Clustering Determines Who Survives for Competing Brownian and Lévy Walkers

The competition between two ecologically similar species that use the same resources and differ from each other only in the type of spatial motion they undergo is studied. The latter is assumed to be described either by Brownian motion or Lévy flights. Competition is taken into account by assuming that individuals reproduce in a density-dependent fashion. It is observed that no influence of the type of motion occurs when the two species are in a well-mixed unstructured state. However, as soon as the species develop spatial clustering, the one forming more concentrated clusters gets a competitive advantage and eliminates the other. A similar competitive advantage would occur between walkers of the same type but with different diffusivities if this leads also to different clustering. The coexistence of both species is also possible under certain conditions.

Figure 1

Spatial configurations of Lévy (blue) and Brownian (magenta) organisms at long times: (a) coexistence without clustering at time $t=1500$, $\kappa =1$, and ${\kappa }_{\mu }=1$ [cf. Fig. 2a for the population sizes]; (b) the Lévy bugs with ${\kappa }_{\mu }=4\times {10}^{-4}$ have extinguished the Brownian ones with $\kappa ={10}^{-5}$; (c) the Brownian bugs with $\kappa =8\times {10}^{-6}$ have extinguished the Lévy ones with ${\kappa }_{\mu }=4\times {10}^{-2}$; (d) coexistence with clustering at time $t=173000$, $\kappa =6\times {10}^{-6}$, and ${\kappa }_{\mu }=4\times {10}^{-3}$ [cf. Fig. 2b for the population sizes].

Figure 2

Time evolution of the population sizes of the Lévy and Brownian walkers $N_B$ and $N_L$ in the two possible cases of coexistence: (a) large diffusion coefficients when no clustering occurs ${\kappa }_{\mu }=1$ and $\kappa =1$ [the same system as in Fig. 1a]; (b) small diffusion coefficients leading to the clustering ${\kappa }_{\mu }=4\times {10}^{-3}$ and $\kappa =6\times {10}^{-6}$ [the same system as in Fig. 1d].

Figure 3

Competition between Brownian and Lévy walkers. Depending on the values of $\kappa$ and ${\kappa }_{\mu }$ either Brownian or Lévy walkers win, or coexistence occurs. Each point reflects the outcome of 25 realizations. The solid line presents the separation line $\kappa =0.0217{\kappa }_{\mu }$ provided by the mean-field description.