1. We are examining a population of turtles in a lake. A biologist tells us that
ID: 3209756 • Letter: 1
Question
1. We are examining a population of turtles in a lake. A biologist tells us that the population is obeying the following differential equation: d P d (0.4-hP) where h is some positive constant that the biologist doesn't know yet. (a) This equation is logistic (you do not need to prove this). Find its carrying capacity. (b) The biologist proposes that h = 0.0005. Suppose that, over two subsequent years. we see 160 turtles, and then 145 turtles in the pond. Is this observation consistent with the biologist's proposal? Explain why or why not.Explanation / Answer
(a)
dP /dt = P(0.4 - hP) = 0.4P (1 - P/(0.4/h) )
compare it with dP/dt = kP(1-P/K) where K is the carrying capacity
carrying capacity K is 0.4 / h
(b) dP/dt = kP(1-P/K)
=> dP/ P(1-P/K) = kdt
=> dP/P + dP/K-P = kdt
Integrate both sides from t0 = 0 to t = 2 and P0 = 160 to P = 145
lnP/P0 - ln(K-P)/(K-P0) = k(t-t0)
=> ln(P*(K-P0)/(P0*(K-P)) = 0.4 * 2 = 0.8
=> 145*(K- 160) = 160 * (K - 145 ) * e0.8
=> 145K - 23200 = 160 * (K - 145 ) 2.22 = 355.2 K - 51504
=> 210.2K = 28304
=> K = 134.6 (actual)
Now K = 0.4 / 0.0005 = 800 (observed )
Hence not consistent
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