Page 122 -

7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
116

Fig. 7.4. Fields in Guanxi Province, China

species of predators with concentration v (here u and v are scalar variables, note
the change of notation!), their interaction dynamics is modeled by:
u t = d 1Δu + au − buv − fu 2
2
v t = d 2Δv − dv + cuv − ev ,
where a, b, f , d, c, e, d 1 and d 2 are positive constants or parameter functions.
These reaction parameters can easily be interpreted. In the absence of predators
and without diffusion, the prey species satisfies a logistic equation, predict-
a
ing saturation at the value , occuring with exponential speed at the rate a.
f
Moreover, in the absence of prey and without diffusion, the predators die
out exponentially, with exponential rate d. We remark that the parameters
f and e account for the strength of intra-species friction among prey and
predators resp. This friction, caused for example by competition for nutri-
ents, stabilizes the prey population even without predators. If both species
are present initially, then their respective rates of change are supposed to
be influenced by the number of their encounters, typically ending badly for
the prey, i.e. b> 0, and good for the predator, i.e. c> 0. Note that due to
the quadratic nature of the Lotka–Volterra system, only two-body interactions
prey-prey, predator-predator and predator-prey are taken into account by this
model.