Creature Sanctuary 2D: Comfort Map, Phase Line and Logistic Growth

Species

Set temp20 °C

Set humidity50 %

Start N1

Speedx1

Presets (same as the 3D habitat lab):

Experiment: measured vs predicted

Runs the RK4 integrator in the background and compares it with two closed forms.

The science

Comfort map. Each species has a preferred temperature T* and humidity RH*. Comfort is the product of two Gaussians: c = exp(-½((T-T*)/σT)²)·exp(-½((RH-RH*)/σRH)²). The rings on the map are the c = 0.5 contours (an ellipse with half-widths 1.177σ). Click or drag on the map to move the set-point.

Climate control. The glass box relaxes towards the set-point (Newton's law of cooling): dT/dt = (T_set-T)/τ, so T(t)=T_set+(T₀-T_set)e^-t/τ (τ_T=1.2 d, τ_RH=0.9 d).

Phase line. Population follows dN/dt = r·c·N(1-N/K) - d(1-c)N with K = 6 and d = 0.25 per day. Plotting dN/dt against N shows the fixed points: N = 0 and N* = K(1 - d(1-c)/(r·c)). Where the curve crosses zero going downward, the equilibrium is stable (arrows point towards it). If r·c ≤ d(1-c) the colony declines to extinction.

Prediction. With a constant comfort the exact solution is logistic with rate a = r·c - d(1-c): N(t) = aN₀e^at/(a + b N₀(e^at-1)), b = rc/K. The dashed curve on the time plot is this prediction from the current state. Happiness relaxes to comfort, dh/dt=(c-h)/1.5 d.

Simplifications. N is continuous; one species per habitat; no age structure or noise; the preferred values are invented for imaginary creatures. The physics module is a verbatim copy of the 3D version, so both give identical numbers.