General Principle

In the Master thesis “Simulation of Biological Two- and Three-Species Systems Using Cellular Automata” different two- and three-species models were developed using cellular automata. The main principle of the models is that individuals spend energy to live and to reproduce and gain energy when they eat. This is  represented by an energy variable called “food reserve” which is increased after eating and decreased each time step and during reproduction. A food reserve equal to zero implies that the individual dies. In order not to die out completely, individuals reproduce depending on (i) the value of their specific food reserve and (ii) on a certain reproduction probability.

Figure 1.1

  • (a) $r=(0.06,0.06)$
  • (b) $r=(0.9,0.9)$
  • (b) $r=(0.06,0.9)$
  • (d) $r=(0.9,0.06)$

Spatial distribution of predators and prey in dynamical equilibrium for different reproduction probabilities $r=(r_{Predator},r_{Prey})$. Predators are blue, prey are yellow, predators and prey sharing a site are red and empty sites are white. The pictures were taken right before the eating phase. Grid size: $N^2=1024^2$ .

Main Results

In Figure 1.1 the spatial distribution of predators and prey is shown for different reproduction probabilities $r = (r_{Predator}, r_{Prey})$. One of the main results that can be deduced from these figures is that the predator density is small for small prey reproduction probabilities and large for high prey reproduction probabilities. Their density only depends to a small degree on $r_{Predator}$ . However, the spatial distribution of predators and prey and the density of prey depend on the reproduction probabilities of both species.

The density ratio of prey to predators (see Figure 1.2) derived for different combinations of reproduction probabilities shows that for an increase in $r_{Prey}$, the amount of prey available per predator decreases. The decrease is fast for $r_{Prey} \le 0.2$ and then levels off. Furthermore, the density ratio is nearly independent of $r_{Predator}$.

Figures 1.1(a) and 1.1(d) show that there are two underlying processes which cause the large density ratios for small $r_{Prey}$: If the reproduction probability of predators is also small, the slow growth of the predator population implies that predators cannot eat up a prey cluster very fast. In the case of large values of $r_{Predator}$ , predators form defined fronts at the edge of large prey clusters. This can be explained in the following way: A predator that eats a prey fills up its food reserve and therefore reproduces with a high probability. For this reason, the local density of predators increases rapidly in the vicinity of a prey cluster. Thus, small prey clusters are eaten up very fast. Afterwards, the chance of predators to die from starvation is high since the distance to the next prey cluster is too large to be reached. As a consequence, predators survive with a higher probability at the edge of large prey clusters. In the case of $r_{Prey} \ge 0.2$ (see Figures 1.1(b) and 1.1(c)), the density ratio stays nearly constant. This is due to the comparatively large amount of newborn prey which correspond to a large amount of food predators find. Hence, less predators die from starvation and they can accumulate at the edge of a prey cluster. This leads to a small ratio of prey to predators. The slight decrease of the density ratio for increasing $r_{Predator}$ results from the fact that predators eat up prey faster. Thus, the prey clusters become smaller and the density ratio decreases.

Figure 1.2

  • Density ratios depending on different reproduction probabilities of prey (x-axis) and predators legend). The mean values and corresponding standard deviations were calculated from 3000 time steps in dynamical equilibrium.