Exercise · ZW-LOGISTIC-MODEL-010 · ★★★ Advanced
For \(0<P<K\) we have \(P''=r\left(1-\tfrac{2P}{K}\right)P'\). When is the graph of \(P(t)\) concave up?
Choose one option.
Show answer
- B. When \(P<K/2\)
Explanation
With \(0<P<K\), \(P'>0\), so the sign of \(P''\) is that of \(1-2P/K\): positive if \(P<K/2\). At \(P=K/2\) there is an inflection point.
Review the concept: Logistic model
Model of a population \(P\) with intrinsic rate \(r>0\), carrying capacity \(K>0\) and initial population \(P_0\): it grows almost exponentially at first and levels off at \(K\).
\[P'=rP\left(1-\frac{P}{K}\right)\;\Longrightarrow\; P(t)=\frac{K}{1+\left(\frac{K}{P_0}-1\right)e^{-rt}}\]
More practice problems: logistic model
- ZW-LOGISTIC-MODEL-001 Find the equilibrium points of P'=0.5 P(1-(P)/(100)).
- ZW-LOGISTIC-MODEL-002 The equation P'=0.2 P-0.002 P^2 is logistic. What is its carrying capacity K?
- ZW-LOGISTIC-MODEL-003 In P'=0.03 P(1-(P)/(500)), what are r and K?
- ZW-LOGISTIC-MODEL-004 In the logistic model, if 0<P<K, how does P evolve?
- ZW-LOGISTIC-MODEL-005 For P'=0.4 P(1-(P)/(200)), at what population is the growth rate P' largest?
- ZW-LOGISTIC-MODEL-006 For P'=P(1-(P)/(10)) with P(0)=1, the solution is P(t)=(10)/(1+9e^-t). What is P(ln 9)?
- ZW-LOGISTIC-MODEL-007 With r=0.1 and K=100, a population P=300 exceeds the carrying capacity. What is P' at that moment?
- ZW-LOGISTIC-MODEL-008 With r=tfrac12, K=1000 and P_0=100, at what time does P reach K/2?