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## Local stability implies global stability for the 2dimensional Ricker map

Venue: | J. Difference Equations and Applications |

Citations: | 1 - 1 self |

### Citations

1405 | Depth-first search and linear graph algorithms
- Tarjan
- 1971
(Show Context)
Citation Context ...ected components. The vertices that form a component by themselves and have no self edges are the ones that are not in directed cycles. To find this decomposition, we will use the algorithm of Tarjan =-=[23]-=-, that runs in linear time. Inner enclosure of the basin of attraction Consider the continuous map f :D f ⊆ Rn→ Rn. (4.2) Definition 4.12. A set O ⊆ D f is called invariant if f (O) = O . An invariant... |

158 |
Stock and recruitment
- Ricker
- 1954
(Show Context)
Citation Context ...ker bifurcation; graph representations; interval analysis; discrete-time single species model 2010 Mathematics Subject Classification: 39A30, 39A28, 65Q10, 65G40, 92D25 1 Introduction In 1954, Ricker =-=[21]-=- introduced the difference equation xk+1 = xkeα−xk (1.1) with a positive parameter α to model the population density of a single species with non-overlapping generations. The function R1 : R 3 x 7→ xe... |

148 |
The Boost Graph Library: User Guide and Reference Manual
- Siek, Lee, et al.
- 2002
(Show Context)
Citation Context ...lly attracting fixed point of the two dimensional Ricker-map Fα for every α ∈ [α]. We implemented our program in C++, using the CAPD Library [5] for rigorous computations, and the Boost Graph Library =-=[22]-=- for handling the directed graphs. The recursion number in Tarjan’s algorithm was very high, therefore we converted it into a sequential program, using virtual stack structures from the Standard Libra... |

106 | A Rigorous ODE Solver and Smale’s 14th Problem.
- Tucker
- 2002
(Show Context)
Citation Context ...s represented as a pair of endpoints [a−,a+]. Having a set S or a number r, we denote their interval enclosures by [S] and [r], respectively. The reader is referred to Moore [19], Alefeld [2], Tucker =-=[25]-=-, [26], Nedialkov, Jackson and Corliss [20] for further details. The structure of the paper is as follows. In Section 2 we construct a compact region S, which is a closed square around (α,α)T , having... |

104 | Validated Solutions of Initial Value Problems for Ordinary Differential Equations
- Nedialkov, Jackson, et al.
- 1999
(Show Context)
Citation Context ...]. Having a set S or a number r, we denote their interval enclosures by [S] and [r], respectively. The reader is referred to Moore [19], Alefeld [2], Tucker [25], [26], Nedialkov, Jackson and Corliss =-=[20]-=- for further details. The structure of the paper is as follows. In Section 2 we construct a compact region S, which is a closed square around (α,α)T , having the property that Fα(S) ⊆ S and every traj... |

93 |
A subdivision algorithm for the computation of unstable manifolds and global attractors
- Dellnitz, Hohmann
- 1997
(Show Context)
Citation Context ...perties of our dynamical system through the study of the graphs. These techniques appeared in many articles, in both rigorous and non-rigorous computations for maps by Hohmann, Dellnitz, Junge, Rumpf =-=[6]-=-, [7], Galias [9], Luzzatto and Pilarczyk [18], and computations for the time evolution of a continuous system with a given timestep by Wilczak [27]. We introduce the general setting and two applicati... |

37 |
Elements of applied bifurcation theory, volume 112
- Kuznetsov
- 1995
(Show Context)
Citation Context ...−α|< 1/22, |y−α|< 1/22} belongs to the basin of attraction of the fixed point α of Fα . Proof. We follow the steps of finding the normal form of the Neimark–Sacker bifurcation, according to Kuznetsov =-=[13]-=-. In our calculations and estimations we use symbolic calculations and built in symbolic interval arithmetic tools of Wolfram Mathematica v. 7 or 8. According to Kuznetsov [13], we are aiming to trans... |

