scholarly journals Comparison principles for impulsive parabolic equations with applications to models of single species growth

1991 ◽  
Vol 32 (4) ◽  
pp. 382-400 ◽  
Author(s):  
L. H. Erbe ◽  
H. I. Freedman ◽  
X. Z. Liu ◽  
J. H. Wu
Keyword(s):  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Single Species ◽  
Reaction Diffusion Equation ◽  
Equation Modeling ◽  
Positive Equilibrium ◽  
Comparison Principles ◽  
Nonlinear Parabolic ◽  
Species Population ◽  
Single Species Population

AbstractThis paper establishes some maximum and comparison principles relative to lower and upper solutions of nonlinear parabolic partial differential equations with impulsive effects. These principles are applied to obtain some sufficient conditions for the global asymptotic stability of a unique positive equilibrium in a reaction-diffusion equation modeling the growth of a single-species population subject to abrupt changes of certain important system parameters.

scholarly journals Stability Analysis of a Population Model with Maturation Delay and Ricker Birth Function

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Chongwu Zheng ◽  
Fengqin Zhang ◽  
Jianquan Li
Keyword(s):  
Population Model ◽  
Sufficient Conditions ◽  
Single Species ◽  
Positive Equilibrium ◽  
Stability Switch ◽  
Birth Function ◽  
Maturation Delay ◽  
Trivial Equilibrium ◽  
Species Population ◽  
Single Species Population

A single species population model is investigated, where the discrete maturation delay and the Ricker birth function are incorporated. The threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium is obtained. The necessary and sufficient conditions ensuring the local asymptotical stability of the positive equilibrium are given by applying the Pontryagin's method. The effect of all the parameter values on the local stability of the positive equilibrium is analyzed. The obtained results show the existence of stability switch and provide a method of computing maturation times at which the stability switch occurs. Numerical simulations illustrate that chaos may occur for the model, and the associated parameter bifurcation diagrams are given for certain values of the parameters.

scholarly journals Survival analysis of a stochastic delay single-species system in polluted environment with psychological effect and pulse toxicant input

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiangjun Dai ◽  
Suli Wang ◽  
Weizhi Xiong ◽  
Ni Li
Keyword(s):  
Sufficient Conditions ◽  
Single Species ◽  
Threshold Value ◽  
Psychological Effect ◽  
Polluted Environment ◽  
Population System ◽  
Stochastic Delay ◽  
Species Population ◽  
The Mean ◽  
Single Species Population

Abstract We propose and study a stochastic delay single-species population system in polluted environment with psychological effect and pulse toxicant input. We establish sufficient conditions for the extinction, nonpersistence in the mean, weak persistence, and strong persistence of the single-species population and obtain the threshold value between extinction and weak persistence. Finally, we confirm the efficiency of the main results by numerical simulations.

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.

Extinction, stable pattern and their transition in a diffusive single species population with distributed maturity

2008 ◽  
Vol 19 (3) ◽  
pp. 285-309
Author(s):  
PEIXUAN WENG
Keyword(s):  
Distributed Delay ◽  
Reaction Diffusion ◽  
Single Species ◽  
Cylindrical Domain ◽  
Stable Pattern ◽  
Structured Population ◽  
Species Population ◽  
Single Species Population ◽  
Spatially Varying ◽  
Robin Boundary

We consider a single-species structured population with distributed maturity and spatial diffusion in a cylindrical domain subject to Neumann and Robin boundary conditions. We first establish the threshold property of the reaction–diffusion system with distributed delay and non-local interaction in a corresponding lower-dimensional domain, so that the system approaches either an extinction state or a stable spatially varying pattern. We then investigate the transition from the extinction state to the stable pattern of the system in the cylindrical domain.

scholarly journals A stage-structured delayed advection reaction-diffusion model for single species

2020 ◽  
Vol 10 (6) ◽  
pp. 6260
Author(s):  
Raed Ali Alkhasawneh
Keyword(s):  
Diffusion Equation ◽  
Time Delays ◽  
Reaction Diffusion ◽  
Single Species ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Reaction Diffusion Model ◽  
Advection Term ◽  
Species Population ◽  
Homogeneous Dirichlet Boundary Conditions

In this paper, we derived a delay advection reaction-diffusion equation with linear advection term from a stage-structured model, then the derived equation is used under the homogeneous Dirichlet boundary conditions u_m (0,t)=0, u_m (L,t)=0, and the initial condition u_m (x,0)=u_m^0 (x)>0,x∈[-τ,0] with u_m^0 (0)>0 in order to find the minimum value of domain L that prevents extinction of the species under the effect of advection reaction diffusion equation. Finally, for the measurement the time lengths from birth to the development of the species population, time delays are integrated.

scholarly journals Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2560
Author(s):  
Zuxiong Li ◽  
Shengnan Fu ◽  
Huili Xiang ◽  
Hailing Wang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
Single Species ◽  
Dimensional System ◽  
Globally Asymptotically Stable ◽  
Species Population ◽  
Existence And Stability ◽  
Single Species Population

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Survival analysis of a stochastic single-species population model with jumps in a polluted environment

2015 ◽  
Vol 09 (01) ◽  
pp. 1650011 ◽  
Author(s):  
Meng Liu ◽  
Ke Wang
Keyword(s):  
Population Model ◽  
Sufficient Conditions ◽  
Single Species ◽  
Polluted Environment ◽  
Stochastic Permanence ◽  
Persistence In The Mean ◽  
Species Population ◽  
The Mean ◽  
Single Species Population ◽  
Lévy Jumps

This paper is concerned with a stochastic single-species system with Lévy jumps in a polluted environment. Some sufficient conditions on extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean, stability in the mean and stochastic permanence are obtained. The threshold between extinction and weak persistence in the mean is established. At the same time, under a simple condition, it is proved that this threshold also is the threshold between extinction and stability in the mean. The results reveal that Lévy jumps have significant effects to the persistence and extinction results.

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