Logistic regrowth
Pasture biomass B (kg dry matter per hectare) grows towards a ceiling K set by light, nutrients and sward health H:
dB/dt = r B (1 - B/(K H)) - G(B) - m B
r = r0 fT(T) fW(theta) (0.35 + 0.65 H)
fT = 1 - ((T - 20)/14)^2 (0 outside 6..34 C)
Growth is slowest when the sward is nearly bare and when it is full (self-shading), and fastest near K/2. With r0 = 0.07 per day and K = 3000 the peak is r0 K/4 = 52 kg DM/ha/day. Without grazing B(t) = K / (1 + ((K - B0)/B0) e^(-r t)).
Water
Each paddock keeps a 100 mm soil-water bucket: d(theta)/dt = (0.85 R - ET)/100. Water stress stops growth when the bucket is nearly empty and standing leaf dies back.
Grazing and stocking rate
Intake follows a saturating (Holling type II) response on forage above the 350 kg/ha stubble: h = (B - Bres)/(B - Bres + Bhalf), intake capacity 3.6 % of live weight per day. Goats take 30 % of their diet as browse from a shrub layer. Hay in the trough displaces pasture.
Reading the 2D views
Map: colour = biomass (brown bare, yellow thin, green lush); a reddish overlay shows lost sward health H; the amber outline marks the flock. Phase plane: for a given stocking rate (head per grazed hectare, log axis) the flock grazes a paddock from 2400 kg/ha down to your residual (or until the max-days rule) taking d days; the other five paddocks then rest 5d days (green curve). To regrow from the residual back to 2400 at the ideal-weather rate takes t = ln[(B1/(K-B1)) / (B0/(K-B0))] / r0 (amber curve). Where green is above amber the rotation is sustainable; the dot is the live state (actual mean rest of the five resting paddocks). Grazing-time regrowth, browse and weather are ignored in the curves, so they are a first-order guide.
Rest period and overgrazing collapse
When B stays below 800 kg/ha, sward health H declines by up to 1 % per day and lowers both r and K; it recovers only slowly (0.4 % per day). A rotation works because resting paddocks sit on the steep part of the logistic curve while the flock never grazes below the protective residual.
Simplifications
Each paddock is one well-mixed biomass pool: no patchy selective grazing, species change, manure return, parasites, lambing or kidding; animal weight stops at 70 % of the start value. Parameters are textbook-scale illustrations, not calibrated to a farm. Not veterinary or farm-management advice.