Fig 1.
(Color online) schematic illustration of local dynamics.
Corresponding dynamics of the excitable system in uv-plane is depicted by its nullclines. The u nullcline includes three separate lines: u = 0, u = 1 and u = (v + b)/a, while the line u = v is the v nullcline. The dotted line indicates the boundary layer δ which all the initial conditions within it decay to the fixed point. The itinerary in the uv- plane starts haphazardly once the value of u exceeds the amount of δ. Eventually, the itinerary returns to the fixed point after passing a large excursion, adapted from [24] (A). The medium consists of excitable cells spatially connected to each other by a diffusion-like coupling. An exited cell (the blue asterisk unite) is capable of triggering excitation through its neighbors with the help of diffusing cAMP molecules (B).
Fig 2.
Emergent patterns on different lattices including randomly distributed firing cells with variant ϕ1.
ϕ1 is population density of the cells which determines the number density of occupied units on the lattices of size 400 × 400. Here, the cells are exclusively of proactive/cooperative type. All parameters of the model are kept the same for these systems and listed in section S1 Appendix.
Fig 3.
(Color online) Propagating wave through a system with ϕ1 = 0.5 and ϕ2 = 0.0 (A), ϕ1 = 0.5 and ϕ2 = 0.1 (D), ϕ1 = 0.5 and ϕ2 = 0.2 (G), ϕ1 = 0.5 and ϕ2 = 0.3 (J) and ϕ1 = 0.5 and ϕ2 = 0.4 (M). There are 60 iteration interval between the illustrated subsequent patterns, here ‘it’ is an abbreviation for iteration. Emerging spiral waves are distinguishable only at (D) See also S1 to S6 Videos. Mean concentration of signaling agent, calculated as where uij(t) is the cAMP concentration of point (i, j) at time t, in a system with ϕ1 = 0.5 and ϕ2 = 0.0 (B), ϕ1 = 0.5 and ϕ2 = 0.1 (E), ϕ1 = 0.5 and ϕ2 = 0.2 (H), ϕ1 = 0.5 and ϕ2 = 0.3 (K) and ϕ1 = 0.5 and ϕ2 = 0.4 (N). Logarithmic scale of standard deviation of u, calculated as
, for a system with ϕ1 = 0.5 and ϕ2 = 0.0 (C), ϕ1 = 0.5 and ϕ2 = 0.1 (F), ϕ1 = 0.5 and ϕ2 = 0.2 (I), ϕ1 = 0.5 and ϕ2 = 0.3 (L) and ϕ1 = 0.5 and ϕ2 = 0.4 (O). Red and black lines are representatives of growing phase and oscilatory one, respectively. In each plot red line is the linear regression of log(σ(t)) for the first 300 iterations of every simulation. Black line in each diagram is the mean value of log(σ(t)) in the last 3000 iterations in each simulation. The magnified window located in right bottom of panels C, F, I, L and O illustrates the crossover time of the system dynamics between these two regimes. The vertical gray dashed line corresponds to the crossover time in a pure system with ϕ1 = 0.5 and ϕ2 = 0.0 which is the intersection point of black and red lines in panel (C).
Table 1.
Comparison of S values of the media with different proportions of reactive/less-cooperative cells whose surface is covered with randomly distributed firing cells with population density ϕ1 = 0.5.
Fig 4.
(Color online) disposition of pattern snapshots on a common instance of time in lattices of 400 × 400 size, covered with mixed populations with different pairs of ϕ1 and ϕ2.
ϕ1 is the population density of the whole cells on each lattice and ϕ2 is the fraction of reactive/less-cooperative cells among the population. The related row and column of each pattern in the array are appointed by their ϕ1 and ϕ2 indicated in vertical and horizontal margins, respectively. For example, the bottom row includes the systems with the same population density of individuals ϕ1 = 0.8 but with a variant fraction of reactive/less-cooperative ones ϕ2. Similar patterns are enclosed with a colored border. Pure systems are arranged in the first column.
Fig 5.
The square root of u concentration along amid row pixels of the networks during the last 2000 iterations of simulations in lattices of 400 × 400 size, covered with mixed populations with different pairs of ϕ1 and ϕ2.
ϕ1 is the population density of the whole cells on each lattice and ϕ2 is the fraction of reactive/less-cooperative cells among the population. The related row and column of each pattern in the array are appointed by their ϕ1 and ϕ2 indicated in vertical and horizontal margins, respectively.