scholarly journals Exchange of equilibria in two species Lotka-Volterra competition models

1982 ◽  
Vol 24 (2) ◽  
pp. 160-170 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
System Of Equations ◽  
Species Competition ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Periodic Functions ◽  
Competition System ◽  
Competition Models ◽  
Stable Periodic

AbstractSufficient conditions are obtained for the existence of a unique asymptotically stable periodic solution for the Lotka-Volterra two species competition system of equations when the intrinsic growth rates are periodic functions of time.

scholarly journals Global asymptotic stability in a periodic Lotka-Volterra system

1985 ◽  
Vol 27 (1) ◽  
pp. 66-72 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Volterra System ◽  
Periodic Coefficients ◽  
Globally Asymptotically Stable ◽  
Stable Periodic

AbstractA set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

scholarly journals Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Liya Yang ◽  
Xiangdong Xie ◽  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Cooperative Effect ◽  
Stable Periodic Solution ◽  
Predator Species ◽  
Predator Prey ◽  
Stable Periodic ◽  
Extinction Property ◽  
Globally Stable

A nonautonomous discrete predator-prey-mutualist system is proposed and studied in this paper. Sufficient conditions which ensure the permanence and existence of a unique globally stable periodic solution are obtained. We also investigate the extinction property of the predator species; our results indicate that if the cooperative effect between the prey and mutualist species is large enough, then the predator species will be driven to extinction due to the lack of enough food. Two examples together with numerical simulations show the feasibility of the main results.

RANK ONE CHAOS IN SWITCH-CONTROLLED PIECEWISE LINEAR CHUA'S CIRCUIT

2007 ◽  
Vol 16 (05) ◽  
pp. 769-789 ◽  
Author(s):  
QIUDONG WANG ◽  
ALI OKSASOGLU
Keyword(s):  
Periodic Solution ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Pspice Simulations ◽  
Rank One ◽  
Stable Periodic

In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

scholarly journals Stability Analysis of Three-Species Almost Periodic Competition Models with Grazing Rates and Diffusions

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Chang-you Wang ◽  
Rui-fang Wang ◽  
Ming Yi ◽  
Rui Li
Keyword(s):  
Periodic Solution ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Almost Periodic Solution ◽  
Species Competition ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Globally Asymptotically Stable ◽  
Positive Space

Almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained by Q. C. Lin (1999), F. D. Chen and X. X. Chen (2003), and Y. Q. Liu, S. L, Xie, and Z. D. Xie (1996).

HOPF BIFURCATION CONTROL OF DELAYED SYSTEMS WITH WEAK NONLINEARITY VIA DELAYED STATE FEEDBACK

2005 ◽  
Vol 15 (05) ◽  
pp. 1787-1799 ◽  
Author(s):  
ZAIHUA WANG ◽  
HAIYAN HU
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Weak Nonlinearity ◽  
Delayed Systems ◽  
Trivial Equilibrium ◽  
Stable Periodic

This paper presents a study on the problem of Hopf bifurcation control of time delayed systems with weak nonlinearity via delayed feedback control. It focusses on two control objectives: one is to annihilate the periodic solution, namely to perform a linear delayed feedback control so that the trivial equilibrium is asymptotically stable, and the other is to obtain an asymptotically stable periodic solution with given amplitude via linear or nonlinear delayed feedback control. On the basis of the averaging method and the center manifold reduction for delayed differential equations, an effective method is developed for this problem. It has been shown that a linear delayed feedback can always stabilize the unstable trivial equilibrium of the system, and a linear or nonlinear delayed feedback control can always achieve an asymptotically stable periodic solution with desired amplitude. The illustrative example shows that the theoretical prediction is in very good agreement with the simulation results, and that the method is valid with high accuracy not only for delayed systems with weak nonlinearity and via weak feedback control, but also for those when the nonlinearity and feedback control are not small.

Stability analysis of n-species Lotka–Volterra almost periodic competition models with grazing rates and diffusions

2014 ◽  
Vol 07 (02) ◽  
pp. 1450011
Author(s):  
Yi-Jin Zhang ◽  
Chang-You Wang
Keyword(s):  
Periodic Solution ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Almost Periodic Solution ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Globally Asymptotically Stable ◽  
Competition Models ◽  
Positive Space

In this paper, almost periodic solution of a n-species Lotka–Volterra competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions under which the strictly positive space homogeneous almost periodic solution of the system is globally asymptotically stable. Moreover, some numerical simulations are given to validate our theoretical analysis.

scholarly journals Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Weibing Wang ◽  
Jianhua Shen ◽  
Juan J. Nieto
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Impulsive Effect ◽  
Two Dimensional ◽  
Stable Periodic Solution ◽  
Predator Prey ◽  
Prey System ◽  
Stable Periodic ◽  
Holling Type Functional Response

We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.

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