Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions

In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation.

Heterogeneity, turn-over rate and karyotype space shape susceptibility to missegregation-induced extinction

The incidence of somatic copy number alterations (SCNAs) per base pair of the genome is orders of magnitudes larger than that of point mutations. This makes SCNAs phenotypically effective. One mitotic event stands out in its potential to significantly change a cell's SCNA burden -- a chromosome missegregation. We have presented a general deterministic framework for modeling whole chromosome missegregations and use it to evaluate the possibility of missegregation-induced population extinction (MIE). The model predicts critical curves that separate viable from non-viable populations as a function of their turnover- and mis-segregation rates. Missegregation- and turnover rates estimated for nine cancer types are then compared to these predictions for various biological assumptions. The assumption of heterogeneous missegregation rates within a tumor was sufficient to explain the observed data. By contrast, when assuming constant mis-segregation rates, several cancers were located in regions predicted as unviable. Intra-tumor heterogeneity, including heterogeneity in mis-segregation rates, increases as tumors progress. Our predictions suggest that this intra-tumor heterogeneity hinders the chance of success of therapies aimed at MIE.

Synchronization and Global Dynamics of a Cournot Model with Nonlinear Demand and R&D Spillovers

In this paper, a dynamical Cournot model with nonlinear demand and R&D spillovers is established. The system is symmetric when the duopoly firms have same economic environments, and it is proved that both the diagonal and the coordinate axes are the one-dimensional invariant manifolds of system. The results show that Milnor attractor of system can be found through calculating the transverse Lyapunov exponents. The synchronization phenomenon is verified through basins of attraction. The effects of adjusting speed and R&D spillovers on the dynamical behaviors of the system are discussed. The topological structures of basins of attraction are analyzed through critical curves, and the evolution process of “holes” in the feasible region is numerically simulated. In addition, various global bifurcation behaviors, such as two kinds of contact bifurcation and the blowout bifurcation, are shown.

Gravitational Lensing by a Massive Object in a Dark Matter Halo. I. Critical Curves and Caustics

We study the gravitational lensing properties of a massive object in a dark matter halo, concentrating on the critical curves and caustics of the combined lens. We model the system in the simplest approximation by a point mass embedded in a spherical Navarro–Frenk–White density profile. The low number of parameters of such a model permits a systematic exploration of its parameter space. We present galleries of critical curves and caustics for different masses and positions of the point in the halo. We demonstrate the existence of a critical mass, above which the gravitational influence of the centrally positioned point is strong enough to eliminate the radial critical curve and caustic of the halo. In the point-mass parameter space we identify the boundaries at which critical-curve transitions and corresponding caustic metamorphoses occur. The number of transitions as a function of the position of the point is surprisingly high, ranging from three for higher masses to as many as eight for lower masses. On the caustics we identify the occurrence of six different types of caustic metamorphoses. We illustrate the peculiar properties of the single radial critical curve and caustic appearing in an additional unusual nonlocal metamorphosis for a critical mass positioned at the halo center. Although we construct the model primarily to study the lensing influence of individual galaxies in a galaxy cluster, it can also be used to study the lensing by dwarf satellite galaxies in the halo of a host galaxy, as well as (super)massive black holes at a general position in a galactic halo.

Phase transitions for spatially extended pinning

We consider a directed polymer of length N interacting with a linear interface. The monomers carry i.i.d. random charges $$(\omega _i)_{i=1}^N$$ ( ω i ) i = 1 N taking values in $${\mathbb {R}}$$ R with mean zero and variance one. Each monomer i contributes an energy $$(\beta \omega _i-h)\varphi (S_i)$$ ( β ω i - h ) φ ( S i ) to the interaction Hamiltonian, where $$S_i \in {\mathbb {Z}}$$ S i ∈ Z is the height of monomer i with respect to the interface, $$\varphi :\,{\mathbb {Z}}\rightarrow [0,\infty )$$ φ : Z → [ 0 , ∞ ) is the interaction potential, $$\beta \in [0,\infty )$$ β ∈ [ 0 , ∞ ) is the inverse temperature, and $$h \in {\mathbb {R}}$$ h ∈ R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $${\mathbb {Z}}$$ Z . We study both the quenched and the annealed free energy per monomer in the limit as $$N\rightarrow \infty $$ N → ∞ . We show that each exhibits a phase transition along a critical curve in the $$(\beta ,h)$$ ( β , h ) -plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $$\beta ,h \downarrow 0$$ β , h ↓ 0 in three different regimes for the tail exponent of $$\varphi $$ φ .

Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function

In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.

Multiplicity of positive solutions for (p,q)-Laplace equations with two parameters

We study the zero Dirichlet problem for the equation [Formula: see text] in a bounded domain [Formula: see text], with [Formula: see text]. We investigate the relation between two critical curves on the [Formula: see text]-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighborhoods of the point [Formula: see text], where [Formula: see text] is the first eigenfunction of the [Formula: see text]-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents [Formula: see text] and [Formula: see text].

Level lines of a polynomial in the plane

We propose a method for computing the position of all level lines of a real polynomial in the real plane. To do this, it is necessary to compute its critical points and critical curves, and then to compute critical values of the polynomial (there are finite number of them). Now finite number of critical levels and one representative of noncritical level corresponding to a value between two neighboring critical ones enough to compute. We propose a scheme for computing level lines based on polynomial computer algebra algorithms: Gröbner bases, primary ideal decomposition. Software for these computations are pointed out. Nontrivial examples are considered.