In this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

中文翻译：

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密度对具有强弱Allee效应的种群动态的影响

在本文中，我们考虑了种群动态中的反应扩散模型，并研究了异种封闭区域中不同类型的Allee效应对逻辑增长的影响。对于强力Allee效应，通常情况是，根据资源和稀疏功能，该效应较弱时，物种会无条件地灭绝，并且会出现灭绝-生存的情况。特别是，当稀疏性为正数或负数时，我们研究了乘法Allee效应对经典扩散的影响。负稀疏意味着较弱的Allee效应，种群在某些区域中生存，而在其他方面则发散。积极的稀疏性具有很强的Allee效应，并且人口无条件灭绝。提出了Allee效应对正稳态的存在和持久性的影响，以及全局分叉图。子超解的方法用于分析方程。给出了稳定性条件和正解的区域（可能存在多个解）。当不存在扩散时，我们考虑有收获和没有收获的模型，它们是初始值问题（IVP），并研究局部稳定性分析和当前的分叉分析。我们提供了许多数值示例来验证分析结果。它们是初始值问题（IVP），并研究局部稳定性分析和当前的分叉分析。我们提供了许多数值示例来验证分析结果。它们是初始值问题（IVP），并研究局部稳定性分析和当前的分叉分析。我们提供了许多数值示例来验证分析结果。