In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.

在这项调查中，我们提供并检查了一个与物种和Beddington-DeAngelis功能反应均成比例的具有避难所的捕食者与食饵相互作用的模型。我们首先证明时间和时空模型的正定性，它们被限制在正不变区域中。然后，对于时间模型，我们分析了其时间动力学，包括所有可行的非负平衡的一致有界性，持久性和稳定性，并表明避难所可以通过Hopf分叉在唯一的正平衡周围引起周期性振荡。对于时空模型，我们不仅研究了它的持久性，非负恒定稳态的稳定性和图灵不稳定性，而且还利用Leray-Schauder度理论研究了非恒定正稳态的存在与不存在。关键的观察结果是，庇护系数在修改当前系统的动力学方面起着重要的作用，并介导了种群的持久性，并存平衡的稳定性，甚至是图灵不稳定性参数空间。最后，给出了一般的数值模拟结果，以说明理论结果的有效性。通过数值模拟，可以观察到模型动力学显示了猎物避难所和自扩散控制时空模式增长到斑点，条斑混合和条带繁殖。结果表明，带有避难所的模型的动力学不是简单的，而是丰富而复杂的。此外，数值模拟表明，其他模型参数对物种的空间非均匀分布也有重要影响，这导致形成斑点图案，斑点和条纹图案的混合，斑点，条纹和环图案的混合以及反斑点图案。这可能会改善反应扩散-捕食者-猎物系统上的避难所模型动力学。