Bacteria and Algebra

Bacteria and Algebra.

Part A
For one strain of bacteria, each bacterium divides into two every minute.
The table shows the number of bacteria present in a particular sample for the first 5 minutes.
Time (minutes) number of bacteria present
0 2
1 4
2 8
3 16
4 32
5 64
Write down an algebraic rule linking the number of bacteria present at a particular time to the number present one minute previously.
= y2t+2*y2t
Write down an expression for the number of bacteria present after t minutes
The solution: 22t, where t stands for time in minutes.
Calculate the number of bacteria present after 2 hours. (State any assumptions you make.)
Given that t is time in minutes, then 2 hours is equivalent to 120 minutes. Therefore the solution is: 22*120=1.76684706477838E723
Calculate the time for the colony to reach 1 million bacteria.
Given that the time taken is t and the initial bacteria are 2, then we have:
22t=1000000,
4t=1000000,
tlog4=log 1000000
t=9.96578428466209 minutes
We therefore say that it takes approximately 10 minutes to have bacteria colony reach 1 million.
Part B
Use algebra to extend this model for the growth of bacteria colonies.
You could investigate:
The relationship between the number of bacteria and the size of the colony
Different rates of replication
Colonies of different sizes at the start
Effect of growth limiting factors (such as build-up of waste products, competition for space)
Explain all the steps of your analysis and state any assumptions you make in constructing your model.

Give references for any data you use.
In order to analyze the growth of bacteria over a period, we have employed the use of the SPSS software to generate models and to analyze the bacteria replication relationship with time. This results provided gives a good description of the data used; correlation explains the relationship that exists between time and the replication of the bacteria. A model for forecasting was also used to forecast the growth of bacteria over the future time. A graph was also used to give a good pictorial representation of this anticipated growth. The results are provided below.
Correlation
Descriptive Statistics
Mean Std. Deviation N
time 2.50 1.871 6
number of bacteria present 21.00 23.723 6
Correlations
time number of bacteria present
time Pearson Correlation 1 .906*
Sig. (2-tailed) .013
Sum of Squares and Cross-products 17.500 201.000
Covariance 3.500 40.200
N 6 6
number of bacteria present Pearson Correlation .906* 1
Sig. (2-tailed) .013 Sum of Squares and Cross-products 201.000 2814.000
Covariance 40.200 562.800
N 6 6
*. Correlation is significant at the 0.05 level (2-tailed).
The correlation analysis performed shows that bacteria replication is positively correlated with time. It shows a stronger relationship between time and replication of the bacteria.
A model for prediction was also created to forecast how the bacteria will replicate with increase in time. This is essential so that it can be known in advance whether the available space will effectively accommodate them. The assumption made is that the replication goes on to infinity, however due to insufficient way of presentation of the findings due to voluminous output, we cannot present it here.
Time series modeler
Model Description
Model Type
Model ID number of bacteria present Model_1 Simple
Model Fit
Fit Statistic Mean SE Minimum Maximum Percentile
5 10 25 50 75 90 95
Stationary R-squared -1.292 . -1.292 -1.292 -1.292 -1.292 -1.292 -1.292 -1.292 -1.292 -1.292
R-squared .515 . .515 .515 .515 .515 .515 .515 .515 .515 .515
RMSE 16.517 . 16.517 16.517 16.517 16.517 16.517 16.517 16.517 16.517 16.517
MAPE 41.667 . 41.667 41.667 41.667 41.667 41.667 41.667 41.667 41.667 41.667
MaxAPE 50.000 . 50.000 50.000 50.000 50.000 50.000 50.000 50.000 50.000 50.000
MAE 10.333 . 10.333 10.333 10.333 10.333 10.333 10.333 10.333 10.333 10.333
MaxAE 32.000 . 32.000 32.000 32.000 32.000 32.000 32.000 32.000 32.000 32.000
Normalized BIC 5.907 . 5.907 5.907 5.907 5.907 5.907 5.907 5.907 5.907 5.907
The forecasting model shows the trend of the bacteria replication, various factors are presented in the table. This includes: mean, the standard error, maximum and the minimum values of the analyzes obtained. The correlation coefficient however, seems to be constant at 0.515. This figure says that the variability explained by the model is 51.5%; this is a fair figure though not strong enough.
The graph below displays the growth of the bacteria over time; this growth is exponential as can be seen in the graph below. This depicts the growth as time goes and is certainly replicating at a faster rate.

The growth limiting factors are the factors that inhibit growth. This leads to a slow growth in the bacteria colony over time and it could eventually lead to a decline and total extinction of the bacteria. These factors could include the constriction of space and therefore bacteria lack room for expansion; it could also be due to competition for available resources by the other organisms for which they occupy the same niche.