Positive Orthant Research Articles

Localization of compact invariant sets and global stability in analysis of one tumor growth model

In this paper, we study some features of global behavior of the four‐dimensional superficial bladder cancer model with Bacillus Calmette‐Guérin (BCG) immunotherapy described by Bunimovich‐Mendrazitsky et al. in 2007 with the help of localization analysis of its compact invariant sets. Its dynamics is defined by the BCG treatment and by densities of three cells populations: effector cells, tumor infected cells by BCG, and tumor uninfected cells. We find upper bounds for ultimate dynamics of the whole state vector in the positive orthant and also under condition that there are no uninfected tumor cells. Further, we prove the existence of the bounded positively invariant domain in both of these two situations. Finally, by using these assertions, we derive our main result: sufficient conditions of global asymptotic stability of the tumor‐free equilibrium point in the positive orthant. Copyright © 2013 John Wiley & Sons, Ltd.

On the Stability of Critical Point for Positive Systems and Its Applications to Biological Systems

The coexistence and extinction of species are important concepts for biological systems and can be distinguished by an investigation of stability. When determining local stability of nonlinear systems, Lyapunov indirect method based on the Jacobian linearization has been widely employed due to its simplicity. Despite such popularity, it is not applicable to singular systems whose Jacobian has at least one eigenvalue that is equal to zero. In such singular cases, an appropriate Lyapunov function should be sought to determine the stability of systems, which is rather difficult and quite involved. In this paper, we seek for a simple criterion to determine stability of the equilibrium that is located at the boundary of the positive orthant, when one of eigenvalues of the Jacobian is zero. The goal of the paper is to present a generalized condition for the equilibrium to attract all trajectories that starting from initial condition in the positive orthant and near the equilibrium. Unlike the Lyapunov direct method, the proposed method requires just a simple algebraic computation for checking the stability of the critical point. Our approach is applied to various biological systems to show the effectiveness of the proposed method.

A class of degenerate stochastic differential equations with non-Lipschitz coefficients

We obtain sufficient condition for SDEs to evolve in the positive orthant. We use arguments based on comparison theorems for SDEs to achieve this. As an application we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.

Density games

The basic idea of evolutionary game theory is that payoff determines reproductive rate. Successful individuals have a higher payoff and produce more offspring. But in evolutionary and ecological situations there is not only reproductive rate but also carrying capacity. Individuals may differ in their exposure to density limiting effects. Here we explore an alternative approach to evolutionary game theory by assuming that the payoff from the game determines the carrying capacity of individual phenotypes. Successful strategies are less affected by density limitation (crowding) and reach higher equilibrium abundance. We demonstrate similarities and differences between our framework and the standard replicator equation. Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation. Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks. For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.

Bayesian Dirichlet mixture model for multivariate extremes: A re-parametrization

The probabilistic framework of extreme value theory is well-known: the dependence structure of large events is characterized by an angular measure on the positive orthant of the unit sphere. The family of these angular measures is non-parametric by nature. Nonetheless, any angular measure may be approached arbitrarily well by a mixture of Dirichlet distributions. The semi-parametric Dirichlet mixture model for angular measures is theoretically valid in arbitrary dimension, but the original parametrization is subject to a moment constraint making Bayesian inference very challenging in dimension greater than three. A new unconstrained parametrization is proposed. This allows for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity of the Markov chain are verified and the algorithm is tested up to dimension five. In this non identifiable setting, convergence monitoring is performed by integrating the sampled angular densities against Dirichlet test functions.

The power normal distribution - modelling, simulation and applications

In the following work we model a random sample having support on the positive orthant using the power normal (PN) distribution. The PN distribution is unimodal, obtained from inverse Box-Cox (BC) transformation and there exists no close form expression for moment estimation. Hence, we simulate from the distribution to estimate its moments and also implement a naive bootstrap, on the random sample prior to the modelling, to account for the uncertainty in estimating the BC transformation parameters. In the real world examples we consider, we find the distribution efficient in modelling the tails of the underlying unknown process. Due to the minimalistic assumptions with PN distribution it can be used to model data observed both on the nominal scale and interval scale. In our simulation studies, wherein we consider samples from a known distribution with support defined on the positive orthant only, we observe that the PN distribution performs better than Kernel density estimation methods based on Gaussian assumptions for sampling and density estimation.

Persistence and Permanence of Mass-Action and Power-Law Dynamical Systems

Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering and are often used to describe the dynamics in interaction networks. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to permanent systems, even if the reaction rate parameters vary in time (to allow for the influence of external signals). These results also apply to power-law systems and other nonlinear dynamical systems. In addition, ideas behind these results allow us to prove the global attractor conjecture for three-species systems.

Global dynamics of the Kirschner–Panetta model for the tumor immunotherapy

In this paper we examine the global dynamics of the Kirschner–Panetta model describing the tumor immunotherapy. We give upper and lower ultimate bounds for densities of cell populations involved in this model. We demonstrate for this dynamics that there is a positively invariant polytope in the positive orthant. We present sufficient conditions on model parameters and treatment parameters under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. We compare our results with Kirschner–Tsygvintsev results and concern biological implications of our assertions.

Fitting Kent models to compositional data with small concentration

Compositional data can be transformed to directional data by the square root transformation and then modelled by using the Kent distribution. The current approach for estimating the parameters in the Kent model for compositional data relies on a large concentration assumption which assumes that the majority of the transformed data is not distributed too close to the boundaries of the positive orthant. When the data is distributed close to the boundaries with large variance significant folding may result. To treat this case we propose new estimators of the parameters derived based on the actual folded Kent distribution which are obtained via the EM algorithm. We show that these new estimators significantly reduce the bias in the current estimators when both the sample size and amount of folding is moderately large. We also propose using a saddlepoint density approximation for the Kent distribution normalising constant in order to more accurately estimate the shape parameters when the concentration is small or only moderately large.

