Yamabe systems and optimal partitions on manifolds with symmetries

<p style='text-indent:20px;'>We prove the existence of regular optimal <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant partitions, with an arbitrary number <inline-formula><tex-math id="M2">\begin{document}$ \ell\geq 2 $\end{document}</tex-math></inline-formula> of components, for the Yamabe equation on a closed Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a compact group of isometries of <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula> with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of <inline-formula><tex-math id="M6">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> equations, related to the Yamabe equation. We show that this system has a least energy <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to <inline-formula><tex-math id="M8">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>, giving rise to an optimal partition. For <inline-formula><tex-math id="M9">\begin{document}$ \ell = 2 $\end{document}</tex-math></inline-formula> the optimal partition obtained yields a least energy sign-changing <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution to the Yamabe equation with precisely two nodal domains.</p>

Dynamic Analysis of a Population Competition Model with Disease in One Species and Group Defense in Another Species

In this paper, we study the dynamics of an ecoepidemic competition system where the individuals of one population gather together in herds with a defensive strategy, showing social behavior, while another predator population is subject to a transmissible disease and behaves individually. By analyzing the existence and stability of the equilibria of the system, we find that the relatively isolated population can be eradicated, or the population with group defense can live alone eventually under some constraints. Infected individuals end up in two possible situations. In the first case, the disease is eventually eliminated, meaning that only healthy and group-defense individuals in the system can survive. In the other case, the spread of the disease is controlled and eventually all three individuals can coexist. We also conduct a correlation analysis using competition parameter and recovery rate of disease as birfurcation parameters in order to study the transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. The long-term dynamics of the boundary and interior equilibria are demonstrated by numerical simulations.

The Dynamic Rent-Seeking Games with Policymaker Cost and Competition Intensity

In this paper, a dynamic rent-seeking game incorporating policymaker cost and competition intensity is considered. On the basis of the political environment and rent-seekers with incomplete information set, the locally asymptotic stability of Nash equilibrium is proved. The competition intensity and policymaker cost could enlarge the stability region of Nash equilibrium. The higher the competition intensity is, the more the opponent’s expenditure reduces the player’s success probability, which is beneficial to the maintenance of Nash equilibrium. The higher the policymaker cost is, the less easily both players succeed and the more stable the rent-seeking market is. As the competition parameter decreases or the expenditure parameter increases, there will be chaos in a rent-seeking market. Chaos control is in order to stabilize the equilibrium of the rent-seeking game.

Partially Disordered State in the Frustrated Heisenberg Ferrimagnet

A model for the description of two-subsystem Heisenberg ferrimagnet with frustrated intersubsystem exchange and competition between exchange interactions in a subsystem is proposed. The conditions of the existence of noncollinear Yafet-Kittel state and partially ordered magnetic structure are investigated. The phase diagram of competition parameter vs temperature is obtained in the mean field approximation. The peculiarities of the succesive magnetic phase transitions are considered.

COMPETING DYNAMICAL BEHAVIOR OF THE ISING MODEL IN ONE DIMENSION

We have studied the behavior of the one-dimensional ferromagnetic Ising model in contact with a heat bath and subject to an external source of energy. The contact with the heat bath is simulated by a process of Glauber type, while the continuous flux of energy into the system by a process of Kawasaki type. We show, within the dynamical pair approximation that the phase diagram exhibits a line of continuous nonequilibrium transitions between the paramagnetic and antiferromagnetic phases. However, detailed Monte Carlo simulations on the same model show clearly that the only stationary state is the paramagnetic one, whatever is the value of the competition parameter between the Glauber and Kawasaki dynamics.