Question

2. Suppose 27 blackberry plants started growing in a yard. Absent constraint, the blackberry plants will spread by 80% a month. If the yard can only sustain 140 plants, use a logistic growth model to estimate the number of plants after 5 months. plants

Answer 1

Step-by-step explanation

The question indicates we should use a logistic model to estimate the number of plants after 5 months.

This can be done using the formula below;

`\begin{gathered} P(t)=(K)/(1+Ae^(-kt));A=\frac{K-P_{0_{}}}{P_0}_{} \\ \text{From the question} \\ P_0=\text{ Initial Plants=27} \\ K=\text{Carrying capacity =140} \end{gathered}`

Workings

Step 1: We would need to get the value of A using the carrying capacity and initial plants that started growing in the yard.

This gives;

`\begin{gathered} A=(140-27)/(27) \\ A=(113)/(27) \end{gathered}`

Step 2: Substitute the value of A into the formula.

`P(t)=(140)/(1+(113)/(27)e^(-kt))`

Step 3: Find the value of the constant k

Kindly recall that we are told that the plants increase by 80% after each month. Therefore, after one month we would have;

`\begin{gathered} P(1)=27+((80)/(100)*27) \\ P(1)=(243)/(5) \end{gathered}`

We can then have that after t= 1month

`\begin{gathered} (140)/(1+(113)/(27)e^(-k*1))=(243)/(5) \\ Flip\text{ the equation} \\ (1+(113)/(27)e^(-k))/(140)=(5)/(243) \\ 243(1+(113)/(27)e^(-k))=700 \\ 243+1017e^(-k)=700 \\ 1017e^(-k)=700-243 \\ 1017e^(-k)=457 \\ e^(-k)=(457)/(1017) \\ -k=\ln ((457)/(1017)) \end{gathered}`

Step 4: Substitute -k back into the initial formula.

`\begin{gathered} P(t)=\frac{140}{1+(113)/(27)e^{\ln ((457)/(1017))t}} \\ =\frac{140}{1+(113)/(27)(e^{\ln ((457)/(1017))})^t} \\ P(t)=\frac{140}{1+(113)/(27)((457)/(1017)^{})^t} \\ \end{gathered}`

The above model is can be used to find the population at any time in the future.

Therefore after 5 months, we can estimate the model to be;

`\begin{gathered} P(5)=\frac{140}{1+(113)/(27)((457)/(1017)^{})^5} \\ P(5)=(140)/(1.07668) \\ P(5)=130.029\approx130 \end{gathered}`

Answer: The estimated number of plants after 5 months is 130 plants.