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As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1 $ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

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