Predator–Prey Models: A Review of Some Recent Advances

In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

Assessment of the Local and Global Stability of the Luzzone Arch Dam Including Visualisation of the Data Analysis

This study investigates the local and global stability of the Luzzone Dam. Two finite element models were built; one with foundation rock, the other without. The purpose of this was to demonstrate a potential gulf between rigid connection modelling, and rock–structure interaction (RSI). Strand7 is not a traditional geotechnical finite element model (FEM) program, though performed well when modelling radial displacement on the Luzzone Dam. Generally, the percentage between a rigid base and RSI model displacement was 10%. This result was validated against previous numerical models on the structure. Static loads produced a radial displacement on the crown structure of 9.01 cm. Uneven stress distributions at the base of the structure were shown to be the most unpredictable result. With rigid base connections, these loads produced peak tensile stresses of 10.7 MPa. This was greater than its dynamic counterpart, asking questions about fully fixed restraints. It is noted that this is above yield and should be investigated further. Special attention will be devoted to determining the failure criteria in the simulated dams to suggest better practical guide lines for the practical engineers on site.

Stability analysis of five-grade Leishmania epidemic model with harmonic mean-type incidence rate

In this paper, we discuss the Anthroponotic Cutaneous Leishmania transmission. In addition, we develop a mathematical model for the Anthroponotic Cutaneous Leishmania transmission and consider its qualitative behavior. We derive the threshold number $R_{0}$ R 0 of the model using the next generation method. In the disease-free case, we carry out the local and global stability under the condition $R_{0}<1$ R 0 < 1 . Moreover, we derive the global stability at the disease-free equilibrium point by utilizing the Castillo-Chavez method. On the other hand, at the endemic equilibrium point, we show the local and global stability to be held under specific conditions and $R_{0}>1$ R 0 > 1 . We also establish the global stability at the endemic equilibrium point with the help of a geometrical approach, which is a generalization of Lyapunov theory, by using a second additive compound matrix. Finally, we take into account the sensitivity analysis of the threshold number with other parameters. We also discuss several graphs of important parameters.

Dynamics of a Predator–Prey Model with the Effect of Oscillation of Immigration of the Prey

In this article, the use of predator-dependent functional and numerical responses is proposed to form an autonomous predator–prey model. The dynamic behaviors of this model were analytically studied. The boundedness of the proposed model was proven; then, the Kolmogorov analysis was used for validating and identifying the coexistence and extinction conditions of the model. In addition, the local and global stability conditions of the model were determined. Moreover, a novel idea was introduced by adding the oscillation of the immigration of the prey into the model which forms a non-autonomous model. The numerically obtained results display that the dynamic behaviors of the model exhibit increasingly stable fluctuations and an increased likelihood of coexistence compared to the autonomous model.

Global Analysis of SIRS Epidemic Model With General Incidence Function and Incomplete Recovery Rates Stochastical Model

In this paper, deterministic and stochastic models are developped for a class of SIRS epidemic models. Firstly, The conditions for the existence, local and global stability of the disease-free equilibrium and endemic equilibrium are obtained. Secondly, we built the stochastic model. The populations are computationally simulated under various conditions. Comparisons are made between the deterministic and stochastic model.

Analysis of the Dynamics of SI-SI-SEIR Avian Influenza A(H7N9) Epidemic Model with Re-infection

The spread of Avian influenza in Asia, Europe and Africa ever since its emergence in 2003, has been endemic in many countries. In this study, a non-linear SI-SI-SEIR Mathematical model with re-infection as a result of continuous contact with both infected poultry from farm and market is proposed. Local and global stability of the three equilibrium points are established and numerical simulations are used to validate the results.

Dynamics of General Class of Difference Equations and Population Model with Two Age Classes

In this paper, we study the qualitative behavior of solutions for a general class of difference equations. The criteria of local and global stability, boundedness and periodicity character (with period 2 k ) of the solution are established. Moreover, by applying our general results on a population model with two age classes, we establish the qualitative behavior of solutions of this model. To support our results, we introduce some numerical examples.