Emergence of target waves in paced populations of cyclically competing species

Abstract

We investigate the emergence of target waves in a cyclic predator-prey model incorporating a periodic current of the three competing species in a small area situated at the center of the square lattice. The periodic current acts as a pacemaker, trying to impose its rhythm on the overall spatiotemporal evolution of the three species. We show that the pacemaker is able to nucleate target waves that eventually spread across the whole population, whereby three routes leading to this phenomenon can be distinguished depending on the mobility of the three species and the oscillation period of the localized current. First, target waves can emerge due to the synchronization between the periodic current and oscillations of the density of the three species on the spatial grid. The second route is similar to the first, the difference being that the synchronization sets in only intermittently. Finally, the third route towards target waves is realized when the frequency of the pacemaker is much higher than that characterizing the oscillations of the overall density of the three species. By considering mobility and the frequency of the current as variable parameters, we thus provide insights into the mechanisms of pattern formation resulting from the interplay between local and global dynamics in systems governed by cyclically competing species.

Adjustment of spiral drift by a travelling wave perturbation

“…in the figure the angle of drift direction changes π/3 counterclockwise each time that the propagation direction of the travelling wave increases π/3. This linear dependence confirms our previous rule in (6). There needs to be annotated in Fig.…”

“…3a. Those simulation data are located very close to the fitted straight line of value β = 0.7π , instead of π/2 as predicted by the (6). This error of numerical result different from the prediction is probably caused by using the approximated kinematical equations (2) which leads to a neglect of calculations of spiral rotating phase around the core in the tip's displacement.…”

“…ums [4,5], populations and species [6,7] to biological tissues and neuronal networks [8,9]. The interest of those patterns has broadened over the last decades due to applications in heart diseases [10] and epileptic seizures [11].…”

Social imitation versus strategic choice, or consensus versus cooperation, in the networked Prisoner's Dilemma

“…They showed that if the PD is played on * daniele.vilone@gmail.com † jramasco@ifisc.uib-csic.es ‡ anxo@math.uc3m.es § maxi@ifisc.uib-csic.es a lattice, frozen configurations can appear, where cooperators have survived and even overwhelmed defectors. Subsequent research established that the effect of the topology is not universal [13] (see, also, [14] for a recent review) and that the enhancement of the cooperative strategies depends on the network properties, the dynamical update rule, the exact entries of the payoff matrix, the size of the system, and the time structure [15][16][17][18][19][20][21]. The theoretical discussion was finally settled by several experiments [22][23][24][25][26], which established that networks did not exhibit a cooperation level significantly higher than a well-mixed population.…”

Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response

“…In recent years, several theoretical aspects of multi-species cyclical dominance have already received much attention. Examples include effects of competition on pattern formation [13], the relevance of noise and pattern formation [22,23], emergence of target waves [14].…”