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Suppose 18 blackberry plants started growing in a yard. Absent constraint, the blackberry plants will spread by 85% a month. If the yard can only sustain 100 plants, use a logistic growth model to estimate the number of plants after 3 months.

Sagot :

Answer

The estimated number of plants after 3 months using the logistic model = 70 blackberry plants

Explanation

If a population is growing in a constrained environment with carrying capacity K, and absent constraint would grow exponentially with growth rate r, then the population behavior can be described by the logistic growth model:

[tex]P_n=P_{n-1}+r(1-\frac{P_{n-1}}{K})P_{n-1}[/tex]

From the question,

[tex]\begin{gathered} P_0=18,r=85\%=0.85,K=100 \\ \\ So, \\ \\ P_n=P_{n-1}=+0.85(1-\frac{P_{n-1}}{100})P_{n-1} \end{gathered}[/tex]

After the first month,

[tex]\begin{gathered} P_{n-1}=P_0=18 \\ \\ \therefore P_1=P_0+0.85(1-\frac{P_0}{100})P_0 \\ \\ P_1=18+0.85(1-\frac{18}{100})18 \\ \\ P_1=18+0.85(1-0.18)18=18+0.85\times0.82\times18 \\ \\ P_1=18+12.546 \\ \\ P_1=30.546\text{ }plants \end{gathered}[/tex]

After the second month,

[tex]\begin{gathered} P_1=30.546 \\ \\ \therefore P_2=P_1+0.85(1-\frac{P_1}{100})P_1 \\ \\ P_2=30.546+0.85(1-\frac{30.546}{100})30.546 \\ \\ P_2=30.546+0.85(1-0.30546)30.546=30.546+0.85\times0.69454\times30.546 \\ \\ P_2=30.546+18.033 \\ \\ P_2=48.579\text{ }plants \end{gathered}[/tex]

So after 3 months,

[tex]\begin{gathered} P_2=48.579 \\ \\ \therefore P_3=P_2+0.85(1-\frac{P_2}{100})P_2 \\ \\ P_3=48.579+0.85(1-\frac{48.579}{100})48.579 \\ \\ P_3=48.579+0.85(1-0.48579)48.579=48.5796+0.85\times0.5142\times48.579 \\ \\ P_3=48.579+21.232 \\ \\ P_3=69.811\text{ }plants \\ \\ P_3\approx70\text{ }blackberry\text{ }plants \end{gathered}[/tex]

The estimated number of plants after 3 months using the logistic model = 70 blackberry plants.

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