arXiv subjects

Jiaxin Jin

Publications and source records attributed to Jiaxin Jin.

29 records · Page 2Linked to original sources

arXiv2023

The structure of the moduli spaces of toric dynamical systems

We consider complex-balanced mass-action systems, or toric dynamical systems. They are remarkably stable polynomial dynamical systems arising from reaction networks seen as Euclidean embedded graphs. We study the moduli spaces of toric dynamical systems, called the toric locus: given a reaction network, we are interested in the topological structure of the set of parameters giving rise to toric dynamical systems. First we show that the complex-balanced equilibria depend continuously on the parameter values. Using this result, we prove that the toric locus of any toric dynamical system is connected. In particular, we emphasize its product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.

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arXiv2023

Weakly reversible deficiency one realizations of polynomial dynamical systems: an algorithmic perspective

Given a dynamical system with polynomial right-hand side, can it be generated by a reaction network that possesses certain properties? This question is important because some network properties may guarantee specific dynamical properties, such as existence or uniqueness of equilibria, persistence, permanence, or global stability. Here we focus on this problem in the context of weakly reversible deficiency one networks. In particular, we describe an algorithm for deciding if a polynomial dynamical system admits a weakly reversible deficiency one realization, and identifying one if it does exist. In addition, we show that weakly reversible deficiency one realizations can be partitioned into mutually exclusive Type I and Type II realizations, where Type I realizations guarantee existence and uniqueness of positive steady states, while Type II realizations are related to stoichiometric generators, and therefore to multistability.

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arXiv2023

Weakly reversible single linkage class realizations of polynomial dynamical systems: an algorithmic perspective

Systems of differential equations with polynomial right-hand sides are very common in applications. In particular, when restricted to the positive orthant, they appear naturally (according to the law of mass-action kinetics) in ecology, population dynamics, as models of biochemical interaction networks, and models of the spread of infectious diseases. On the other hand, their mathematical analysis is very challenging in general; in particular, it is very difficult to answer questions about the long-term dynamics of the variables (species) in the model, such as questions about persistence and extinction. Even if we restrict our attention to mass-action systems, these questions still remain challenging. On the other hand, if a polynomial dynamical system has a weakly reversible single linkage class ($W\!R^1$) realization, then its long-term dynamics is known to be remarkably robust: all the variables are persistent (i.e., no species goes extinct), irrespective of the values of the parameters in the model. Here we describe an algorithm for finding $W\!R^1$ realizations of polynomial dynamical systems, whenever such realizations exist.

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arXiv2022

Exponential Mixing of Vlasov equations under the effect of Gravity and Boundary

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in $L^\infty$. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in $L^\infty$. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere.

math.AP↗

arXiv2022

An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding $WR_0$ realizations of polynomial dynamical systems, whenever such realizations exist.

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arXiv2021

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversibly deficiency zero ($\textit{WR}_\textit{0}$) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, $WR_0$ systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

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arXiv2021

Damping of kinetic transport equation with diffuse boundary condition

We prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as $t^{-3}$ when the domain is any general strictly convex subset of $\mathbb{R}^3$ with the smooth boundary of the diffuse boundary condition. We prove the theorem by establishing a novel $L^1$-$L^\infty$ framework via stochastic cycles.

math.AP↗

arXiv2021

Single-Target Networks

We characterize the dynamics of all single-target networks under mass-action kinetics: either the system is (i) globally stable for all choice of rate constants (in fact, dynamically equivalent to a detailed-balanced system) or (ii) has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

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arXiv2020

Instability of boundary equilibrium for Reaction-Diffusion system of a complex-balanced reaction network

In this paper we study the local instability to the boundary equilibria and the local stability to the positive equilibria for some chemical reaction-diffusion systems. We first analyze a three-species system with boundary equilibria in some stoichiometric classes. Then We prove the convergence to the positive equilibria for a general reversible reaction-diffusion network as long as the initial data is closed enough to the the positive equilibria.

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arXiv2019

An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems

Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.

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arXiv2018

Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.

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