Chaotic Population Dynamics

Chaos in Tritrophic Food Webs

Below are videos exploring chaotic dynamics in the Tritrophic Rosenzweig-Macarthur model created by a group of Masters students at Oxford.

The video on the left demonstrates sensitivity to initial data for a chaotic attractor coming from this tritrophic model at a particular parameter point. Two initial points (initial population densities) were selected with a difference of approximately 10⁻¹⁰, and were evolved in time for a long period of time (this video has cut this long period out). Separation of these trajectories occurs on the timescale correlated to the maximal Lyapunov exponent of the orbit.

The video on the right demonstrates period doubling of the attractor. Trajectories were computed for a very long time (and truncated to find the attractor) for a range of a specific parameter, with the initial trajectory being a limit cycle (periodic orbit). This orbit folds in on itself as the parameter is increased, and this folding eventually cascades into the formation of a chaotic attractor. Within the region of parameter space, small "islands of stability" can be observed wherein periodic orbits or quasiperiodic attractors can be seen. Note that the colour corresponds to an arbitrary time to demonstrate movement on the attractor, but the initial point at time zero is on the attractor.

Wave Initiation of Spatiotemporal Chaos in a Rosenzweig-MacArthur model of predator-prey dynamics