## Stationary patterns of strongly coupled prey–predator models.(English)Zbl 1160.35325

The author considers the strongly coupled system $-\text{div} (K(u)\nabla u)= G(u) \quad\text{in }\Omega, \qquad \frac {\partial u}{\partial n}= 0 \quad\text{on }\partial\Omega. \tag{1}$ The author establishes a priori upper and lower bounds for positive solutions of (1), and studies the non-existence of nonconstant positive solutions. Moreover the author considers the bifurcation and the global existence with respect to diffusion terms of nonconstant positive solutions. The analysis of the author shows that the cross-diffusions may be source for the stationary patterns.

### MSC:

 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 92D25 Population dynamics (general)
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### References:

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