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4.3- Spatial TopologyFig. 4d shows the populations of A and B for a typical simulation run on a 100-by-100 square lattice (wrapped around in both dimensions to form a torus). Although the birth and death rates are identical to those used in the other three topologies presented in Fig. 4, the population dynamics are remarkably different. As might be expected from our experience of the previous section, the initial growth in the populations of A and B is quadratic rather than exponential. However, a completely unexpected difference is also strongly evident -- the populations exhibit large undamped oscillations centered about markedly lower equilibria. Simulations indicate that, after many oscillations, the fractional populations of A and B usually settle to values substantially less than half of the homogeneous-mixing equilibria. About 20% of the time, the oscillations persist beyond t=1000. By viewing a graphical display of the simulation as it is running, it is possible to get some insight into the nature of the oscillations. Initially, A expands in a roughly circular pattern (as in the simple model of the previous section). Soon, B follows the trail blazed by A, resulting in a fairly solid core of Bs surrounded by a ring of As. After a while, the Bs in the core near the site of the cell initially occupied with A start to die, having exhausted their supply of As. This situation is illustrated in Fig. 5a, which shows a snapshot of the system at time t=18.1 for the simulation run of Fig. 4b. The first peaks in the populations of A and B in Fig. 4b occur roughly when the expanding circle has fully wrapped around and filled the space. A is now hemmed in on all sides by B, which quickly consumes A, leaving just a few small pockets of surviving As. The safest haven for A is near the originally-occupied cell, where most of the B have already died. A fringe of the original population explosion wraps around and ekes its way through a phalanx of Bs into this central area, giving birth to a new expanding circle which emerges from it, and a second population cycle begins. Fig. 5b displays the situation at t=47.4, when the populations of A and B are headed towards their second peaks. Note the similarity to Fig. 5a -- the confinement of A to the edges of the expanding region and the empty inner core, which will eventually serve as a breeding ground for A and the third population cycle. Note also in Fig. 5b the shards of B left over from the first population explosion.
Figure 5: Snapshots of spatial distribution of A (black cells) and B (gray cells) at a) t=18.1 and b) t=47.4, taken from the simulation run portrayed in Fig. 4b. 
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