One equation for the whole animal. The core temperature Tb follows a lumped heat balance: C dTb/dt = M + G_fur (T_env - Tb) - Q_ears - E_pant - E_base, with heat capacity C = m c (1.5 kg, 3470 J/kg/K), resting metabolic heat M = a * 0.7 * 3.4 m^0.75 watts (Kleiber's law scaled to 70 % for desert mammals, times an activity factor a), and the fur conductance G_fur = A / (L/k + 1/h) for fur depth L = 1.5 cm, k = 0.04 W/m/K and outside coefficient h = h_conv + h_rad, with h_conv = 5.7 + 3.8 v (wind speed v) and h_rad = 4 eps sigma T^3.
Ears as a radiator. The ears are thin, nearly hairless fins fed by warm blood. With blood flow V, a counter-current efficiency e and rho c = 3.6 MJ/m3/K the blood-side conductance is G_b = e rho c V. The ear surface temperature Te solves G_b (Tb - Te) + Q_sun = A_e [ h_c (Te - Ta) + eps sigma (Te^4 - Tr^4) ] (Newton iteration, exact radiation) and the ears lose Q_ears = G_b (Tb - Te). In series with the skin conductance G_s = A_e (h_c + h_r) the total is G_b G_s / (G_b + G_s). Ear area is 4 x 58 cm2 x size (two faces, two ears: 232 cm2 at size 1). The fox opens the ear vessels when the core passes 38.4 C and the surroundings are cooler than the core; if the air and sand are hotter, the ears would gain heat, so the model keeps them nearly closed.
Panting and water. Above 39.2 C panting rises linearly to a maximum of 12 W at 40.4 C. Evaporating water removes 2.42 MJ/kg, so 12 W costs about 18 g of water per hour. Metabolic water from burning food adds about 0.03 g per kJ. The day is a cosine in air temperature (peak at 15:00) with a half-sine sun (1000 W/m2 times your sun setting); the burrow stays at Tmean - 0.4 * amplitude, dark and still.
What is simplified. One body node (no gradients, no fur thermal mass), a single ear conductance, fixed sun-absorption fractions, no humidity (evaporation is always possible), no sweating, no change of the ear temperature through the thickness, a smooth day cycle, no cloud noise. The thresholds, the 12 W panting maximum, the fur depth, the ear area and the blood-flow range are rounded typical or illustrative values chosen so that the thermoneutral zone falls near 25-32 C and the burrow saves most of the water; they are not measurements of a single animal and should be spot-checked against the literature before being quoted. The ears matter most in warm nights and mild days; in the sunny midday heat they cannot help because the surroundings are hotter than the fox.
Checks on this page. (1) The locked-controller steady state against Ta + (M - E0) / (G_fur + G_series); (2) the step-response time constant against C / G; (3) the energy balance C dTb = sum(heat in) - sum(heat out) over a full day; (4) the panting-band steady state against the linear closed form Tb = (M - E0 + G Te_env + k Tp0) / (G + k); (5) the radiation coefficient 4 eps sigma T^3 against a numerical derivative of the Stefan-Boltzmann law.