Biophysics, Fluids & Geoscience
Neural excitation, porous flow, atmospheric and ocean dynamics, and solid-Earth wave cartoons
Enzyme Inhibition Kinetics
Competitive, noncompetitive, and uncompetitive inhibition: Michaelis-Menten curves, apparent Km/Vmax shifts, and a Lineweaver-Burk sketch.
Hardy-Weinberg Population Genetics
Allele frequency p, p²/2pq/q² genotype frequencies, plus selection, mutation, and genetic drift toy trajectories.
Logistic Harvest & Maximum Sustainable Yield
Logistic growth with constant harvest: MSY = rK/4, stable stock, unstable collapse threshold, and population trajectory.
Leslie Matrix Age-Structured Population
Fertility and survival matrix model: total population trajectory, dominant eigenvalue lambda, and stable age distribution.
SIR Vaccination Threshold
Herd-immunity threshold simulator: R_eff = R0(1-v), SIR outbreak curves, peak infected, and final size vs vaccine coverage.
Reaction-Diffusion Turing Patterns
Activator-inhibitor reaction-diffusion toy with spots, stripes, worms presets, diffusion ratio, and schematic dispersion relation.
Hodgkin–Huxley Action Potential
Squid-axon HH ODEs: membrane voltage V(t) and gating (m, h, n); step stimulus and Na⁺/K⁺ conductance traces.
Cardiac Action Potential
FitzHugh–Nagumo/Noble-style cardiac toy: fast upstroke, plateau-like repolarization, refractory recovery gate, and S1-S2 premature-stimulus capture vs block.
Calcium Waves / IP3 Oscillator
Li-Rinzel-style IP3 receptor oscillator: cytosolic Ca, ER store, calcium-induced calcium release, SERCA refilling, flux traces, and a Ca–h phase portrait.
Gene Regulatory Toggle Switch
Two mutually repressing genes with Hill cooperativity: bistable A-high/B-high attractors, phase-plane nullclines, hysteresis under inducer sweeps, and noise-driven switching.
Circadian Oscillator (Goodwin Clock)
Goodwin-style molecular clock: delayed negative transcriptional feedback, mRNA/protein/repressor phase lag, period readout, and light-pulse phase advance or delay.
Pharmacokinetics: 1-/2-Compartment
Linear IV PK simulator: bolus, infusion, and repeated dosing; central/peripheral concentration curves, terminal half-life, AUC, Cmax/Cmin, loading dose, and maintenance rate.
Oxygen-Hemoglobin Dissociation Curve
Hill-equation Hb saturation vs pO2 with effective P50 shifts from pH, CO2, and temperature; arterial/venous points, Bohr effect, and O2 content.
Hodgkin-Huxley / FHN Neuron Network
Multi-node excitable network using FHN reductions of HH dynamics: chain/ring topology, sigmoidal synaptic coupling, propagation delay, pacer-driven waves, and bursting-like activity.
Cable Equation on an Axon
1D passive cable ∂V/∂t = D∂²V/∂x² − (V−V_r)/τ + I; compare uniform fiber vs myelinated (higher D in internodes).
Coupled FitzHugh–Nagumo Neurons
Two excitable FHN units with diffusive coupling: phase plots, time traces, and synchronization vs coupling strength.
Nernst Potentials & Membrane
Nernst E for K⁺ and Na⁺ from inside/outside concentrations; optional Goldman–Hodgkin–Katz resting estimate.
Kelvin–Helmholtz Shear Instability
Two parallel flows with slip U: Fourier-mode growth rate sketch and animated interface billows (linear toy model).
Taylor–Couette Flow (Criterion)
Gap d, cylinders Ω_i, Ω_o, kinematic ν: Taylor number Ta and critical Ta_c for toroidal rolls — cartoon + formula.
Darcy Flow & Penetration Depth
1D head h(z,t): ∂h/∂t = K ∂²h/∂z²; recharge at surface, fixed head at depth — relaxation time and diffusion length.
Groundwater Well Drawdown (Theis)
Confined-aquifer Theis solution: cone of depression s(r,t) = (Q/(4πT)) W(u), u = r²S/(4Tt); transmissivity, storativity, observation well.
Flood Frequency (Gumbel)
Annual maxima ~ Gumbel EV1: return period T, exceedance probability, design flood x_T, and method-of-moments fit to a synthetic sample.