16 |
Introduction to interval analysis
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...terval [a] is represented as a pair of endpoints [a−,a+]. Having a set S or a number r, we denote their interval enclosures by [S] and [r], respectively. The reader is referred to Moore [19], Alefeld =-=[2]-=-, Tucker [25], [26], Nedialkov, Jackson and Corliss [20] for further details. The structure of the paper is as follows. In Section 2 we construct a compact region S, which is a closed square around (α... |

14 |
A delay-recruitment model of population dynamics with an application to baleen whale populations
- Clark
- 1976
(Show Context)
Citation Context ...tive equilibrium of a single species model implies its global stability.’ This claim was recently disproven by a counterexample of Jiménez López [11, 12] on global attractivity for Clark’s equation =-=[4]-=- when the delay is at least 3. Liz, Tkachenko and Trofimchuk [17] proved that if 0 < α < 3 2(d+1) (1.3) then the fixed point (α, . . . ,α)T ∈ Rd+1 of Rd+1 is globally asymptotically stable, where glob... |

14 |
S.: Global stability in discrete population models with delayed-density dependence
- Liz, Tkachenko, et al.
- 2006
(Show Context)
Citation Context ...ability.’ This claim was recently disproven by a counterexample of Jiménez López [11, 12] on global attractivity for Clark’s equation [4] when the delay is at least 3. Liz, Tkachenko and Trofimchuk =-=[17]-=- proved that if 0 < α < 3 2(d+1) (1.3) then the fixed point (α, . . . ,α)T ∈ Rd+1 of Rd+1 is globally asymptotically stable, where globally means that the region of attraction of (α, . . . ,α)T is Rd+... |

13 | Exploring invariant sets and invariant measures.
- Dellnitz, Hohmann, et al.
- 1997
(Show Context)
Citation Context ...es of our dynamical system through the study of the graphs. These techniques appeared in many articles, in both rigorous and non-rigorous computations for maps by Hohmann, Dellnitz, Junge, Rumpf [6], =-=[7]-=-, Galias [9], Luzzatto and Pilarczyk [18], and computations for the time evolution of a continuous system with a given timestep by Wilczak [27]. We introduce the general setting and two applications i... |

8 |
Rigorous investigation of the Ikeda map by means of interval arithmetic. Nonlinearity
- Galias
- 2002
(Show Context)
Citation Context ...namical system through the study of the graphs. These techniques appeared in many articles, in both rigorous and non-rigorous computations for maps by Hohmann, Dellnitz, Junge, Rumpf [6], [7], Galias =-=[9]-=-, Luzzatto and Pilarczyk [18], and computations for the time evolution of a continuous system with a given timestep by Wilczak [27]. We introduce the general setting and two applications in particular... |

8 |
A global attractivity criterion for nonlinear non-autonomous difference equations
- Tkachenko, Trofimchuk
- 2006
(Show Context)
Citation Context ... the region of attraction of (α, . . . ,α)T is Rd+1+ . They also suggested that condition (1.3) can be replaced by 0 < α < 3 2(d+1) + 1 2(d+1)2 , (1.4) which was proven by Tkachenko and Trofimchuk in =-=[24]-=-. This result is a strong support of the conjecture of Levin and May, and it is proven for a class of maps, not only for Rd+1. For the 1dimensional Ricker map R1, condition (1.4) with d = 0 gives the ... |

7 | Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin
- Liz
- 2007
(Show Context)
Citation Context ...mensional Ricker map R1, condition (1.4) with d = 0 gives the region 0 < α < 2. For d = 1, i.e., for the 2-dimensional Ricker map R2, condition (1.4) is equivalent to 0 < α < 0.875. See also [16] and =-=[15]-=- in the topic. Linearising R2 at the fixed point (α,α)T shows that local exponential stability of (α,α)T holds for 0 < α < 1, and (α,α)T is unstable for α > 1. As α passes the value 1, a Neimark–Sacke... |

6 |
A note on difference-delay equations,”
- Levin, May
- 1976
(Show Context)
Citation Context ...if 0 < α ≤ 2, and, for 0 < α ≤ 2, x= α attracts all points from (0,∞); or equivalently, the equilibrium x= α of equation (1.1) is globally stable provided it is locally stable. In 1976, Levin and May =-=[14]-=- considered the case when there are explicit time lags in the density dependent regulatory mechanisms. This leads to the difference-delay equation of order d+1: xk+1 = xkeα−xk−d , (1.2) ∗Corresponding... |

