4. Gause's logistic model In an experiment with the protozoan Paramecium the biologist G.F. Gause modeled the protozoan population size with the logistic function \[ P(t)=\frac{61}{1+31 e^{-0.7944 t}} \] Where \( t \) is measured in days. According to this model, how fast was the population growing after 8 days?

To determine how fast the population was growing after 8 days using Gause's logistic model, we need to find the derivative of the function \( P(t) \) and evaluate it at \( t = 8 \). The function is given by: \[ P(t) = \frac{61}{1 + 31 e^{-0.7944 t}} \] First, we differentiate \( P(t) \) with respect to \( t \): Using the quotient rule: \[ P'(t) = \frac{(0)(1 + 31 e^{-0.7944 t}) - 61 \cdot (-31)(-0.7944)e^{-0.7944t}}{(1 + 31 e^{-0.7944 t})^2} \] This simplifies to: \[ P'(t) = \frac{61 \cdot 31 \cdot 0.7944 e^{-0.7944 t}}{(1 + 31 e^{-0.7944 t})^2} \] Next, we evaluate \( P'(t) \) at \( t = 8 \): First, calculate \( e^{-0.7944 \cdot 8} \): \[ e^{-6.3552} \approx 0.0017 \] Now, substituting this back into the logistic function \( P(t) \): \[ P(8) = \frac{61}{1 + 31 \cdot 0.0017} \approx \frac{61}{1 + 0.0527} \approx \frac{61}{1.0527} \approx 57.9 \] Now, use this \( e^{-6.3552} \) in the derivative: \[ P'(8) \approx \frac{61 \cdot 31 \cdot 0.7944 \cdot 0.0017}{(1 + 31 \cdot 0.0017)^2} \] Calculating the denominator: \[ (1 + 0.0527)^2 \approx 1.0527^2 \approx 1.108 \] Putting it all together for \( P'(8) \): \[ P'(8) \approx \frac{61 \cdot 52.4 \cdot 0.0017}{1.108} \approx \frac{1.74}{1.108} \approx 1.57 \] Thus, the population was growing at approximately **1.57 individuals per day** after 8 days.