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...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...

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...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...

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...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...

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...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...

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...]. 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...

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...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...

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...−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...

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...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...

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...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...

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...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+...

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...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...

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...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...

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... 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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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) ...

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... 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...

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...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...

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... 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...