Global Stability and Hopf Bifurcation for a Stage Structured Model with Competition for Food

2021 ◽  
Vol 31 (10) ◽  
pp. 2150145
Author(s):  
Yunfei Lv ◽  
Yongzhen Pei ◽  
Rong Yuan
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Global Stability ◽  
Delay Differential Equation ◽  
Complete Analysis ◽  
Global Dynamics ◽  
Structured Model ◽  
State Dependent ◽  
State Dependent Delay ◽  
Delay Differential

Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model.

scholarly journals Singular Hopf bifurcation in a differential equation with large state-dependent delay

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130596 ◽  
Author(s):  
G. Kozyreff ◽  
T. Erneux
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Gradual Change ◽  
Classical State ◽  
Sustained Oscillations ◽  
State Dependent ◽  
Analytical Construction ◽  
State Dependent Delay ◽  
Delay Differential ◽  
Sawtooth Oscillations

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

scholarly journals An analysis on the stability of a state dependent delay differential equation

2016 ◽  
Vol 14 (1) ◽  
pp. 425-435 ◽  
Author(s):  
Sertaç Erman ◽  
Ali Demir
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Delay Term ◽  
State Dependent ◽  
State Dependent Delay ◽  
The Stability ◽  
Delay Differential ◽  
Equation With Delay

AbstractIn this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term $\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

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