Figure 1.
(a) Spatial distribution of species B (green) and C (red) are shown on a snapshot taken after 24 periods T of the flow. They both cover a fractal curve and are present at all times in the wake of the cylinder. The initial position of species B and C is a rectangle of linear size 0.07 × 0.76 centered around x = −1.36, x = −1.10, and y = 0, respectively. (The length is measured in units of cylinder radius R.) (Inset) The percentage nB (nC) of species B (C) present in the wake is shown by green (red) line as a function of time, measured in units of T. Note the steady time-periodic behavior reached after about 40 time units. The simulation was performed on a rectangular grid of size ɛ = 2 × 10−3 times the cylinder radius. Species B and C are passively advected during time lags τB = 0.6T, and τC = 0.8T, respectively, then they reproduce instantaneously, with new individuals being “born” within a distance σB = σC = 1/500 of the “parent” if there is resource A available there [γB ∼ σB/τB = 1/(300T), γC ∼ σC/τC = 1/(400T)]. Additionally, at each time lag τB (τC) B (C) individuals die with a probability τBδB (τCδC) [δB = δC = 1/(10T)]. In our model σB (σC) also plays the role of a diffusive length scale. There is efficient diffusive mixing within this distance, and there is thus no need to introduce an additional stochastic noise to mimick diffusion (33). (b) A high-resolution image of the small rectangular region from a indicates self-similarity. The grid size is ɛ = 8 × 10−4. The total area covered by B and C in the wake follows a fractal scaling with dimension D ≈ 1.6, the same as in the autocatalytic chemical model (23). Note that the increase of the resolution does not alter the coexistence, only reveals more details of the fractal structure.