Average conditions for permanence and extinction in nonautonomous single-species Kolmogorov systems

2017 ◽  
Vol 10 (02) ◽  
pp. 1750028
Author(s):  
Jiandong Zhao ◽  
Zhenzhen Chen
Keyword(s):  
Global Attractivity ◽  
Single Species ◽  
Logistic Equation ◽  
Math Biol ◽  
Kolmogorov System ◽  
Generalized Logistic Equation

The nonautonomous single-species Kolmogorov system is studied in this paper. Average conditions are obtained for permanence, global attractivity and extinction in the system. Applications of our main results to logistic equation and generalized logistic equation are given. It is shown that our average conditions are improvement of those in Vance and Coddington [J. Math. Biol. 27 (1989) 491–506] and some published literature on the system.

Permanence and extinction of a single species model in polluted environment

2020 ◽  
Vol 13 (04) ◽  
pp. 2050031
Author(s):  
Jiandong Zhao ◽  
Tonghua Zhang
Keyword(s):  
Sufficient Conditions ◽  
Single Species ◽  
Logistic Equation ◽  
Polluted Environment ◽  
Math Model ◽  
Generalized Logistic Equation ◽  
Appl Math ◽  
Single Species Model

Under the assumption that the growth of the population satisfies the generalized logistic equation, a new single species model in polluted environment is proposed in this work. Sufficient conditions for permanence and extinction of the species in the model are given respectively. It is shown that our model and the results are improvements of those in He and Wang [Appl. Math. Model. 31 (2007) 2227–2238].

OPTIMAL IMPULSIVE HARVESTING OF A SINGLE-SPECIES WITH GOMPERTZ LAW OF GROWTH

2006 ◽  
Vol 14 (02) ◽  
pp. 303-314 ◽  
Author(s):  
YUJUAN ZHANG ◽  
ZHILONG XIU ◽  
LANSUN CHEN
Keyword(s):  
Global Attractivity ◽  
Population Level ◽  
Single Species ◽  
Maximum Sustainable Yield ◽  
Discrete Dynamical Systems ◽  
Optimal Harvesting ◽  
Sustainable Yield ◽  
Gompertz Law ◽  
Optimal Population ◽  
Impulsive Harvesting

In this paper we investigate the optimal harvesting problems of a single species with Gompertz law of growth. Based on continuous harvesting models, we propose impulsive harvesting models with constant harvest or proportional harvest. By using the discrete dynamical systems determined by the stroboscopic map, we discuss existence, stability and global attractivity of positive periodic solutions, and obtain the maximum sustainable yield and the corresponding optimal population level. At last, we compare the maximum sustainable yield of impulsive harvest with that of continuous harvest, and point out that proportional harvest is superior to constant harvest.

Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity

2010 ◽  
Vol 140 (5) ◽  
pp. 1081-1109 ◽  
Author(s):  
Zhi-Cheng Wang ◽  
Wan-Tong Li
Keyword(s):  
Diffusion Equation ◽  
Global Attractivity ◽  
Reaction Diffusion ◽  
Single Species ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Positive Equilibrium ◽  
Cauchy Type ◽  
Delayed Reaction ◽  
Non Local

AbstractThis paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.

scholarly journals Analysis of a Periodic Single Species Population Model Involving Constant Impulsive Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Ronghua Tan ◽  
Zuxiong Li ◽  
Shengliang Guo ◽  
Zhijun Liu
Keyword(s):  
Lyapunov Functions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Single Species ◽  
Different Types ◽  
Species Population ◽  
Impulsive Perturbations ◽  
Single Species Population ◽  
Single Species Model

This is a continuation of the work of Tan et al. (2012). In this paper a periodic single species model controlled by constant impulsive perturbation is investigated. The constant impulse is realized at fixed moments of time. With the help of the comparison theorem of impulsive differential equations and Lyapunov functions, sufficient conditions for the permanence and global attractivity are established, respectively. Also, by comparing the above results with corresponding known results of Tan et al. (2012) (i.e., the above model with linear impulsive perturbations), we find that the two different types of impulsive perturbations have influence on the above dynamics. Numerical simulations are presented to substantiate our analytical results.

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