5 |
Methods and applications of interval analysis, volume 2
- Moore
- 1979
(Show Context)
Citation Context ...om that. An interval [a] is represented as a pair of endpoints [a−,a+]. Having a set S or a number r, we denote their interval enclosures by [S] and [r], respectively. The reader is referred to Moore =-=[19]-=-, Alefeld [2], Tucker [25], [26], Nedialkov, Jackson and Corliss [20] for further details. The structure of the paper is as follows. In Section 2 we construct a compact region S, which is a closed squ... |

4 |
Validated numerics
- Tucker
- 2011
(Show Context)
Citation Context ...esented as a pair of endpoints [a−,a+]. Having a set S or a number r, we denote their interval enclosures by [S] and [r], respectively. The reader is referred to Moore [19], Alefeld [2], Tucker [25], =-=[26]-=-, Nedialkov, Jackson and Corliss [20] for further details. The structure of the paper is as follows. In Section 2 we construct a compact region S, which is a closed square around (α,α)T , having the p... |

3 | Stability of non-autonomous difference equations: simple ideas leading to useful results
- Liz
(Show Context)
Citation Context ...r the 1dimensional Ricker map R1, condition (1.4) with d = 0 gives the region 0 < α < 2. For d = 1, i.e., for the 2-dimensional Ricker map R2, condition (1.4) is equivalent to 0 < α < 0.875. See also =-=[16]-=- and [15] in the topic. Linearising R2 at the fixed point (α,α)T shows that local exponential stability of (α,α)T holds for 0 < α < 1, and (α,α)T is unstable for α > 1. As α passes the value 1, a Neim... |

3 | Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system
- Wilczak
- 2010
(Show Context)
Citation Context ...utations for maps by Hohmann, Dellnitz, Junge, Rumpf [6], [7], Galias [9], Luzzatto and Pilarczyk [18], and computations for the time evolution of a continuous system with a given timestep by Wilczak =-=[27]-=-. We introduce the general setting and two applications in particular. One to enclose the non-wandering points and the other one to estimate the basin of attraction. Both methods (Algorithms 1 and 2) ... |

2 |
negative Schwarzian derivative do not imply G.A.S. in Clark’s equation, Universidad de
- López, Parreño, et al.
(Show Context)
Citation Context ... theorem that ‘The local stability of the unique positive equilibrium of a single species model implies its global stability.’ This claim was recently disproven by a counterexample of Jiménez López =-=[11, 12]-=- on global attractivity for Clark’s equation [4] when the delay is at least 3. Liz, Tkachenko and Trofimchuk [17] proved that if 0 < α < 3 2(d+1) (1.3) then the fixed point (α, . . . ,α)T ∈ Rd+1 of Rd... |

2 |
Luzzatto and Paweł Pilarczyk. Finite resolution dynamics
- Stefano
(Show Context)
Citation Context ...tudy of the graphs. These techniques appeared in many articles, in both rigorous and non-rigorous computations for maps by Hohmann, Dellnitz, Junge, Rumpf [6], [7], Galias [9], Luzzatto and Pilarczyk =-=[18]-=-, and computations for the time evolution of a continuous system with a given timestep by Wilczak [27]. We introduce the general setting and two applications in particular. One to enclose the non-wand... |

1 | A counterexample on global attractivity for Clark’s equation
- López
- 2011
(Show Context)
Citation Context ... theorem that ‘The local stability of the unique positive equilibrium of a single species model implies its global stability.’ This claim was recently disproven by a counterexample of Jiménez López =-=[11, 12]-=- on global attractivity for Clark’s equation [4] when the delay is at least 3. Liz, Tkachenko and Trofimchuk [17] proved that if 0 < α < 3 2(d+1) (1.3) then the fixed point (α, . . . ,α)T ∈ Rd+1 of Rd... |