Polyhedral combinatorics of UPGMA cones

Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the spherical volumes of the UPGMA cones for small n.

The underlying linear dynamics of some positive polynomial systems

The conditions of structural dynamical similarity of two special classes of positive polynomial nonlinear systems, the class of quasi-polynomial systems (Brenig (1988) [1]) and that of reaction kinetic networks with mass action kinetics (Horn and Jackson (1972) [2]) are investigated in this Letter. It is shown that both system classes have an underlying reduced linear dynamics. By applying the theory of X-factorable systems (Samardzija et al. (1989) [9]), it can be shown that the reduced linear dynamics is qualitatively similar to the original one within the positive orthant when the original nonlinear system has a unique positive equilibrium point.

Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations

We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose

Let \documentclass[12pt]{minimal}\begin{document}${\cal E}^{\prime }$\end{document}E′ denote the set of non-normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). We show that the set of stochastic local operations and classical communications (SLOCC) equivalence classes of states in \documentclass[12pt]{minimal}\begin{document}${\cal E}^{\prime }$\end{document}E′, equipped with the quotient topology, is homeomorphic to the quotient R/A5 of the open rectangular box R⊂ R4 by an action of the alternating group A5. We construct an explicit map \documentclass[12pt]{minimal}\begin{document}$\omega :\Omega \rightarrow {\cal E}^{\prime }$\end{document}ω:Ω→E′, where Ω is the open positive orthant in R4, whose image ω(Ω) meets every SLOCC equivalence class \documentclass[12pt]{minimal}\begin{document}$E\subseteq {\cal E}^{\prime }$\end{document}E⊆E′. Although the intersection ω(Ω) ∩ E is not necessarily a singleton set, it is always a finite set of cardinality at most 60. By abuse of language, we say that any state in ω(Ω) ∩ E is a canonical form of any ρ ∈ E. In particular, we show that all checkerboard PPT entangled states can be parametrized up to SLOCC equivalence by only two real parameters. We also summarize the known results on two-qutrit extreme PPT states and edge states, and examine which other interesting properties they may have. Thus we find the first examples of extreme PPT states whose rank is different from the rank of its partial transpose.

Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

Stability analysis of quasi-polynomial dynamical systems with applications to biological network models

We study asymptotic stability properties of a class of quasi-polynomial dynamical systems. This class of nonlinear systems is a special class of interconnected systems arising in several biochemical and biological system applications and can be represented using quasi-polynomial dynamical systems. It is known that a special class of such systems can be embedded into a higher dimensional space and cast in Lotka–Volterra canonical form. We characterize a class of quasi-polynomial dynamical systems with asymptotic stability properties for all initial conditions in the positive orthant. The key advantage of the proposed method is that it is algebraic such that asymptotic stability conditions can be derived in terms of (as they are usually in biological network models) parameters of the system. We apply our results to parameterized models of three different biological systems: the generalized mass action (GMA) model, an oscillating biochemical network, and a reduced order model of the glycolysis pathway, and show that one can apply our proposed method to verify asymptotic stability for each case in terms of underlying parameters.

An Index Theory for Positive Dynamical Systems with an Application to Lotka–Voterra Systems

In this paper we develop an index theory of positive dynamical systems in planar case. We present a formula of index sum of isolated zeros a positive dynamical system on the positive orthant of R2. As an application, we present some sufficient conditions on existence of zeros of Lotka–Voterra systems in the positive orthant of R2.

Bounds on exponential moments of hitting times for reflected processes on the positive orthant

We first consider a multi-dimensional reflected fractional Brownian motion process on the positive orthant with the Hurst parameter H∈(0,1). In particular, when H>1/2, this model serves to approximate fluid stochastic network models fed by a big number of heavy tailed ON/OFF sources in heavy traffic. Assuming the initial state lies outside some compact set, we establish that the exponential moment of the first hitting time to the compact set has a lower bound with an exponential growth rate in terms of the magnitude of the initial state. We extend this result to the case for reflected processes driven by a class of stable Lévy motions, which arise as approximations to cumulative network traffic over a time period. For the case of H=1/2, under a natural stability condition on the reflection directions and drift vector, we obtain a matching upper bound on exponential moments of hitting times, which grows at an exponential rate in terms of the initial condition of the process. We also show that such an upper bound is valid for reflected processes driven by general light-tailed Lévy processes.

A Maximum Principle for the Stability Analysis of Positive Bilinear Control Systems with Applications to Positive Linear Switched Systems

We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron--Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.

Optimal arbitrage under model uncertainty

In an equity market model with “Knightian” uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton–Jacobi–Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.

Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone $${\mathcal{C}}$$ of all completely positive symmetric n × n matrices (which can be factorized into $${FF^\top}$$ , where F is a rectangular matrix with no negative entry), and its dual cone $${\mathcal{C}^*}$$ , which coincides with the cone of all copositive matrices (those which generate a quadratic form taking no negative value over the positive orthant). We provide structural algebraic properties of these cones, and numerous (counter-)examples which demonstrate that many relations familiar from semidefinite optimization may fail in the copositive context, illustrating the transition from polynomial-time to NP-hard worst-case behaviour. In course of this development we also present a systematic construction principle for non-attainability phenomena, which apparently has not been noted before in an explicit way. Last but not least, also seemingly for the first time, a somehow systematic clustering of the vast and scattered literature is attempted in this paper.