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4.2- Random GraphFig. 4b shows the populations of A and B on a 10000-node random graph in which a typical node has 8 neighbors. The birth and death rates are the same as in the homogeneous case depicted in Fig. 4a. The dynamics are substantially similar to the deterministic solution for the homogeneous mixing model displayed in Fig. 4a. Note that even the amplitudes of the decaying predator-prey oscillations are quite similar. Furthermore, an average over 100 simulation runs shows that the probability that A and B will survive for a long time (beyond t=1000) is approximately 80%, as in the homogeneous case.
Figure 4: The effect of various topologies on the population dynamics of species A and B: a) homogeneous mixing model, b) 10000-node random graph with d=8.0, c) 10000-node random graph with d=2.0, and d) 100-by-100 square lattice wrapped around to form a torus. The birth and death rates are  ,
 ,
 , and
 in all cases. The homogeneous mixing
curves are numerical solutions of a coupled pair of differential equations in which the
initial fractional populations of A and B were 0.0001 and 0.0, respectively.
The 2-D square lattice and the random graph curves were obtained from typical simulation runs
in which initially just one cell (out of 10000 total) was occupied with A and none with B.
(Recall that a B is born whenever an A dies.)
In Fig. 4c, the only parameter which has been changed from those of
Fig. 4c is the average number of neighbors,
which has been decreased from
8 to 2 
|
, and becomes extinct near
t=89.1. In all 200 simulation runs that were conducted under these conditions, A and B never
came close to surviving up to the time limit t=1000. Thus it appears that the sparsity of the
graph has plunged the system below the epidemic threshold.
More extensive simulation studies show that, for values of d
intermediate between 2 and 8, the survival probability and the stable equilibria for surviving
populations increase monotonically with d, rising very dramatically in the range 3 < d < 5.