scholarly journals On the Study of Chemostat Model with Pulsed Input in a Polluted Environment

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Zhong Zhao ◽  
Xinyu Song
Keyword(s):  
Periodic Solution ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Chemostat Model ◽  
Globally Asymptotically Stable ◽  
Floquet Theorem

Chemostat model with pulsed input in a polluted environment is considered. By using the Floquet theorem, we find that the microorganism eradication periodic solution is globally asymptotically stable if the impulsive periodTis more than a critical value. At the same time, we can find that the nutrient and microorganism are permanent if the impulsive periodTis less than the critical value.

scholarly journals Extinction and permanence of two-nutrient and one-microorganism chemostat model with pulsed input

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Xinyu Song ◽  
Zhong Zhao
Keyword(s):  
Periodic Solution ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Chemostat Model ◽  
Globally Asymptotically Stable ◽  
Floquet Theorem

A chemostat model with periodically pulsed input is considered. By using the Floquet theorem, we find that the microorganism eradication periodic solution(u1∗(t),v1∗(t),0)is globally asymptotically stable if the impulsive periodTis more than a critical value. At the same time we can find that the nutrient and microorganism are permanent if the impulsive periodTis less than the critical value.

ANALYSIS OF A CHEMOSTAT MODEL WITH PULSED INPUT

2006 ◽  
Vol 14 (04) ◽  
pp. 583-598 ◽  
Author(s):  
XIANGYUN SHI ◽  
XINYU SONG
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Continuous System ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Chemostat Model ◽  
Globally Asymptotically Stable

In this paper, we consider a chemostat model with pulsed input. We find a critical value of the period of pulses. If the period is more than the critical value, the microorganism-free periodic solution is globally asymptotically stable. If less, the system is permanent. Moreover, the nutrient and the microorganism can co-exist on a periodic solution of period τ. Finally, by comparing the corresponding continuous system, we find that the periodically pulsed input destroys the equilibria of the continuous system and initiates periodic solutions. Our results are valuable for the manufacture of products by genetically altered organisms.

Analysis of nonautonomous two species system in a polluted environment

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
G. Samanta
Keyword(s):  
Periodic Solution ◽  
Population Growth ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Volterra Model ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Globally Asymptotically Stable ◽  
Bounded Functions

AbstractIn this paper, a two-species nonautonomous Lotka-Volterra model of population growth in a polluted environment is proposed. Global asymptotic behaviour of this model by constructing suitable bounded functions has been investigated. It is proved that each population for competition, predation and cooperation systems respectively is uniformly persistent (permanent) under appropriate conditions. Sufficient conditions are derived to confirm that if each of competition, predation and cooperation systems respectively admits a positive periodic solution, then it is globally asymptotically stable.

The Effects of Impulsive Toxicant Input on a Population in a Polluted Environment

2003 ◽  
Vol 11 (03) ◽  
pp. 265-274 ◽  
Author(s):  
Bing Liu ◽  
Lansun Chen ◽  
Yujuang Zhang
Keyword(s):  
Continuous System ◽  
Positive Periodic Solution ◽  
Single Species ◽  
Critical Value ◽  
Point Of View ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Globally Asymptotically Stable ◽  
Closed Environment ◽  
Input Amount

In this paper, we investigate a single species model in a polluted closed environment with pulse toxicant input at fixed moment. We show that the population is extinct when the impulsive period is less than some critical value, otherwise the population is permanent, and the permanent condition also assure that there exists a unique positive periodic solution which is globally asymptotically stable. By comparing with the continuous system, we show that pulse toxicant input cause a periodic behavior in our system and oscillation of the solution. From the biological point of view, it is easy to protect species by changing impulsive period of the exogenous input of toxicant, and toxicant input amount, etc. Further, the effects of environmental noise on the population are studied by using numerical methods.

scholarly journals An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yanyan Hu ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Stage Structure ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Birth Pulse ◽  
Impulsive Equation ◽  
Predator Model ◽  
Predator System

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

scholarly journals Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Chang Tan ◽  
Jun Cao
Keyword(s):  
Discrete System ◽  
Numerical Experiments ◽  
Critical Value ◽  
Euler Method ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

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 Global Dynamics Behaviors of Viral Infection Model for Pest Management

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Chunjin Wei ◽  
Lansun Chen
Keyword(s):  
Small Amplitude ◽  
Critical Value ◽  
Infection Model ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Virus Particles ◽  
All Solutions ◽  
Pest Eradication ◽  
Viral Infection Model

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet's theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the amount of virus particles released is larger than some critical value. When the amount of virus particles released is less than some critical value, the system is shown to be permanent, which implies that the trivial pest-eradication solution loses its stability. Further, the mathematical results are also confirmed by means of numerical simulation.

The effect of diffusion loss on the time-varying giant Panda population

2016 ◽  
Vol 09 (04) ◽  
pp. 1650062
Author(s):  
Meng Zhang ◽  
Guohua Song
Keyword(s):  
Mathematical Model ◽  
Periodic Solution ◽  
Giant Panda ◽  
Allee Effect ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Giant Pandas ◽  
Diffusion Loss ◽  
Positive Formula

It has been certificated that corridors can help giant pandas to keep their habitat from fragmenting. However there are still losses during the process of moving along corridors. In this study, a mathematical model with Allee effect is carried out to describe the diffusion of giant pandas between n patches. Some criteria are obtained to keep the system persisting. It is proved that the system has a unique positive [Formula: see text]-periodic solution which is globally asymptotically stable. The ecological meanings of these findings are discussed following the results. And some numerical simulations in the Qinling Mountain giant panda nature reservation area are also presented in the end.

scholarly journals Some Discussions on the Difference Equationxn+1=α+(xn-1m/xnk)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Awad A. Bakery
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk),  n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1,  x0to be bounded,α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm=k=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.

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