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In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility functionf(a) and the maturation function$$\beta (a)$$$\beta \left(a\right)$are assumed to be piecewise functions related to their maturation period$$\tau $$$\tau$. Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period$$\tau $$$\tau$as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.

Arid ecosystems are known to be sensitive to climate change. The Jornada Basin in the USA, as one representative of arid land, has suffered from land degradation in recent decades. In order to disentangle the climate–vegetation feedback, we analyzed the vegetation dynamics under the effects of climate change via a mathematical model based on the reaction–diffusion mechanism. Using this model, we conducted a sensitive analysis of climate factors and concluded that the ecosystem might experience a catastrophic shift with the climatic deterioration. We considered the non-local interaction term to explain the competition among plants. Additionally, the PLR (power law range) metric was used to quantify the extent of the degradation and to compare the results of the vegetation patterns from the remote sensing data and the simulations. From the results, this model could simulate the trends of land degradation in this area. We found that the land degradation could be mainly attributed to climate changes in recent years. This approach suggests that vegetation patterns can provide hints as to whether the ecosystem is approaching desertification. These results can help with mapping vulnerable arid areas around the world through model simulation and satellite images.