Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast and the diffusion (white noise) term is small. The system is modeled by $$dX^{\epsilon,\delta}(t)=f(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dt+\sqrt{\delta}\sigma(X^{\epsilon,\delta}(t), \alpha^\epsilon(t))dW(t) , \ X^\epsilon(0)=x,$$ where $\alpha^\epsilon(t)$ is a finite state space Markov chain with irreducible generator $Q=(q_{ij})$. The relative changing rates of the switching and the diffusion are highlighted by the two small parameters $\epsilon$ and $\delta$. We associate to the system the averaged ODE $$d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x,$$ where $\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)\nu_i$ and $(\nu_1,\dots,\nu_{m_0})$ is the unique invariant probability measure of the Markov chain with generator $Q$. Suppose that for each pair $(\epsilon,\delta)$ of parameters, the process has an invariant probability measure $\mu^{\epsilon,\delta}$, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure $\mu^0$ for the averaged equation. We are able to prove that if $\bar f$ has finitely many unstable or hyperbolic fixed points, then $\mu^{\epsilon,\delta}$ converges weakly to $\mu^0$ as $\epsilon\to 0$ and $\delta \to 0$. Our results generalize to the setting of state-dependent switching $$\mathbb{P}\{\alpha^\epsilon(t+\Delta)=j~|~\alpha^\epsilon=i, X^{\epsilon,\delta}(s),\alpha^\epsilon(s), s\leq t\}=q_{ij}(X^{\epsilon,\delta}(t))\Delta+o(\Delta),~~ i\neq j$$ as long as the generator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible for all $x\in\mathbb{R}^d$. We conclude our analysis by studying a predator-prey model.Comment: 40 page

Intertemporal Risk Management Decisions of Farmers under Preference, Market, and Policy Dynamics

This paper adapts a generalized expected utility (GEU) maximization model (Epstein and Zin, 1989 and 1991) to examine the intertemporal risk management of wheat producers in the Pacific Northwest. Optimization results based on simulated data indicate the feasibility of the GEU optimization as a modeling framework. It further extends the GEU model by incorporating a welfare measure, the certainty equivalent, to investigate the impacts of U.S. government programs and market institutions on farmers' risk management decisions and welfare. A comparison between the GEU and other expected utility models further implies GEU has the advantage of specifying farmers' intertemporal preferences separately and completely. Impact analysis results imply that farmers' optimal hedging is sensitive to changes in the preferences and the effects of these preference changes are intertwined. Target price and loan rate levels, offered by certain government payment programs, can lead to the substitution of government programs for hedging. The evaluation of current risk management tools shows both crop insurance and government payments can improve farmers' welfare significantly. Government payment programs have a greater effect on farmers' welfare than crop insurance and crop insurance outperforms hedging.generalized expected utility, risk management, multi-period production, dynamic optimization, intertemporal preference, market institution, government payments, Risk and Uncertainty, Q14, D9, C61,

Dynamical Behavior of a stochastic SIRS epidemic model

In this paper we study the Kernack - MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold $\lambda$ which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of $\lambda$, the system can globally tend towards an endemic case or a disease free case. The aim of this work is also to describe completely the omega-limit set of all positive solutions to the model. Moreover, the attraction of the omega-limit set and the stationary distribution of solutions will be pointed out.Comment: 16 page

Robustness of Network of Networks with Interdependent and Interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (J.X. Gao et. al., Nature Physics, 2012) and only for connectivity coupling (E.A. Leicht and R.M. D Souza, arxiv:0907.0894). The case of network of n networks with both interdependent and interconnected links is more complicated, and also more closely to real-life coupled network systems. Here we develop a framework to study analytically and numerically the robustness of this system. For the case of starlike network of n ER networks, we find that the system undergoes from second order to first order phase transition as coupling strength q increases. We find that increasing intra-connectivity links or inter-connectivity links can increase the robustness of the system, while the interdependency links decrease its robustness. Especially when q=1, we find exact analytical solutions of the giant component and the first order transition point. Understanding the robustness of network of networks with interdependent and interconnected links is helpful to design resilient infrastructures

Cloning and expression of porcine β1,4 N-acetylgalactosaminyl transferase encoding a new xenoreactive antigen

Xenograft rejection of pigs organs with an engineered mutation in the GGTA-1 gene (GTKO) remains a predominantly antibody mediated process which is directed to a variety of non-Gal protein and carbohydrate antigens. We previously used an expression library screening strategy to identify six porcine endothelial cell cDNAs which encode pig antigens that bind to IgG induced after pig-to-primate cardiac xenotransplantation. One of these gene products was a glycosyltransferase with homology to the bovine β1,4 N-acetylgalactosaminyltransferase (B4GALNT2). We now characterize the porcine B4GALNT2 gene sequence, genomic organization, expression, and functional significance

Comment on ``Evidence for Anisotropic State of Two-Dimensional Electrons in High Landau Levels''

In a recent letter M. Lilly et al [PRL 82, 394 (1999)] have shown that a highly anisotropic state can arise in certain two dimensional electron systems. In the large square samples studied, resistances measured in the two perpendicular directions are found to have a ratio that may be 60 or larger at low temperature and at certain magnetic fields. In Hall bar measurements, the anisotropy ratio is found to be much smaller (roughly 5). In this comment we resolve this discrepancy by noting that the anisotropy of the underlying sheet resistivities is correctly represented by Hall bar resistance measurements but shows up exponentially enhanced in resistance measurements on square samples due to simple geometric effects. We note, however, that the origin of this underlying resistivity anisotropy remains unknown, and is not addressed here.Comment: 1 page, minor calculational error repaire