Local stability under evolutionary game dynamics

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

End-to-end rate-based congestion control with random loss: convergence and stability

A general system of difference equations is presented for multispecies communities with density dependent population growth and delayed maturity. Interspecific competition, mutualism, predation, commensalism, and amensalism are accommodated. A sufficient condition for the local asymptotic stability of a coexistence equilibrium in this system is then proven. Using this system, the generalisation of the Beverton-Holt and Leslie-Gower models of competition to multispecies systems with possible maturation delays is presented and shown to yield interesting stability properties. The stability of coexistence depends on the relative abundances of the species at the unique interior equilibrium. A sufficient condition for local stability is derived that only requires intraspecific competition to outweigh interspecific competition. The condition does not depend on maturation delays. The derived stability properties are used to develop a novel estimation approach for the coefficients of interspecific competition. This approach finds an optimal configuration given two conjectures. First, coexisting species strive to outcompete competitors. Second, persisting species are more likely in stable systems with strong dampening of perturbations and high ecological resilience. The optimal solution is compared to estimates of niche overlap using an empirical example of malaria mosquito vectors with delayed maturity in the Anopheles gambiae sensu lato species complex.

Evolutionary game dynamics in finite size populations can be described by a fitness-dependent Wright-Fisher process. We consider symmetric 2×2 games in a well-mixed population. In our model, two parameters to describe the level of player's rationality and noise intensity in environment are introduced. In contrast with the fixation probability method that used in a noiseless case, the introducing of the noise intensity parameter makes the process an ergodic Markov process and based on the limit distribution of the process, we can analysis the evolutionary stable strategy (ESS) of the games. We illustrate the effects of the two parameters on the ESS of games using the Prisoner's dilemma games (PDG) and the snowdrift games (SG). We also compare the ESS of our model with that of the replicator dynamics in infinite size populations. The results are determined by simulation experiments.

In this paper, a delayed predator-prey model for worms inside a computer system is proposed. Local stability and Hopf bifurcation of the model is analyzed by regarding the time delay as the bifurcation parameter. Sufficient conditions for local stability and occurrence of the Hopf bifurcation are obtained. Numerical simulations are presented in order to support the obtained theoretical results.

We analyze some local stability and Hopf bifurcation properties of the Adaptive Virtual Queue (AVQ) and the Exponential-RED (E-RED) queue policy. Using a fluid model for flow control, coupled with these queue management policies, we first determine the necessary and sufficient conditions for local stability. Then, for both queue policies, we show that the underlying coupled system undergoes a Hopf bifurcation. Further, for the E-RED policy, using Poincaré normal forms and the center manifold theory, we outline an analytical basis to determine the asymptotic orbital stability of the emerging limit cycles. Finally, using packet-level simulations we highlight that the existence of limit cycles leads to a loss in link utilization. Such limit cycles can hurt network performance and should be avoided. Some stability charts, and numerical computations, complement our analysis.

Based on a result of stability for linear time varying systems, a sufficient condition for global stability that does not use Lyapunov theory has been established for a class of discrete Takagi–Sugeno fuzzy systems. The condition involves testing the norms of products of the matrices of the consequences. The approach seeks to determine if at an instant of time the system becomes a contraction mapping. Because the computation burden associated with these tests may become prohibitive, a necessary and sufficient condition for local stability is derived. In a way, this decreases the computational burden and allows to customize the test. A two-step iterative stabilization algorithm is then introduced. The problem involves norm minimization and a posteriori stability test. Numerical examples are presented to demonstrate the applicability of the approach.

Evolutionary game dynamics with non-uniform interaction rates

Evolutionary game dynamics with impulsive effects

Based on the two-party Game model between the government and the supplier, or between suppliers, a three-group Game income model of government, suppliers and consumers is constructed; the evolutionary dynamics of government incentives, low-price supply from suppliers, and consumer purchases are analyzed. ESS (Evolutionary Stable Strategy) of the three-group Evolutionary Game under different parameter values is solved; and the numerical simulation is used to verify the parameter conditions for the stability of the main subject strategy of the three-group Game. The research results show that the three groups of government, enterprises and consumers differ in their willingness of participation in the supply of public goods, and enterprises are more sensitive to government incentives, consumers more rational in purchasing public goods, and the government incentive strategies more reasonable.

Evolutionary game dynamics in a finite asymmetric two-deme population and emergence of cooperation

The stability of nonlinear autonomous systems with nonlinearity representable in polynomial form is investigated. For the case of locally stable systems the following theorem is presented: A sufficient condition for local stability of the system x˙ = X(x) is the existence of a definite function v = φ(x) such that dv/dt = θ(x)g[ξ(x)], where θ(x) is a semidefinite function not identically equal to zero on a solution of x˙ = X(x), g(x) is such that g(0) = 0 and sign g(u) ≠ sign g(−u), and ξ(x) = 0 is a closed surface. A procedure for constructing Liapunov functions based upon the use of a generating v-function is developed. Such a generating v-function may have the form: v(x)=x′A(x)x where A(x) = {aij(xi, xj)}, and aij = aji. The coefficients aij (xi, xj) can be computed in order to obtain dv/dt of the wanted form. Particular emphasis is given to the case of systems with limit cycles and, as an example, the limit cycle of the van der Pol equation is identified with good approximation. It is also analytically proved that outside a closed algebraic curve, circumscribing the limit cycle, the system is asymptotically stable.

A within host viral infection model with both virus to cell and cell to cell transmissions and three discrete delays using reverse transcriptase inhibitors is investigated. The effect of time delay on stability of the equilibria of the infection model has been studied. By choosing the immune delay as the bifurcation parameter, we establish the sufficient condition for local stability of chronic infection equilibrium, Hopf bifurcation occurs when the immune delay passes a critical value. The global stability analysis of the model is carried out in terms of the basic reproduction number $$R_0$$ . If $$R_0 < 1$$ , the infection-free (semi-trivial) equilibrium is globally stable for both ODE and DDE model; if $$R_0>1$$ , the chronic infection (positive) equilibrium exists and is globally stable for both ODE and DDE model. Numerical simulations are obtained to validate our analytical findings by varying the system parameter.

Is a Unique Cournot Equilibrium Locally Stable?

An observer design problem for a class of nonlinear time-delay systems subject to white Gaussian noise is studied. The problem addressed is the design of a full-state observer based on the extended Kalman filter (EKF), in which two terms are added to the Riccati differential equation. One of those makes the system robust to noise influences, and the other one is utilized for dealing with the delayed state. A Lyapunov-Krasovskii functional is used for giving sufficient conditions for local stability of the reconstruction error dynamics. A hybrid chaotic cryptosystem procedure based on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -shift cipher is combined with synchronization based on the EKF for developing a secure communication system. A semiconductor laser with optical feedback is employed as a chaotic signal generator in which a full-state estimation via the EKF is achieved.