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In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem. [ABSTRACT FROM AUTHOR]

We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation −u′1+u′2′=λu1+up,−L0L,p\gt 0 are two evolution parameters. We prove that on the (λ,‖u‖∞)\left(\lambda ,\Vert u{\Vert }_{\infty })-plane, for 01p\gt 1, the bifurcation curve is reversed ε\varepsilon -like shaped bifurcation if L>L∗L\gt {L}^{\ast }, and is exactly decreasing for λ>λ∗\lambda \gt {\lambda }^{\ast } if 0

A knowledge graph is a repository that represents a vast amount of information in the form of triplets. In the training process of completing the knowledge graph, the knowledge graph only contains positive examples, which makes reliable link prediction difficult, especially in the setting of complex relations. At the same time, current techniques that rely on distance models encapsulate entities within Euclidean space, limiting their ability to depict nuanced relationships and failing to capture their semantic importance. This research offers a unique strategy based on Gibbs sampling and connection embedding to improve the model's competency in handling link prediction within complex relationships. Gibbs sampling is initially used to obtain high-quality negative samples. Following that, the triplet entities are mapped onto a hyperplane defined by the connection. This procedure produces complicated relationship embeddings loaded with semantic information. Through metric learning, this process produces complex relationship embeddings imbued with semantic meaning. Finally, the method's effectiveness is demonstrated on three link prediction benchmark datasets FB15k-237, WN11RR and FB15k. [ABSTRACT FROM AUTHOR]

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically. [ABSTRACT FROM AUTHOR]