Turing patterns and long-time behavior in a three-species food-chain model

- The food chain model is found to exhibit diffusion induced chaos.
- Various Turing patterns are observed in 1d and 2d.
- Existence of finite dimensional global attractor in L2(Ω) is shown.
- Lower bound of attractor dimension is estimated numerically via time series analysis.
- The system dynamics strongly depends on the top predator.

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator-prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L2(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.