Soil Water Retention Curve
van Genuchten θ(ψ), effective saturation S_e, Mualem K(θ), field capacity, wilting point, and plant-available water.
Watershed Runoff Hydrograph
Rainfall hyetograph, initial abstraction, infiltration losses, triangular unit hydrograph convolution, peak discharge, and time-to-peak.
Quadrupole / Penning-Style Trap
2D motion in a hyperbolic electric potential Φ ∝ x²−y² with axial magnetic field B: epicycloid-like bounded orbits.
Layered Medium: P/S Fronts & Hodograph
P and S in a two-layer straight-ray toy: reflected legs and optional head branches when the lower layer is faster; hodograph plus minimum-time first arrivals (no conversions or amplitudes).
Geomagnetic Reversals (Toy Dynamo)
Low-order α–Ω cartoon: axial dipole B(t) from coupled amplitudes with symmetry breaking — stochastic flips, not MHD.
Greenhouse: One vs Two Slabs
Compare gray-atmosphere surface temperature for one IR slab vs two coupled slabs; albedo α and absorptivities.
0-D Energy Balance & Ice–Albedo
Budyko-style (1−α)S/4 = εσT⁴ with icy vs clear albedo, freeze/melt hysteresis when S is swept, and optional smooth α(T) relaxation.
Urban Heat Island Energy Balance
Two 0-D surface slabs: albedo, heat capacity, anthropogenic heat, radiative/turbulent cooling; urban–rural ΔT and day/night lag.
Daisyworld (Climate Regulation)
Black and white daisies compete on bare ground, shifting planetary albedo; compare T(L) with a lifeless rock and watch homeostasis in a toy Watson–Lovelock model.
ENSO Delayed Oscillator (Toy)
Scalar delay differential equation ẋ = a·x(t−τ) − b·x − c·x³ + seasonal forcing: explore El Niño– / La Niña–like swings vs damping in a Suarez–Schopf–style cartoon.
Rossby Waves on a Beta-Plane
Linear planetary wave ψ = A cos(kx+ly−ωt) with ω = −βk/(k²+l²+Rd⁻²): streamfunction bands and geostrophic arrows show westward phase propagation.
Geostrophic Balance & Thermal Wind
Pressure-gradient force balanced by Coriolis: isobars set geostrophic wind, while meridional temperature gradients produce vertical shear between 1000 and 300 hPa.
Atmospheric Stability / Parcel Diagram
Toy sounding with environmental lapse rate, dry/moist parcel ascent, LCL, LFC, equilibrium level, CAPE, CIN, and Lifted Index.
Groundwater Contaminant Plume
Advection-dispersion pulse in groundwater with longitudinal/transverse spreading, retardation factor R, and a monitoring-well breakthrough curve.
Infinite Slope Stability
Limit-equilibrium factor of safety for a shallow planar slide: slope angle, cohesion, friction, depth, and rainfall-driven pore pressure set the landslide threshold.
Earthquake Aftershocks: Omori + Gutenberg-Richter
Modified Omori aftershock decay n(t)=K/(t+c)^p combined with Gutenberg-Richter magnitude-frequency curves, b-value, and a synthetic catalog.
Mantle Convection Cell (Toy)
High-Prandtl-number mantle convection cartoon: Rayleigh-number vigor, thermal boundary layers, hot upwelling, and a cold subducting slab in one viscous cell.
Carbon Cycle (4-Box Model)
Atmosphere, ocean mixed layer, deep ocean, and land biosphere exchange linearly; add fossil emissions or GtC pulses and watch inventory split vs a toy airborne fraction.
Mid-Ocean Ridge: Magnetic Stripes
Symmetric stripes accrete at a spreading center as polarity flips — a 2D cartoon of Vine–Matthews–Morley marine magnetic anomalies.
Glacier Flow 1D (Glen’s Law)
Shallow-ice thickness H(x,t) on a tilted bed: nonlinear flux q ∝ H^{n+2}|∂s/∂x|^{n−1}∂s/∂x plus accumulation; τ_b ≈ ρgH sinα readout.
River Meandering (Toy)
Pinned centerline y(x,t): toy migration ∂y/∂t ≈ k₁ ∂κ/∂s + λ y_xx with κ ≈ y_xx — bend sharpening cartoon, not full sediment hydraulics.
Ekman Spiral in the Ocean
Wind stress, Coriolis f, eddy viscosity A_z: classical Ekman spiral of horizontal velocity with depth; deflection angle.