Dynamics of a Leslie-Gower type predator-prey system with herd behavior and constant harvesting in prey

In this paper, the dynamics of a Leslie-Gower type predator-prey system with herd behavior and constant harvesting in prey are investigated. Earlier work has shown that the herd behavior in prey merely induces a supercritical Hopf bifurcation in the classic Leslie-Gower predator-prey system in the absence of harvesting. However, the work in this paper shows that the presence of herd behavior and constant harvesting in prey can give rise to numerous kinds of bifurcation at the non-hyperbolic equilibria in the classic Leslie-Gower predator-prey system such as two saddle-node bifurcations and one Bogdanov-Takens bifurcation of codimension two at the degenerate equilibria and one degenerate Hopf bifurcation of codimension three at the weak focus. Hence, the research results reveal that the herd behavior and constant harvesting in prey have a strong influence on the dynamics and also contribute to promoting the ecological diversity and maintaining the long-term economic benefits.Comment: 20 pages, 10 figure

Dynamics Analysis of Modified Leslie-Gower Model with Simplified Holling Type IV Functional Response

In this paper, the modified Leslie-Gower predator-prey model with simplified Holling type IV functional response is discussed. It is assumed that the prey population is a dangerous population. The equilibrium point of the model and the stability of the coexistence equilibrium point are analyzed. The simulation results show that both prey and predator populations will not become extinct as time increases. When the prey population density increases, there is a decrease in the predatory population density because the dangerous prey population has a better ability to defend itself from predators when the number is large enough.Dalam tulisan ini dibahas modifikasi model mangsa pemangsa Leslie-Gower dan fungsi respon Holling tipe IV yang disederhanakan. Diasumsikan bahwa populasi mangsa adalah populasi yang berbahaya. Titik-titik kesetimbangan model dan kestabilan dari titik kesetimbangan koeksistensi dianalisis. Selanjutnya, dilakukan simulasi numerik pada titik kesetimbangan koeksistensi. Hasil simulasi menunjukkan bahwa kedua populasi mangsa dan pemangsa tidak akan punah pada saat waktu semakin membesar. Pada saat kepadatan populasi mangsa meningkat terjadi penurunan terhadap kepadatan populasi pemangsa karena populasi mangsa yang berbahaya memiliki kemampuan yang lebih baik untuk mempertahankan diri dari pemangsa ketika jumlahnya cukup besar

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R+2. Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system

Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model

In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one

Bifurcation analysis of Leslie-Gower predator-prey system with harvesting and fear effect

In the paper, a Leslie-Gower predator-prey system with harvesting and fear effect is considered. The existence and stability of all possible equilibrium points are analyzed. The bifurcation dynamic behavior at key equilibrium points is investigated to explore the intrinsic driving mechanisms of population interaction modes. It is shown that the system undergoes various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations. The numerical simulation results show that harvesting and fear effect can seriously affect the dynamic evolution trend and coexistence mode. Furthermore, it is particularly worth pointing out that harvesting not only drives changes in population coexistence mode, but also has a certain degree delay. Finally, it is anticipated that these research results will be beneficial for the vigorous development of predator-prey system

Bogdanov-Takens bifurcation of codimension 33 in the Gierer-Meinhardt model

Bifurcation of the local Gierer-Meinhardt model is analyzed in this paper. It is found that the degenerate Bogdanov-Takens bifurcation of codimension 3 happens in the model, except that teh saddle-node bifurcation and the Hopf bifurcation. That was not reported in the existing results about this model. The existence of equilibria, their stability, the bifurcation and the induced complicated and interesting dynamics are explored in detail, by using the stability analysis, the normal form method and bifurcation theory. Numerical results are also presented to validate theoretical results

Complexity of a Discrete-Time Predator-Prey Model Involving Prey Refuge Proportional to Predator

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied