Abstract
We consider a predator-prey system of Leslie type with generalized Holling type III functional response p(x)=mx2ax2+bx+1. By allowing b to be negative (b>-2a), p(x) is monotonic for b>. 0 and nonmonotonic for b<. 0 when x≥. 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov-Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov-Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>-2a are much more complex and far richer than the case when b>. 0 but also provide new bifurcation phenomena for predator-prey systems.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1721-1752 |
| Number of pages | 32 |
| Journal | Journal of Differential Equations |
| Volume | 257 |
| Issue number | 6 |
| DOIs | |
| State | Published - Sep 15 2014 |
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Keywords
- Bogdanov-Takens bifurcation
- Degenerate focus type BT bifurcation of codim 3
- Holling type-III functional response
- Hopf bifurcation
- Predator-prey system
ASJC Scopus subject areas
- Analysis
Cite this
Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. / Huang, Jicai; Ruan, Shigui; Song, Jing.
In: Journal of Differential Equations, Vol. 257, No. 6, 15.09.2014, p. 1721-1752.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response
AU - Huang, Jicai
AU - Ruan, Shigui
AU - Song, Jing
PY - 2014/9/15
Y1 - 2014/9/15
N2 - We consider a predator-prey system of Leslie type with generalized Holling type III functional response p(x)=mx2ax2+bx+1. By allowing b to be negative (b>-2a), p(x) is monotonic for b>. 0 and nonmonotonic for b<. 0 when x≥. 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov-Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov-Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>-2a are much more complex and far richer than the case when b>. 0 but also provide new bifurcation phenomena for predator-prey systems.
AB - We consider a predator-prey system of Leslie type with generalized Holling type III functional response p(x)=mx2ax2+bx+1. By allowing b to be negative (b>-2a), p(x) is monotonic for b>. 0 and nonmonotonic for b<. 0 when x≥. 0. The model has two non-hyperbolic positive equilibria (one is a multiple focus of multiplicity one and the other is a cusp of codimension 2) for some values of parameters and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3 for other values of parameters. When there exist a multiple focus of multiplicity one and a cusp of codimension 2, we show that the model exhibits subcritical Hopf bifurcation and Bogdanov-Takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria, respectively. Different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have: (i) a stable limit cycle enclosing two non-hyperbolic positive equilibria; (ii) a stable limit cycle enclosing an unstable homoclinic loop; (iii) two limit cycles enclosing a hyperbolic positive equilibrium; (iv) one stable limit cycle enclosing three hyperbolic positive equilibria; or (v) the coexistence of three stable states (two stable equilibria and a stable limit cycle). When the model has a Bogdanov-Takens singularity of codimension 3, we prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3. These results not only demonstrate that the dynamics of this model when b>-2a are much more complex and far richer than the case when b>. 0 but also provide new bifurcation phenomena for predator-prey systems.
KW - Bogdanov-Takens bifurcation
KW - Degenerate focus type BT bifurcation of codim 3
KW - Holling type-III functional response
KW - Hopf bifurcation
KW - Predator-prey system
UR - http://www.scopus.com/inward/record.url?scp=84903761342&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84903761342&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2014.04.024
DO - 10.1016/j.jde.2014.04.024
M3 - Article
AN - SCOPUS:84903761342
VL - 257
SP - 1721
EP - 1752
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 6
ER -