Asymptotic behavior of a nonhomogeneous predator‐prey system with one diffusing and one sedentary species

摘要

AbstractIn general the predator‐prey system studied will have many equilibrium states with different “patches”, i.e. regions where the sedentary species vanishes. If one equilibrium is given, a Ljapunov functional can be constructed for a set of initial data with patches satisfying a certain compatibility condition. By the methods of Alikakos 1 and Rothe 28this implies uniform bounds for the diffusing species in a Hölder space and the sedentary species in aLp‐space. Considering pieces of the trajectory for time in some bounded interval and using the weak*‐topology for the sedentary species, one defines an ω‐limit set which is compact and nonvoid. Using the Ljapunov functional an asymptotic version of the differential equations is derived. It turns out that they admit only constant solutions except for a very special case. Thus convergence to the equili