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 (little leaf to intercept light) and when it is full (self-shading), and fastest near K/2. With r0 = 0.07 per day and K = 3000 the peak growth is r0 K/4 = 52 kg DM/ha/day, a realistic spring figure. The closed form without grazing is B(t) = K / (1 + ((K - B0)/B0) e^(-r t)), used for the numerical check below.
Water
Each paddock keeps a 100 mm bucket of plant-available soil water: d(theta)/dt = (0.85 R - ET)/100. Evapotranspiration follows temperature (ET0 = 1.5 + 0.2 T mm/day) and falls as the soil dries. Water stress fW drops to 0 when the bucket is nearly empty, and standing leaf then dies back at up to 3 % per day. Rain falls as random events, with the chance and mean depth set by the scenario.
Grazing and stocking rate
Intake per head follows a saturating (Holling type II) functional response on the forage above the ungrazeable stubble (350 kg/ha):
I = c W (1 - phi s/(cW)) h(B), h = (B - Bres)/(B - Bres + Bhalf)
G = sum(I)/A (kg DM/ha/day removed)
c = 3.6 % of live weight per day is the dry-matter intake capacity, A the paddock area. Goats take 30 % of their diet as browse from a separate shrub layer. Supplement s displaces pasture by a substitution rate phi (0.6 for hay, 0.35 for barley, 0.5 for pellets).
Weight gain per head
Metabolisable energy in is I q + s ME; maintenance (including grazing activity) is 0.52 W^0.75 MJ/day. The surplus becomes gain at 32 MJ per kg, a deficit is paid from body reserves at 24 MJ per kg. Forage quality falls as the sward matures, q = 11 - 3 B/K MJ/kg DM, so lush short grass feeds better than rank tall grass.
Rest period and overgrazing collapse
When B stays below 800 kg/ha, sward health H (crown reserves, ground cover, root mass) declines by up to 1 % per day and lowers both r and K; it recovers only slowly (0.4 % per day) when the sward is allowed to grow above 1300. A rotation works because paddocks spend most of the time resting at the steep part of the logistic curve, where regrowth per day is largest, while the flock never grazes below the residual that protects H.
Simplifications
Each paddock is one well-mixed biomass pool: no patchy selective grazing, no species change, no manure nutrient return, no parasites, lambing or kidding, and animals never die (their weight stops at 70 % of the starting value, a real flock would be fed or sold). Parameters are textbook-scale illustrations, not calibrated to a farm. The paddocks are drawn 24 m wide for readability; the area used in the maths is the slider value. Not veterinary or farm-management advice.