Species Interaction Phase Portrait

Interaction Type

B→A: 0 A→B: 0

Parameters

Interaction strength 0.60

Simulation speed 1.0×

Live Readouts

Population A 🟢

20.0

Population B 🟠

15.0

Elapsed generations

0.0

Interior equilibrium (A*, B*)

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Linear stability at A*, B*

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How it works

This is the phase-portrait companion to the 3D swarm version: instead of watching two crowds of dots, you watch the two population sizes as a single point moving through (A, B) state space, driven by the same generalized Lotka–Volterra equation:

dA/dt = A·(rA − sA·A + iAB·B)
dB/dt = B·(rB − sB·B + iBA·A)

The grey arrows are the vector field — at every point in the plane they show the instantaneous direction (dA/dt, dB/dt) a population at that point would move next. The dashed teal curve is the A-nullcline (dA/dt = 0), the dashed amber curve is the B-nullcline (dB/dt = 0); wherever they cross away from the axes is an interior equilibrium, marked with a white diamond.

Type	B → A	A → B
Mutualism	+	+
Competition	−	−
Predation	−	+
Parasitism	− (weak)	+ (weak)
Commensalism	0	+
Amensalism	−	0
Neutralism	0	0

Linear stability is computed from the Jacobian at the interior equilibrium: eigenvalue signs classify it as a stable node/spiral, an unstable node/spiral, or a saddle. With this model's parameters (self-limitation sA = sB = rA/K is small next to typical interaction strengths), competition and predation usually turn out to be a saddle — the interior point exists on paper but is unstable, so the trajectory slides past it toward a boundary where one species is driven to extinction (competitive exclusion / prey extinction). That is the textbook outcome for strong asymmetric interaction, not a rendering artifact.
The bottom strip is the same two populations plotted as ordinary time series A(t), B(t), so you can read off oscillation, monotonic decline, or blow-up directly against elapsed time.
Integration uses a semi-implicit (symplectic) Euler step: A is advanced first, then B is advanced using the already-updated A, 10 substeps per frame — this is genuinely semi-implicit, matching what the equations call for near stiff regions.
Interaction strength scales |iAB| and |iBA|; higher values push the pair further from independent logistic growth and further from the tiny self-limitation term, which is exactly what tips the interior equilibrium from stable to a saddle.

Population A Population B A-nullcline (dA/dt=0) B-nullcline (dB/dt=0)