scholarly journals POSITIVE EQUILIBRIUM SOLUTIONS TO GENERAL POPULATION MODEL

2013 ◽  
Vol 85 (6) ◽  
Author(s):  
J.H. Kang
Keyword(s):  
General Population ◽  
Population Model ◽  
Positive Equilibrium ◽  
Equilibrium Solutions

Qualitative Analysis of Three-Dimensional Single Food-Chain Model with Variable Consumption Rate in Chemostat

2014 ◽  
Vol 955-959 ◽  
pp. 463-470
Author(s):  
Jing Liu ◽  
Hong Wei Jiang ◽  
Chao Liu
Keyword(s):  
Food Chain ◽  
Consumption Rate ◽  
Three Dimensional ◽  
Quadratic Function ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Equilibrium Solutions ◽  
Existence And Stability ◽  
Variable Consumption ◽  
Food Chain Model

The paper studies three-dimensional food-chain model with variable consumption rate in Chemostat. Assume the prey population's consumption rate of the nutrients is quadratic function, and the predator's consumption rate of the prey population is linear function. Use qualitative theory of ordinary differential equation to analyze the equilibrium solution of the model, especially the existence and stability of positive equilibrium solutions and Hopf bifurcation solutions. Finally,several numerical simulations illustrating the theoretical analysis are also given.

scholarly journals A multi-class extension of the mean field Bolker–Pacala population model

2018 ◽  
Vol 26 (3) ◽  
pp. 163-174 ◽  
Author(s):  
Mariya Bessonov ◽  
Stanislav Molchanov ◽  
Joseph Whitmeyer
Keyword(s):  
Population Model ◽  
Limit Theorems ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Geometric Ergodicity ◽  
Equilibrium Solutions ◽  
The Mean ◽  
Number Of Classes ◽  
Population Suppression

Abstract We extend our earlier mean field approximation of the Bolker–Pacala model of population dynamics by dividing the population into N classes, using a mean field approximation for each class but also allowing migration between classes as well as possibly suppressive influence of the population of one class over another class. For {N\geq 2} , we obtain one symmetric nontrivial equilibrium for the system and give global limit theorems. For {N=2} , we calculate all equilibrium solutions, which, under additional conditions, include multiple nontrivial equilibria. Lastly, we prove geometric ergodicity regardless of the number of classes when there is no population suppression across the classes.

Stability and Bifurcation in a Diffusive Logistic Population Model with Multiple Delays

2015 ◽  
Vol 25 (08) ◽  
pp. 1550107 ◽  
Author(s):  
Shanshan Chen ◽  
Junjie Wei
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Dirichlet Boundary Condition ◽  
Population Model ◽  
Dirichlet Boundary ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Stability And Bifurcation ◽  
The Stability

A diffusive logistic population model with multiple delays and Dirichlet boundary condition is considered in this paper. The stability/instability of the positive equilibrium and delay induced Hopf bifurcation are investigated. Moreover, we show which kind of delay could actually affect the dynamics.

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

2015 ◽  
Vol 25 (10) ◽  
pp. 1550132 ◽  
Author(s):  
Lingyan Zhao ◽  
Haihong Liu ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Virus Production ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
Viral Therapy ◽  
The Stability ◽  
Three Delays ◽  
Production Period ◽  
Hiv 1

In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.

scholarly journals Dynamics of a delayed population model with feedback control

2000 ◽  
Vol 41 (4) ◽  
pp. 451-457 ◽  
Author(s):  
Wang Wendi ◽  
Tang Chunlei
Keyword(s):  
Feedback Control ◽  
Population Growth ◽  
Population Model ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Population Model

This paper studies a system proposed by K. Gopalsamy and P. X. Weng to model a population growth with feedback control and time delays. Sufficient conditions are established under which the positive equilibrium of the system is globally attracting. The conjecture proposed by Gopalsamy and Weng is here confirmed and improved.

scholarly journals The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. R. S. Kulenović ◽  
Connor O’Loughlin ◽  
E. Pilav
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Invariant Manifolds ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
The Difference

We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

scholarly journals Stability analysis of two predators and one prey population model with harvesting in fisheries management

2021 ◽  
Vol 921 (1) ◽  
pp. 012005
Author(s):  
D Didiharyono ◽  
S Toaha ◽  
J Kusuma ◽  
Kasbawati
Keyword(s):  
Equilibrium Point ◽  
Population Model ◽  
Fishery Management ◽  
Stable Equilibrium ◽  
Prey Population ◽  
Positive Equilibrium ◽  
Stable Equilibrium Point ◽  
Maximum Profit ◽  
Positive Equilibrium Point ◽  
Prey Populations

Abstract The discussion is focussed in the interaction between two predators and one prey population model in fishery management. Mathematically model is built by involving harvesting with constant efforts in the two predators and one prey populations. The positive equilibrium point of the model is analyzed via linearization and Routh-Hurwitz stability criteria. From the analysis, there exists a certain condition that makes the positive equilibrium point is asymptotically stable. The stable equilibrium point is then related to the maximum profit problem. With suitable value of harvesting efforts, the maximum profit is reached and the predator and prey populations remain stable. Finally, a numerical simulation is carried out to find out how much the maximum profit is obtained and to visualize how the trajectories of predator and prey tend to the stable equilibrium point.

Spatial dynamics of a lattice population model with two age classes and maturation delay

2014 ◽  
Vol 26 (1) ◽  
pp. 61-91 ◽  
Author(s):  
SHI-LIANG WU ◽  
PEIXUAN WENG ◽  
SHIGUI RUAN
Keyword(s):  
Population Model ◽  
Global Attractivity ◽  
Spatial Dynamics ◽  
Sufficient Conditions ◽  
Growth Function ◽  
Spreading Speed ◽  
Positive Equilibrium ◽  
Structured Population Model ◽  
Minimal Wave Speed ◽  
Age Structured

This paper is concerned with the spatial dynamics of a monostable delayed age-structured population model in a 2D lattice strip. When there exists no positive equilibrium, we prove the global attractivity of the zero equilibrium. Otherwise, we give some sufficient conditions to guarantee the global attractivity of the unique positive equilibrium by establishing a series of comparison arguments. Furthermore, when those conditions do not hold, we show that the system is uniformly persistent. Finally, the spreading speed, including the upward convergence, is established for the model without the monotonicity of the growth function. The linear determinacy of the spreading speed and its coincidence with the minimal wave speed are also proved.

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