The use of spread sheeting and graphical calculators are encouraged on this assi
ID: 1719401 • Letter: T
Question
The use of spread sheeting and graphical calculators are encouraged on this assignment. Although it might be true to say that a population of mice with unlimited food and space would grow exponentially, this does not occur in practice. For any particular closed environment, there is obviously some maximum population of mice that can be supported. This maximum is g called the carrying capacity. Instead of modelling the growth rate as being a constant multiple of the population, it would obviously be better to model the growth rate as being dependent on the population and t e excess o carrying capacity over the population. Thus we would have dN/dT = k_1N (C - N) Where N = population, C = carrying capacity and k_1 is a growth rate. Solving this equation gives the formula Where R =-_o/C - N_o and N_o = initial population at time t = 0 For field mice on a small farm, the parameters are estimated as k_1 and C. You will be given unique values for these parameters. You will also be given a unique initial population of N_o. t is the time measured in weeks. Using the data values supplied, find your value for R and then construct a table of the population vs time as determined by the formula (N_1) over a period of 10 weeks. Draw a line graph of your results. Construct a table of the population vs time using the natural exponential model N_2 = N_0e^k_2t-over a period of 10 weeks. You will be given a value for k_2 by your teacher. Draw a graph of your results.(on same axes as in part a) Use technology to produce graphs for 30 weeks for each model. Comment on the graphs for the two models. What do you notice? Explain what is happening and why. For what values of t (if any) do you believe second, simpler natural exponential model is a suitable approximation of the carrying capacity model? Justify your answer and identify any assumptions made. Determine algebraically the time for the population model and the carrying capacity model.Explanation / Answer
Given k1=0.000034, N0=70
The population v/s time determine the formula for N1
C
T
R=N0/C-N0
N1=Rcek1Ct/1+Rek1Ct
1
1
1.0144
0.5035
2
2
-1.0294
69.796
3
3
-1.0447
69.666
4
4
-1.0606
69.391
5
5
-1.0769
69.267
6
6
-1.0937
69.16
7
7
-1.1111
69.004
8
8
-0.8857
-63.23
9
9
-1.1475
68.73
10
10
-1.1666
68.65
The population vs time using natural exponential determine the value of N2
Given k2=0.27,N0=70
t
N2=N0ek2t
1
91.697
2
120.120
3
157.353
4
206.127
5
270.019
6
353.716
7
463.355
8
606.979
9
795.121
10
1041.581
C
T
R=N0/C-N0
N1=Rcek1Ct/1+Rek1Ct
1
1
1.0144
0.5035
2
2
-1.0294
69.796
3
3
-1.0447
69.666
4
4
-1.0606
69.391
5
5
-1.0769
69.267
6
6
-1.0937
69.16
7
7
-1.1111
69.004
8
8
-0.8857
-63.23
9
9
-1.1475
68.73
10
10
-1.1666
68.65
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