On the Two-Parameter Bifurcation in a Predator-Prey System of Ivlev Type (original) (raw)
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Discrete-time bifurcation behavior of a prey-predator system with generalized predator
Advances in Difference Equations, 2015
In the present study, keeping in view of Leslie-Gower prey-predator model, the stability and bifurcation analysis of discrete-time prey-predator system with generalized predator (i.e., predator partially dependent on prey) is examined. Global stability of the system at the fixed points has been discussed. The specific conditions for existence of flip bifurcation and Neimark-Sacker bifurcation in the interior of R 2 + have been derived by using center manifold theorem and bifurcation theory. Numerical simulation results show consistency with theoretical analysis. In the case of a flip bifurcation, numerical simulations display orbits of period 2, 4, 8 and chaotic sets; whereas in the case of a Neimark-Sacker bifurcation, a smooth invariant circle bifurcates from the fixed point and stable period 16, 26 windows appear within the chaotic area. The complexity of the dynamical behavior is confirmed by a computation of the Lyapunov exponents.
Bifurcations of a predator-prey model with non-monotonic response function
Comptes Rendus Mathematique, 2005
A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra-Lotka system by a non-monotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on time-periodic perturbations by several examples of strange attractors. c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Bifurcations dans un système prédateur-proie avec réponse fonctionnelle nonmonotone Résumé. On considère un modèle prédateur-proie en dimension 2 dépendant de cinq paramètres adapté du système Volterra-Lotka par une réponse fonctionnelle non-monotone. Une description des différents domaines de stabilité structurelle est présentée ainsi que leurs bifurcations. La structure de l'ensemble de bifurcation se réduità quatre centres organisateurs de codimension 3. Nous présentons quelques examples d'attracteursétranges obtenus par une pertubation périodique non autonome. c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS 1
On the Bifurcation Pattern and Normal Form in a Modified Predator–Prey Nonlinear System
Journal of Computational and Nonlinear Dynamics, 2007
Detailed bifurcation pattern and stability structure is studied in a modified predator-prey system, with nonmonotonic response function. It is observed that almost all the parameters of the system have a positive influence as far as bifurcation is concerned. The analysis is done with the help of the package MATCONT. In the second stage of the analysis the detailed structure of the normal form is obtained after the corresponding position of Hopf bifurcation and Bogdanov-Takens bifurcation are identified with the help of a modified approach recently proposed by Kuznetsov (1995, 2 where ␣ ജ 0, ␦ Ͼ 0, ജ 0, and  Ͼ −2 ͱ ␣ are parameters of the system. The coefficient "a" represents the intrinsic growth rate of the prey, ␦ is the natural death rate of the predator, and stands then proceeding as in Ref. ͓17͔ we get, a 3 = −1194.389148, b 3 = −1898.968608, and a 4 = 35131.19539, b 4 = −53953.71848, which indicates that a 2 b 4 Ͻ 0, showing that we have a BT bifurcation which is nondegenerate and for which a 2 b 2 0. Therefore, the normal form for nondegenerate codimension 3 BT bifurcation is
Global stability and sliding bifurcations of a non-smooth Gause predator–prey system
Applied Mathematics and Computation, 2013
A non-smooth Gause predator-prey model with a constant refuge is proposed and analyzed. Firstly, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between the existence of a regular equilibrium and a pseudo-equilibrium are studied, and the results indicate that the two types of equilibria cannot coexist. The sufficient and necessary conditions for the global stability of limit cycle, sliding touching cycle, canard cycle, focus point and pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node (focus) and touching bifurcations were investigated by employing theoretical and numerical techniques. Finally, we compare our results with previous studies on a non-smooth Gause predator-prey model without involving a carrying capacity for the prey population, and some biological implications are discussed.
Bifurcation and complex dynamics of a discrete-time predator–prey system
In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-I type in the closed first quadrant 2 R . The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of 2 R by using bifurcation theory. It has been found that the dynamical behavior of the model is very sensitive to the parameter values and the initial conditions. Numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamic behaviors, including phase portraits, period-9, 10, 20-orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance of the chaotic dynamics and attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.
2001
Abstract. We consider the existence of limit cycles for a predator-prey system with a functional response. The system has two or more parameters that represent the intrinsic rate of the predator population. A necessary and sufficient condition for the uniqueness of limit cycles in this system is presented. Such result will usually lead to a bifurcation curve. 2000 Mathematics Subject Classification. 92D40. 1. Introduction. The
Bifurcation Analysis of Prey-Predator Model with Harvested Predator
This paper aims to study the effect of harvested predator species on a Holling type IV Prey-Predator model involving intra-specific competition. Prey-predator model has received much attention during the last few decades due to its wide range of applications. There are many kind of prey-predator models in mathematical ecology. The Prey-predator models governed by differential equations are more appropriate than the difference equations to describe the prey-predator relations. Harvesting has a strong impact on the dynamic evolution of a population. This model represents mathematically by non-linear differential equations. The locally asymptotic stability conditions of all possible equilibrium points were obtained. The stability/instability of non-negative equilibrium and associated bifurcation were investigated by analysing the characteristic equations. Moreover, bifurcation diagrams were obtained for different values of parameters of proposed model.
Bifurcations and feedback control of a stage-structure exploited prey-predator system
International Journal of Engineering, Science and Technology, 2011
The present paper describes a bioeconomic modelling of a stage-structure prey-predator system with differential algebraic equations. The criterion for coexistence of the equilibrium points and their stability nature are investigated. Singularity induced bifurcation are studied for zero economic profit and in this perspective, feedback control is designed to preserve the persistence property of the system. In contrast to zero profit, an interior equilibrium point remains stable for positive economic profit. The reasons behind the different nature of the interior equilibriums for zero and positive profit are discussed in conclusion section. Some numerical simulations are given to verify the analytical results. How the maximum profit hampers the system is provided through saddle-node bifurcation in the last subsection of numerical simulation. 4 3 J Det J Det ≠ The characteristic polynomial of 3 J follows the expansion of ), ( 3 3 J I Det − λ but in the case of 4 J it is of ), ( 4 J A Det −
Bifurcations of a predator-prey system with weak Allee effects
We formulate and study a predator-prey model with nonmonotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001Math. 61 ( ), no. 4, 1445Math. 61 ( -1472 but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).
Stability and Hopf-bifurcation in a general Gauss type two-prey and one-predator system
Applied Mathematical Modelling, 2016
Highlights: Local and global stability behavior of a Gauss type general two-prey and one-predator model. Stability and direction of Hopf-bifurcation is studied. The intra-specific interference coefficient of the predator plays an important role in governing the dynamics of the system. Numerical simulation experiments have been conducted to examine the behavior of the system with different parameters.