Biology · Introductory biology · Worked example
Find where logistic growth is fastest
A fish population follows logistic growth with r = 0.5 per year and K = 12,000. At what population size does it grow fastest, and how fast is that?
Write the growth rate as a function of N
The logistic growth rate is G(N) = rN(1 − N/K) = 0.5N(1 − N/12,000). It is 0 at N = 0 and at N = K, and positive in between.
Find the peak
G(N) is a downward parabola in N, so its peak lies halfway between its zeros, at N = K/2. Setting the derivative r(1 − 2N/K) equal to 0 gives the same answer.
Evaluate the peak rate
At N = 6,000 half of the carrying capacity is unused, so the population adds rK/4 fish a year.
Check both sides of the peak
At 3,000 and at 9,000 fish the rate is the same, 1,125 a year: the smaller population has room but few breeders, and the larger one has breeders but little room.
Connect to harvesting
Removing 1,500 fish a year could be balanced by growth only while the population stays at 6,000: the idea behind maximum sustainable yield. Real values of r and K are uncertain and change from year to year, so harvesting at the calculated maximum risks a collapse.
Result
Growth is fastest at N = K/2 = 6,000 fish, where the population adds rK/4 = 1,500 fish a year.
Your turn
A population has r = 0.3 per year and K = 800. Find the largest growth rate and the population size where it occurs.
Show the answer and explanation
60 individuals a year, at N = 400.
The peak is at N = K/2 = 400, and the rate there is rK/4 = 0.3 × 800/4 = 60.
Keep exploring
The population growth tool opens with 500 fish, r = 0.5 and K = 12,000. The curve is steepest where it crosses 6,000.
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