4.1- Homogeneous MixingAs was the case for the one-species model, placement of the two-species model just described on a fully-connected graph effectively eliminates topology from the problem. Deterministic analysis (valid for sufficiently large systems) leads to the following coupled pair of nonlinear differential equations:
Model 2 can be solved numerically to yield a(t) and b(t), the fractional populations of A and B, respectively.
Analysis of the solution shows that the epidemic threshold for A
is unaffected by B's birth
and death rates; it remains at
Figure 4a illustrates the population dynamics of A and B for a
particular set of birth and death rates chosen such that A is above the epidemic threshold:
|
.
B has no intrinsic epidemic threshold; it
can survive as long as there are As upon which to feed,
regardless of the relative values
of B's birth and death rates.
,
,
, and
. Thus
,
but the life cycle of B is ten times that of A.
When the initial populations
are very small, they increase exponentially at first.
At intermediate times, there may be some damped
predator-prey oscillations -- the existence, frequency, and decay rate of which depend upon the
birth and death rates.
Eventually, the populations settle to stable
values determined by the birth and death rates.
The presence of B can strongly suppress the
equilibrium population of A. For the set of
parameters illustrated in Fig.