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7 Reaction-Diffusion Equations – Homogeneous and Heterogeneous Environments
117

The coefficients d 1 and d 2 determine the strength of diffusion of prey and
predators, resp. Obviously, the environment dependence can be accounted for
by making the coefficients x-dependent in an appropriate way.
Themathematicalliteratureonreaction-diffusionequationsisvast.Asastan-
dard text we reference the textbook [9].
Typical mathematical questions in the theory of reaction-diffusion equations
deal with existence of solutions, global boundedness of solutions by means
of maximum and invariant region methods, large-time asymptotics, travelling
waves and geometry and topology of attracting sets, singular limits etc.
Themaybemostbasicmathematicalquestionreferstothestabilityproperties
of the reaction-diffusion system under consideration. For this, assume that, for
simplicity’s sake, D is independent of u and t, F independent of t and that
u 0 = u 0 (x) is a stationary state of the reaction-diffusion system, i.e.

0 = div (D(x)grad u 0 )+ F(x, u 0 ).

An important question concerns the behaviour of solutions of the nonlinear
system in comparism with the solutions of the linearized system, when the
linearization is performed at the stationary state u 0 , in direction of a function v:
v t = div (D(x)grad v)+ D u F(x, u 0 )v .

Clearly, this is still a difficult problem in full generality (and also for many
particularly interesting cases). Thus, it seems intruiging to neglect diffusion
and to analyse the linear ODE system instead

w t = D u F(x, u 0 )w .
At least in the homogeneous case, where D, F and consequently u 0 are inde-
pendent of x, it suffices to calculate the eigenvalues of DF(u 0 )todecide about
linearized, diffusionless stability. If all eigenvalues have negative real parts, then
only exponentially decaying modes of w exist, eigenvalues with positive real
parts generate exponential instabilities and more information is required if
eigenvalues with zero real part occur. In the first two cases the linearized be-
haviour carries over to the diffusionless nonlinear ODE system locally around
stationary points.
For example, take the predator-prey model formulated above. Then, neglect-
ing diffusion, a simple calculation shows the existence of two stationary states,
namely (0, 0) corresponding to extinction of both species and a state (u 0 , v 0 )with
ae + db
u 0 =
ef + bc
ac − df
v 0 = .
ef + bc
Note that the predator-equilibrium value becomes negative unless ac − df > 0.