WebGL is not available in this browser. The numbers and charts on the right still run.
Year 0 Flowers 0 /m² Bees 0 /100 m² Visits 0 /s stable
Drag to orbit, wheel or pinch to zoom. Each bee is one pollinator (about 4 times life size, so it can be seen). The number of flowers and bees on screen follows the model; the wooden hive and the bare strip on the right (habitat loss) are scenery. Flower positions and flight paths are artistic.
Pollinator Decline Lab
Flowers need bees to set seed, and bees need flowers for nectar. This two-way dependence is robust until stress pushes it over a tipping point, after which both vanish and relief alone does not bring them back.
Pause Reset Restore (add flowers and bees) Quality: High
Meadow Hive Flower close-up From above
Time speed
Stress
Stress changes take effect immediately on the running meadow. Presets are mild on purpose: each one is just below its tipping point, so a second stress tips it.
Readouts
Populations over the last 4 years Pink: flowering plants per m² (scale 0 to 50). Yellow: pollinators per 100 m² (scale 0 to 150).
Phase plane Yellow curve: bees stay constant. Pink curve: flowers stay constant. Where they cross the system rests: green dot = stable meadow, red dot = the Allee saddle (the edge of the safe basin). White trail: the actual path. Start below and left of the red dot and the meadow dies.
Experiment: ramp the stress up, then back down
Slowly raises one stressor from zero to its maximum (state follows the ODE, 2500 days per step), then lowers it again. Compares the collapse with the analytic tipping point (red dashed line) and shows whether the meadow comes back.
Stressor Pesticide Habitat loss Phenology mismatch Run
Cyan: ramp up. Orange: ramp down. If the orange line stays at zero, the meadow has no memory of its past health.
Independent numerical checks (run in this page)
Legend and controls Presets only change the stress; the populations keep running. Reset restores the healthy state, Restore adds 18 flowers per m² and 45 bees per 100 m² (a re-planting and re-introduction). The status chip says whether a stable meadow exists and whether you are inside its basin. The science
Model. P = flowering plants per m², B = pollinators per 100 m², time in days. A bee visits flowers at the type-II (Holling) rate f(P) = aP/(1+ahP): more flowers help, but handling time h saturates it. Per flower, visits are v = a₂ B/(1+ahP) and the seed-set fraction is s = v/(v+vh ).
dP/dt = P [ ρ s (1 − P/Keff ) − μ ], dB/dt = B [ e f(P) − m − cB ]. Both species need the other: with no pollinators plants cannot recruit, with no flowers pollinators starve.
Stressors. Pesticide raises the pollinator death rate, m = m0 + pesticide. Habitat loss lowers the carrying capacity, Keff = K (1 − loss), and hides the lost strip of flowers. Phenology mismatch (warming moves flowering away from bee emergence) scales both visit rates by (1 − mismatch).
Tipping point. Nullclines are B = (e f(P) − m)/c and the curve where seed-set balances plant mortality. Their intersections are equilibria. At baseline there are three: the empty state (always stable), a saddle at (P, B) about (10, 19), and a stable meadow at about (32, 100). Increasing stress moves the saddle and the stable state together until they annihilate (a saddle-node bifurcation): the meadow vanishes abruptly. Because the empty state stays stable, undoing the stress later does not bring plants and bees back (hysteresis). The Allee saddle is the minimum population that can sustain itself.
Checks. Equilibria are found by root-finding on the scan of P; stability is confirmed independently from Jacobian eigenvalues (finite differences); time integration (RK4, 0.5 d) from nearby starts must return to or leave the equilibrium accordingly; the analytic collapse stress is compared with a slow ODE ramp.
Honest simplifications. Well-mixed meadow with no space, seasons, bee castes, colony cycle, wild and managed bees together as one pool, one plant species, no other food or nesting limits. The parameter values (a, h, e, c, a₂, vh , ρ, μ, m0 ) are round illustrative numbers chosen so the model behaves like a plausible meadow; the stress sizes are not calibrated to a particular pesticide or region. Real declines have many interacting causes. The qualitative lesson (mutualism can collapse abruptly and irreversibly) is well documented in theory and in some field cases; the numbers here are not forecasts. Bees are drawn about 4 times life size. The 3D models are made with Blender.
Independent educational model, not for pest-control, beekeeping or conservation decisions.