Lotka–Volterra Attractor

Attractor Builder (Blender add-on)
Equations:
ẋ = x - x*y + c*x**2 - a*z*x**2
ẏ = -y + x*y
ż = -b*z + a*z*x**2
Parameters:
| a = 2.9851 | b = 3.0 | c = 2.1 |
Simulation settings:
Initial state: x₀ = 0.9, y₀ = 0.9, z₀ = 0.5 
Method: RK4  
Time Step: 0.01 
Steps: 15000
Burn-in: 50   
Scale: 10

The classical Lotka–Volterra model (1925–1926) describes the cyclic dynamics of a predator–prey system in two dimensions. The trajectories of this system are closed orbits around a fixed point, representing stable and periodic oscillations of population sizes. In 1988, Nikola Samardzija and Lawrence D. Greller proposed a three-dimensional generalization of this model, extending the classical formulation by introducing an additional species and a nonlinear coupling between variables. In this new version, the system generates complex oscillations, bifurcations, and transitions to chaos. The original system of equations (1988, p. 466) is given as:

\(\dot{X} = X - XY + CX^{2} - AZX^{2}\)

\(\dot{Y} = -Y + XY\)

\(\dot{Z} = -BZ + AZX^{2}\)

The parameters A, B, and C are positive (A, B, C > 0). They control the strength of interactions between species: A defines the nonlinear coupling between predators, B describes the damping rate of the Z population, and C represents the self-reinforcing growth of X. The authors demonstrated that this system can produce chaotic trajectories and a fractal torus, providing an example of an “explosive route to chaos.”

Sources:
Lotka, A. J. (1925). Elements of Physical Biology. Williams & Wilkins, Baltimore.
Samardzija, N., & Greller, L. D. (1988). Explosive route to chaos through a fractal torus in a generalized Lotka–Volterra model. Bulletin of Mathematical Biology, 50(5), 465–491. DOI: 10.1016/S0092-8240(88)80003-X
Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie della Reale Accademia Nazionale dei Lincei, 2(6), 31–113.