Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.
c. After how many hours does the population reach half of the carrying capacity

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Identify the logistic growth model formula: \(P(t) = \frac{K}{1 + Ae^{-rt}}\), where \(P(t)\) is the population at time \(t\), \(K\) is the carrying capacity, \(A\) and \(r\) are parameters to be determined.

Use the initial condition at \(t=0\) where \(P(0) = 20\) to find \(A\). Substitute \(t=0\) and \(P(0)=20\) into the formula: \(20 = \frac{1600}{1 + A}\), then solve for \(A\).

Use the population at \(t=20\) hours, \(P(20) = 80\), to find the growth rate \(r\). Substitute \(t=20\), \(P(20)=80\), \(K=1600\), and the previously found \(A\) into the logistic equation and solve for \(r\).

Set \(P(t)\) equal to half the carrying capacity, which is \(\frac{1600}{2} = 800\), and write the equation: \(800 = \frac{1600}{1 + Ae^{-rt}}\).

Solve the equation from step 4 for \(t\) to find the time when the population reaches half the carrying capacity.

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Logistic Growth Model

The logistic growth model describes how a population grows rapidly at first and then slows as it approaches a maximum limit called the carrying capacity. It is often expressed by the equation P(t) = K / (1 + Ae^(-rt)), where P(t) is the population at time t, K is the carrying capacity, r is the growth rate, and A is a constant related to initial conditions.

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Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely given the available resources. In logistic growth, it acts as an upper bound, causing the growth rate to decrease as the population nears this limit, preventing unlimited exponential growth.

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Solving for Time in Logistic Growth

To find the time when the population reaches a specific value, such as half the carrying capacity, you substitute that population value into the logistic growth equation and solve for t. This typically involves algebraic manipulation and taking natural logarithms to isolate the time variable.

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