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Coexistence for species sharing a predator. (English) Zbl 1054.34079

A class of equations describing the dynamics of two preys sharing a common predator is considered. Even though the boundary and internal dynamics can exhibit oscillatory behavior, it is shown that these equations are permanent if and only if they admit a positive equilibrium. Going beyond permanence, a subclass of equations are constructed that are almost surely permanent but not permanent; there exists an attractor in the positive orthant that attracts Lebesgue almost every (but not every) initial condition.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34D45 Attractors of solutions to ordinary differential equations
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