, the carrying capacity ! is the number of rabbits that can be sustained by the resources present. One way to take the carrying capacity into account is to use the logistic model. In this model, the rate of change in the population is directly proportional to the product of the population, ! , and to the difference between the carrying capacity and the population, ! − !, which we can write as follows: (a) __________________________. If the initial number in the population at time ! = 0, ! 0 = !! , then the solution to the IVP is ! ∙ !!
! ! =
. ! − !! ! !!"# + !!
Example continued. Recall from above, initially there are 24 rabbits. Suppose the carrying capacity of the rabbit population at the proposed site is ! = 100. Then (leaving k unknown) the logistic model for this particular rabbit population has solution: (b) ! ! = Modeling A Population (c) Use the information ! 1 2 = 48 in you solution to part (b) to find ! . As a self check you should get ! = 0.021453. Notice that this value for k is different than the value of the constant of proportionality for the unihibited growth model. Show your work here: 3. Analysis a. Graph !1 ! = 24! !.!"#! , and !2
!"##
! = !"! !!.!"#$! !!" at the same time on your graphing calculator in the Window [0,5]x[0,100]. Sketch your result below. (label axes) b. For each model, find the instantaneous % − growth rate for the rabbit population at time 0 , !′(0)/! 0 and after two years (! = 2), find !′(2)/! 2 . For the uninhibited growth model, find !′(0)/! 0 and !′(2)/!(2) using pencil and paper only. For the logistic model use a numerical method on your graphing calculators to find !′(0) and !′(2) to use in the ratios above. Modeling A Population c. For the logistic model, how many rabbits are there predicted to be in 18 months? Write a report addressing the following questions. (It should be ½ to 1 page in length) a) Compare the assumptions used in the two models for the rabbit population. b) How did you use data in formulating each of the models? c) Compare the solutions to the two models. Use your results from 1(a), 3(a), 3(b), and 3(c) in your discussion, as well as, lim!→! !(!) for both model solutions. Which results seem the most realistic? Which model would you recommend using and why?