Convergence to equilibrium in a genetic model with differential viability between the sexes

Selgrade, James F.; Ziehe, Martin

doi:10.1007/bf00276194

Cited by 33 publications

(45 citation statements)

References 18 publications

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“…Directional selection favoring A 1 in both sexes: When selection favors A 1 in both sexes, the A 1 fixation is globally stable (Karlin and Lessard 1986;Selgrade and Ziehe 1987). In females, we assume that the viabilities in Table 1 .…”

Section: Modelsmentioning

confidence: 99%

“…With those viabilities for which both fixation states are unstable, the polymorphism is globally stable. Global stability is assured due to the fact that the one-locus two-sex model has the nice property of being strongly monotone (Selgrade and Ziehe 1987). In strongly monotone systems, equilibria are ordered in a special way and the local stability of equilibria alternates with respect to this ordering.…”

Section: Modelsmentioning

confidence: 99%

“…Since the assumption of directional selection excludes the possibility of three polymorphic equilibria, local instability at both fixation equilibria means there must be only one polymorphic equilibrium and it must be locally stable. As long as all equilibria have eigenvalues of nonunit magnitude, the polymorphic equilibrium is also globally stable (Selgrade and Ziehe 1987).…”

Section: Modelsmentioning

confidence: 99%

See 2 more Smart Citations

Sex-Specific Viability, Sex Linkage and Dominance in Genomic Imprinting

Cleve

Feldman

2007

Genetics

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Genomic imprinting is a phenomenon by which the expression of an allele at a locus depends on the parent of origin. Two different two-locus evolutionary models are presented in which a second locus modifies the imprinting status of the primary locus, which is under differential selection in males and females. In the first model, a modifier allele that imprints the primary locus invades the population when the average dominance coefficient among females and males is .

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Section: Modelsmentioning

confidence: 99%

Section: Modelsmentioning

confidence: 99%

Section: Modelsmentioning

confidence: 99%

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Sex-Specific Viability, Sex Linkage and Dominance in Genomic Imprinting

Cleve

Feldman

2007

Genetics

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“…Competition between 2-species with rational transition functions has been studied by Hassell and Comins [7], Franke and Yakubu [5,6], Selgrade and Ziehe [16], Smith [17], and others. A simple competitive model that allows unbounded growth of a population size has been discussed in [1,2]:…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of a competitive system of linear fractional difference equations

Kulenović

Nurkanović

2006

Adv. Differ Equ.

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We investigate the global asymptotic behavior of solutions of the system of difference equations x n+1 = (a + x n )/(b + y n ), y n+1 = (d + y n )/(e + x n ), n = 0,1,..., where the parameters a,b,d, and e are positive numbers and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

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“…Analyzing the behavior of solutions of a higher-order nonlinear difference equation is very interesting and attracted many researchers in recent times. Behavior of solutions means studying the equilibrium point, boundedness and persistence, existence and uniqueness of positive equilibrium point, local and global stability, periodicity nature of such difference equations or systems of difference equations (see [8][9][10][11][12][13][14][15][16] and references cited therein).…”

Section: Introductionmentioning

confidence: 99%

Global Dynamics and Bifurcations Analysis of a Two‐Dimensional Discrete‐Time Lotka‐Volterra Model

Khan

Qureshi

2018

Complexity

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In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant R 2 . It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria (0, 0), (( 1 − 1)/ 3 , 0), (0, ( 4 −1)/ 6 ) and the unique positive equilibrium ((, by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria (( 1 − 1)/ 3 , 0), (0, ( 4 − 1)/ 6 ) and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium ((( 1 − 1) 6 − 2 ( 4 − 1))/( 3 6 − 2 5 ), ( 3 ( 4 − 1) + 5 (1 − 1 ))/( 3 6 − 2 5 )). Further it is shown that every positive solution of the discrete model is bounded and the set [0, 1 / 3 ] × [0, 4 / 6 ] is an invariant rectangle. It is proved that if 1 < 1 and 4 < 1, then equilibrium (0, 0) of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium ((( 1 − 1) 6 − 2 ( 4 − 1))/( 3 6 − 2 5 ), ( 3 ( 4 − 1) + 5 (1 − 1 ))/( 3 6 − 2 5 )) is a global attractor. Some numerical simulations are presented to illustrate theoretical results.

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Convergence to equilibrium in a genetic model with differential viability between the sexes

Cited by 33 publications

References 18 publications

Sex-Specific Viability, Sex Linkage and Dominance in Genomic Imprinting

Sex-Specific Viability, Sex Linkage and Dominance in Genomic Imprinting

Asymptotic behavior of a competitive system of linear fractional difference equations

Global Dynamics and Bifurcations Analysis of a Two‐Dimensional Discrete‐Time Lotka‐Volterra Model

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