How many fish to restock if you may only sell half the pond this year — sustainable restocking (Capitulary de Villis context)

Short answer (idea): after you sell half the pond, count how many remain, decide the minimum number you want at the start of next season (a breeding/buffer population), then restock enough so that after natural mortality and reproduction you meet that minimum. The exact restock number depends on three things you must choose or estimate: initial fish count, expected net growth (reproduction minus mortality) before next season, and the minimum desired population.

Step-by-step method and formula

1. Define variables:
   

   P0 = number of fish now (start of this season)
   s = fraction you sell this year (here s = 0.5)
   R = number of fish you will restock (what we want)
   g = net growth factor from after-restock now to the start of next season (g = 1 + net percentage change; e.g. g = 0.9 if you expect a 10% net loss, or g = 1.2 if you expect 20% net gain)
   B = desired minimum population at the start of next season (a breeding stock or buffer)

2. Calculate how many remain immediately after selling half:
   

   Remaining after sale = (1 - s) * P0 = 0.5 * P0

3. Add restocking R to that immediate remainder to get the immediate post-restock total:
   

   Immediate total = 0.5 * P0 + R

4. Apply expected net growth to get next-season population:
   

   P1 = (0.5 * P0 + R) * g

5. Require P1 to be at least the desired minimum B and solve for R:
   

   P1 >= B  -->  (0.5 * P0 + R) * g >= B
   So  R >= (B / g) - 0.5 * P0
   If that right-hand side is negative, you don't need to restock (R = 0) to reach B.

Worked numeric examples

Example A — simple, no net growth or loss (g = 1):

P0 = 100 fish, s = 0.5 -> 50 left after sale.
If you want B = 50 at next season and g = 1,
R >= (50 / 1) - 50 = 0.
So you do not need to restock to meet B = 50.

Example B — some winter loss, g = 0.8 (20% net loss):

P0 = 100, 0.5*P0 = 50 remain after sale.
Want B = 50 next season.
R >= (50 / 0.8) - 50 = 62.5 - 50 = 12.5 -> round up to 13 fish to restock.
Check: immediate total = 50 + 13 = 63; after loss g=0.8 -> P1 = 63*0.8 = 50.4 >= 50.

Example C — you want to allow the same sale fraction next year (sustainable steady state):

Suppose you aim for steady-state population P* that lets you sell s each year and come back to P* the next season. Then steady-state solves
P* = (1 - s) * P* * g + R  ->  R = P* [1 - (1 - s) g].
If you put R = 0 and want a positive P*, you need (1 - s) g = 1 (rare). Usually you choose P* and compute R.

Practical guidance and medieval context (Capitulary de Villis)

• Pick a sensible minimum B: in practice you want a breeding stock large enough to avoid crashes. For small ponds, that might be several dozen fish; for larger commercial ponds it will be larger. Avoid thinking "1 fish"—you need many breeders.
• Estimate g realistically: include expected winter mortality, disease losses, and reproduction. If you don’t know g, be conservative (assume net loss) and restock more.
• Round R up to whole fish and add a safety buffer (a few extra fish) to protect against unexpected losses.
• Historically, rules like those in the Capitulary de Villis encouraged careful estate management (keeping brood animals, not over-harvesting). The algebra above is a modern way to express the same principle: do not harvest so much that you cannot recover to a healthy breeding population.

If you give me numbers for P0, your estimate for g, and the minimum B you want, I’ll compute the exact restocking number R and show the arithmetic.