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Keywords = Poincaré–Hopf theorem

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20 pages, 2678 KiB  
Article
The Effects of the Weak Allee Effect and Disease on the Dynamics of a Predator–Prey System: Stability and Bifurcation Properties
by Yurong Dong, Hua Liu, Jianhua Ye, Gang Ma and Yumei Wei
Axioms 2025, 14(7), 531; https://doi.org/10.3390/axioms14070531 - 12 Jul 2025
Viewed by 212
Abstract
In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation [...] Read more.
In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation are proven using Sotomayor’s theorem and Poincare–Andronov–Hopf theorems. In addition, the correctness of the theoretical analysis is verified by numerical simulation. The numerical simulation results show that the eco-epidemiological model with a weak Allee effect has complex dynamics. If the prey population is not affected by disease, the predator becomes extinct due to a lack of food. Under low infection rates, all populations are maintained in a coexistent state. The Allee effect does not influence this coexistence. At high infection rates, if the prey population is not affected by the Allee effect, the infected prey is found to coexist in an oscillatory state. The predator population and the susceptible prey population will be extinct. If the prey population is affected by the Allee effect, all species will be extinct. Full article
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16 pages, 3792 KiB  
Article
A Study of Differential Topology on the Magnetically Induced Isotropically Averaged Lorentz Force Density of a Few Simple Molecules
by Michele Orza, Francesco F. Summa, Riccardo Zanasi and Guglielmo Monaco
Molecules 2024, 29(18), 4502; https://doi.org/10.3390/molecules29184502 - 23 Sep 2024
Viewed by 860
Abstract
Quantum chemical topology addresses the study of the chemical structure by applying the tools of differential topology to scalar and vector fields obtained by quantum mechanics. Here, the magnetically induced isotropically averaged Lorentz force density was computed and topologically analyzed for 11 small [...] Read more.
Quantum chemical topology addresses the study of the chemical structure by applying the tools of differential topology to scalar and vector fields obtained by quantum mechanics. Here, the magnetically induced isotropically averaged Lorentz force density was computed and topologically analyzed for 11 small molecules. Critical points (attractors, repellers, and saddles) were determined and trajectories connecting the attractors computed. It is shown that kinds and numbers of the critical points are to some extent transferable in similar molecules. CC bonds of different orders are endowed with critical points of different kinds close to their center. The sum of topological indices of the isolated critical points is influenced by the presence of repellers on the outer part of the molecules. Full article
(This article belongs to the Special Issue Feature Papers in Computational and Theoretical Chemistry)
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26 pages, 9560 KiB  
Article
Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest
by Zuxiong Li, Shengnan Fu, Huili Xiang and Hailing Wang
Mathematics 2021, 9(20), 2560; https://doi.org/10.3390/math9202560 - 13 Oct 2021
Cited by 2 | Viewed by 1890
Abstract
In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability [...] Read more.
In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper. Full article
(This article belongs to the Section C1: Difference and Differential Equations)
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