About deterministic extinction in ratio-dependent predator-prey models |
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Authors: | Christian Jost Ovide Arino Roger Arditi |
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Institution: | (1) Ecologie des populations et communautés, 2154, Université Paris-Sud XI, Bat. 362, 91405 Orsay cedex, France;(2) Institut national agronomique, Paris-Grignon, 75231 Paris cedex 05, France;(3) Mathématiques appliquées, Université de Pau et des Pays de l’Adour, 64000 Pau, France |
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Abstract: | Ratio-dependent predator-prey models set up a challenging issue regarding their dynamics near the origin. This is due to the
fact that such models are undefined at (0, 0). We study the analytical behavior at (0, 0) for a common ratio-dependent model
and demonstrate that this equilibrium can be either a saddle point or an attractor for certain trajectories. This fact has
important implications concerning the global behavior of the model, for example regarding the existence of stable limit cycles.
Then, we prove formally, for a general class of ratio-dependent models, that (0, 0) has its own basin of attraction in phase
space, even when there exists a non-trivial stable or unstable equilibrium. Therefore, these models have no pathological dynamics
on the axes and at the origin, contrary to what has been stated by some authors. Finally, we relate these findings to some
published empirical results. |
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