Global stability of genetic systems governed by mutation and selection

1976 ◽  
Vol 80 (2) ◽  
pp. 331-336 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Global Stability ◽  
Stable State ◽  
Single Locus ◽  
Genetic Systems ◽  
Infinite Set ◽  
Globally Stable ◽  
Mutation And Selection

AbstractThis paper considers the behaviour of infinite haploid genetic populations under the influence of mutation and selection depending on a single locus. Under wide conditions the Perron–Frobenius theory of non-negative matrices and its generalization by Vere-Jones are used to prove that there is a single globally stable state of the population when there is a finite or, under more restrictive conditions, an infinite set of possible alleles.

Global stability of genetic systems governed by mutation and selection. II

1977 ◽  
Vol 81 (3) ◽  
pp. 435-441 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Global Stability ◽  
Evolutionary Theory ◽  
Infinite Number ◽  
Infinite Sequence ◽  
Stable State ◽  
Single Locus ◽  
Stationary States ◽  
Genetic Population ◽  
Nearest Neighbours ◽  
Mutation And Selection

AbstractAn infinite genetic population of haploid particles is considered in which selection is controlled by a single locus at which there are an infinite number of possible alleles. These alleles are arranged in an infinite sequence and mutation occurs only to nearest neighbours. This is the ‘ladder model’ of Ohta and Kimura which was put forward as a possible explanation of the distributions of electromorphs in electrophoretic observations. Following an earlier paper, conditions are obtained on the selection coefficients which ensure that a stationary stable state exists. One such model is solved explicitly. The problem, important in evolutionary theory, of the rate of approach to such stationary states starting from some other state, is also discussed briefly.

Global stability of an SEI model for plant diseases

2016 ◽  
Vol 66 (1) ◽  
Author(s):  
Yuming Chen ◽  
Junyuan Yang
Keyword(s):  
Global Stability ◽  
Epidemic Model ◽  
Total Population ◽  
Endemic Equilibrium ◽  
Plant Diseases ◽  
Globally Stable ◽  
Sei Model

AbstractWe propose an SEI epidemic model for plant diseases, which incorporates disease latency, disease-caused removal, and constant recruitment in both susceptible and exposed classes. Because of the recruitment and disease-caused removal, the total population is varying. It is shown that the model only has an endemic equilibrium and the equilibrium is globally stable.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

2008 ◽  
Vol 01 (04) ◽  
pp. 503-520 ◽  
Author(s):  
ZHIQI LU ◽  
JINGJING WU
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Response ◽  
Qualitative Properties ◽  
Chemostat Model ◽  
Stability And Instability ◽  
Discrete Delays ◽  
Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

scholarly journals Global Stability of Vector-Host Disease with Variable Population Size

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Saeed Islam ◽  
Sher Afzal Khan ◽  
Gul Zaman
Keyword(s):  
Global Stability ◽  
Population Size ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Host Disease ◽  
Variable Population ◽  
Globally Stable ◽  
Disease Free ◽  
Disease Persistence ◽  
Variable Population Size

The paper presents the vector-host disease with a variability in population. We assume, the disease is fatal and for some cases the infected individuals become susceptible. We first show the local and global stability of the disease-free equilibrium, for the case when, the disease free-equilibrium of the model is both locally as well as globally stable. For , the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for . Numerical results are presented for the justifications of theoratical results.

scholarly journals On Using Curvature to Demonstrate Stability

2008 ◽  
Vol 2008 ◽  
pp. 1-7
Author(s):  
C. Connell McCluskey
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Large Amplitude ◽  
Minimum Distance ◽  
New Approach ◽  
Sudden Appearance ◽  
Globally Stable ◽  
Rule Out

A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.

On the properties of bilinear models for the balance between genetic mutation and selection

1977 ◽  
Vol 81 (3) ◽  
pp. 443-453 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Final Section ◽  
Stable Equilibrium ◽  
Genetic Mutation ◽  
Sufficient Condition ◽  
Bilinear Models ◽  
Selective Neutrality ◽  
Special Cases ◽  
Existence And Stability ◽  
Globally Stable ◽  
Mutation And Selection

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.

Global asymptotic stability for a SEI reaction–diffusion model of infectious diseases with immigration

2018 ◽  
Vol 11 (03) ◽  
pp. 1850044
Author(s):  
Salem Abdelmalek ◽  
Samir Bendoukha
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Reaction Diffusion ◽  
Stability Of Solutions ◽  
Reaction Diffusion Model ◽  
Globally Stable

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

Map dynamics versus dynamics of associated delay reaction–diffusion equations with a Neumann condition

2010 ◽  
Vol 466 (2122) ◽  
pp. 2955-2973 ◽  
Author(s):  
Taishan Yi ◽  
Xingfu Zou
Keyword(s):  
Steady State ◽  
Global Stability ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Neumann Condition ◽  
Main Concern ◽  
Homogeneous Neumann Boundary Condition ◽  
Globally Stable

In this paper, we consider a class of delay reaction–diffusion equations (DRDEs) with a parameter ε >0. A homogeneous Neumann boundary condition and non-negative initial functions are posed to the equation. By letting , such an equation is formally reduced to a scalar difference equation (or map dynamical system). The main concern is the relation of the absolute (or delay-independent) global stability of a steady state of the equation and the dynamics of the nonlinear map in the equation. By employing the idea of attracting intervals for solution semiflows of the DRDEs, we prove that the globally stable dynamics of the map indeed ensures the delay-independent global stability of a constant steady state of the DRDEs. We also give a counterexample to show that the delay-independent global stability of DRDEs cannot guarantee the globally stable dynamics of the map. Finally, we apply the abstract results to the diffusive delay Nicholson blowfly equation and the diffusive Mackey–Glass haematopoiesis equation. The resulting criteria for both model equations are amazingly simple and are optimal in some sense (although there is no existing result to compare with for the latter).

scholarly journals Best experienced payoff dynamics and cooperation in the centipede game

2019 ◽  
Vol 14 (4) ◽  
pp. 1347-1385 ◽  
Author(s):  
William H. Sandholm ◽  
Segismundo S. Izquierdo ◽  
Luis R. Izquierdo
Keyword(s):  
Stable State ◽  
Fixed Number ◽  
Cooperative Play ◽  
Game Dynamics ◽  
Testing Strategies ◽  
Study Population ◽  
Population Game ◽  
Globally Stable

We study population game dynamics under which each revising agent tests each of his strategies a fixed number of times, with each play of each strategy being against a newly drawn opponent, and chooses the strategy whose total payoff was highest. In the centipede game, these best experienced payoff dynamics lead to cooperative play. When strategies are tested once, play at the almost globally stable state is concentrated on the last few nodes of the game, with the proportions of agents playing each strategy being largely independent of the length of the game. Testing strategies many times leads to cyclical play.

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