**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor Wei-Ming Ni, School of Mathematics,
University of Minnesota

*Diffusion, Cross-Diffusion and their Spike-Layer Steady States*

Although, for single equations, diffusion can be viewed as a
smoothing and trivializing process, the situation becomes
drastically different when we come to systems of diffusion
equations. For example, in a system of equations modeling
two interactive substances, *different diffusion rates*
could lead to nonhomogeneous distribution of such reactants.
Using an activator-inhibitor system, with slowly diffusing
activator and rapidly diffusing inhibitor, we will briefly
introduce the current mathematical research on those highly
concentrated solutions (i.e. solutions whose graphs display
narrow peaks or spikes, also known as *point-condensation
solutions* or, *spike-layers*). In this example,
not only is gap between the diffusion rates important, the
reaction terms are also essential in producing the patterns.
In contrast to this example, we are going to discuss the weak
competition case in the classical competition-diffusion
Lotka-Voterra system which does not have any nontrivial
solutions no matter what the diffusion rates are. However,
in modeling two competing species in population dynamics,
since individuals are *not* moving around just
randomly, one must take into account the population pressure
created by each of the competitors. Thus we are lead to
"cross-diffusion" systems. It is interesting to note that
spike-layers also appear in cross-diffusion systems. In
addition to discussing spike-layers, we shall focus on the
effect of cross-diffusion versus that of diffusion in those
systems.

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