Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice

2012 ◽  
Vol 13 (4) ◽  
pp. 1873-1890 ◽  
Author(s):  
Cui-Ping Cheng ◽  
Wan-Tong Li ◽  
Zhi-Cheng Wang
Keyword(s):  
Population Model ◽  
Stage Structure ◽  
Two Dimensional ◽  
Spatial Lattice ◽  
Delayed Population Model

Traveling Waves Connecting Equilibrium and Periodic Orbit for a Delayed Population Model on a Two-Dimensional Spatial Lattice

2016 ◽  
Vol 26 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Cui-Ping Cheng ◽  
Wan-Tong Li ◽  
Zhi-Cheng Wang ◽  
Shenzhou Zheng
Keyword(s):  
Periodic Orbit ◽  
Traveling Waves ◽  
Population Model ◽  
Traveling Wave Solution ◽  
Heteroclinic Orbits ◽  
Two Dimensional ◽  
Regular Perturbation ◽  
Perturbation Problem ◽  
Spatial Lattice ◽  
Delayed Population Model

This paper is concerned with the existence of fast traveling waves connecting an equilibrium and a periodic orbit in a delayed population model with stage structure on a two-dimensional spatial lattice, under the assumption that the corresponding ODEs have heteroclinic orbits connecting an equilibrium point and a periodic solution. In this work, we rewrite the mixed functional differential equation as an integral equation in a Banach space and analyze the corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by using the Liapunov–Schmidt method and implicit function theorem.

Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function

2015 ◽  
Vol 25 (12) ◽  
pp. 1550170 ◽  
Author(s):  
Li Fan ◽  
Sanyi Tang
Keyword(s):  
Bifurcation Analysis ◽  
Population Model ◽  
Birth Rate ◽  
Global Bifurcation ◽  
Stage Structure ◽  
Phase Portraits ◽  
Approximation Techniques ◽  
Codimension Two ◽  
Codimension One ◽  
Stage Transition

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

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