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?

We are given the population function \[ P(t)=\frac{61}{1+31e^{-0.7944t}}, \] and we want to find the growth rate \( P'(t) \) at \( t=8 \) days. **Step 1.** Write the function in terms of a denominator \( D(t) \): \[ P(t)=\frac{61}{D(t)} \quad \text{where} \quad D(t)=1+31e^{-0.7944t}. \] **Step 2.** Differentiate \( P(t) \) using the quotient rule. Since the numerator is constant, we have \[ P'(t)=-\frac{61 \, D'(t)}{(D(t))^2}. \] **Step 3.** Differentiate \( D(t) \): \[ D'(t)=31 \cdot \frac{d}{dt}\Bigl(e^{-0.7944t}\Bigr)=31\cdot(-0.7944)e^{-0.7944t}=-31\cdot0.7944\,e^{-0.7944t}. \] **Step 4.** Substitute \( D'(t) \) into the derivative of \( P(t) \): \[ P'(t)=-\frac{61(-31\cdot0.7944e^{-0.7944t})}{(1+31e^{-0.7944t})^2} =\frac{61\cdot31\cdot0.7944\,e^{-0.7944t}}{(1+31e^{-0.7944t})^2}. \] **Step 5.** Evaluate at \( t=8 \) days. First, compute the exponent: \[ 0.7944\cdot8=6.3552, \] so \[ e^{-6.3552}. \] An approximate value is \[ e^{-6.3552}\approx0.00174. \] **Step 6.** Substitute back into the derivative: \[ P'(8)=\frac{61\cdot31\cdot0.7944\,e^{-6.3552}}{\Bigl(1+31e^{-6.3552}\Bigr)^2}. \] First compute the constant product: \[ 61\cdot31=1891,\quad \text{and then}\quad 1891\cdot0.7944\approx1502.2. \] Thus the numerator is \[ 1502.2\cdot e^{-6.3552}\approx1502.2\cdot0.00174\approx2.62. \] **Step 7.** Now compute the denominator: \[ 1+31e^{-6.3552}=1+31\cdot0.00174\approx1+0.05394\approx1.05394. \] Square the denominator: \[ (1.05394)^2\approx1.111. \] **Step 8.** Finally, compute \[ P'(8)\approx\frac{2.62}{1.111}\approx2.36. \] Thus, according to Gause's logistic model, after 8 days the population was growing at a rate of approximately \( 2.36 \) individuals per day.