scholarly journals Pairwise interact-and-imitate dynamics

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ennio Bilancini ◽  
Leonardo Boncinelli ◽  
Nicola Campigotto

AbstractThis paper introduces and studies a class of evolutionary dynamics—pairwise interact-and-imitate dynamics (PIID)—in which agents are matched in pairs, engage in a symmetric game, and imitate the opponent with a probability that depends on the difference in their payoffs. We provide a condition on the underlying game, named supremacy, and show that the population state in which all agents play the supreme strategy is globally asymptotically stable. We extend the framework to allow for payoff uncertainty, and check the robustness of our results to the introduction of some heterogeneity in the revision protocol followed by agents. Finally, we show that PIID can allow the survival of strictly dominated strategies, leads to the emergence of inefficient conventions in social dilemmas, and makes assortment ineffective in promoting cooperation.

2021 ◽  
Author(s):  
Ennio Bilancini ◽  
Leonardo Boncinelli ◽  
Nicola Campigotto

Abstract This paper introduces and studies a class of evolutionary dynamics --- Pairwise Interact-and-Imitate Dynamics (PIID) --- in which agents are matched in pairs, engage in a symmetric game, and imitate the opponent with a probability that depends on the difference in their payoffs. We provide a condition on the underlying game, named supremacy, and show that the population state in which all agents play the supreme strategy is globally asymptotically stable. We extend the framework to allow for payoff uncertainty, and check the robustness of our results to the introduction of some heterogeneity in the revision protocol followed by agents. Finally, we show that PIID can allow the survival of strictly dominated strategies, leads to the emergence of inefficient conventions in social dilemmas, and makes assortment ineffective in promoting cooperation.


2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.


2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Meirong Xu ◽  
Yuzhen Wang

The difference equationyn+1−yn=−αyn+∑j=1mβje−γjyn−kjis studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.


2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Awad A. Bakery

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk),  n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1,  x0to be bounded,α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm=k=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.


2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chenquan Gan ◽  
Xiaofan Yang ◽  
Wanping Liu

This paper aims to investigate the global stability of negative solutions of the difference equationxn+1=(α+βxn-k)/(γ+xn),n=0,1,2,…, where the initial conditionsx-k,…,x0∈-∞,0,kis a positive integer, and the parametersβ,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.


2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lin-Xia Hu ◽  
Xiu-Mei Jia

The main goal of this paper is to investigate the global asymptotic behavior of the difference equationxn+1=β1xn/A1+yn,yn+1=β2xn+γ2yn/xn+yn,n=0,1,2,…withβ1,β2,γ2,A1∈(0,∞)and the initial value(x0,y0)∈[0,∞)×[0,∞)such thatx0+y0≠0. The major conclusion shows that, in the case whereγ2<β2, if the unique positive equilibrium(x-,y-)exists, then it is globally asymptotically stable.


2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ihsan Ullah ◽  
Saeed Ahmad ◽  
Qasem Al-Mdallal ◽  
Zareen A. Khan ◽  
Hasib Khan ◽  
...  

Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0}<1$ ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.


2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.


2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.


Sign in / Sign up

Export Citation Format

Share Document