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Persistence and permanence of mass-action and power-law dynamical systems
Gheorghe Craciun, Fedor Nazarov, Casian Pantea
TL;DR
The paper asks when mass-action and related power-law systems avoid the boundary of the positive orthant and remain bounded. It proves persistence and permanence results for two-species networks, extends permanence to endotactic systems with time-varying rates, and applies related ideas to three-species global attraction.
Problem
The paper addresses how to establish persistence, permanence, and global attraction for nonlinear interaction systems, including mass-action and power-law models.
Method
The authors analyze network structure, proving results for weakly reversible and endotactic systems and constructing invariant-set arguments for three-species systems.
Results
Two-species weakly reversible mass-action systems are persistent and permanent, while two-species endotactic κ-variable systems are permanent; related ideas prove the Global Attractor Conjecture for three-species systems.
Takeaways & Limitations
Network structure can guarantee robust long-term behavior for two-species mass-action systems even when reaction rates vary, with consequences extending to broader nonlinear systems.
Takeaways & Limitations
The Persistence Conjecture remains open for systems with three or more species, and κ-variable Lotka–Volterra systems are not generally persistent.
Abstract
from arXiv · showhide
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 we allow the reaction rate parameters to vary in time. 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.
1. Introduction.
The paper studies persistence and permanence in mass-action systems, proves key results for two-species networks, and connects persistence to the Global Attractor Conjecture. It shows two-species endotactic κ-variable systems are permanent while the corresponding higher-species conjectures remain open.
- Persistence means positive trajectories have no boundary ω-limit points, whereas permanence requires eventual entry into a compact subset of the positive orthant.These properties matter for whether chemical species remain available, ecological species avoid extinction, and infections die out or persist.
- The paper proves the Persistence Conjecture for two-species weakly reversible mass-action networks.Weak reversibility means each connected component of the directed reaction graph is strongly connected.
- Any two-species endotactic κ-variable reaction system is permanent, extending the result beyond weakly reversible networks and allowing reaction rates to vary within a compact positive set.Endotactic networks strictly contain weakly reversible networks, and permanence implies both strong persistence and uniform boundedness for these systems.
- The Persistence Conjecture remains open for systems with three or more species, while the paper conjectures endotactic κ-variable permanence for any number of species.The paper also develops ideas toward the Global Attractor Conjecture for three-species networks.
- The Global Attractor Conjecture asks whether each positive equilibrium of a complex-balanced system is a global attractor within its stoichiometric compatibility class.Complex-balanced systems already have unique positive equilibria and strict Lyapunov functions giving local asymptotic stability.
- Because complex-balanced systems are weakly reversible and their trajectories converge to equilibria, proving persistence would imply the Global Attractor Conjecture.The Persistence Conjecture and the Global Attractor Conjecture therefore remain closely linked, although both are generally open.
2. Definitions and notation.
The paper introduces reaction-network and κ-variable mass-action terminology, then defines persistence and permanence for dynamical systems on stoichiometric compatibility classes.
- Chemical reaction networks: A chemical reaction network consists of species, complexes, and directed reactions; its reaction graph uses complexes as vertices and reactions as edges.The paper’s two-species example has species A1 and A2, three complexes, and six reactions represented as three reversible pairs.
- Chemical reaction networks: Weak reversibility means every connected component of the directed reaction graph is strongly connected.
- κ-variable mass-action systems: κ-variable mass-action kinetics allows rate constants to vary piecewise differentiably within a fixed positive bounded interval, generalizing constant-rate mass action.Results for κ-variable systems therefore also apply to ordinary mass-action systems with constant positive rate constants.
- κ-variable mass-action systems: The concentration vector remains in the nonnegative orthant and in the affine stoichiometric compatibility class determined by its initial condition.Forward invariance follows because negative terms in each concentration equation contain that concentration as a factor and vanish on the corresponding boundary face.
- Persistence and permanence: Persistence excludes boundary ω-limit points, whereas permanence requires trajectories to eventually enter a compact subset of the positive orthant within each compatibility class.Permanence therefore implies persistence, and for κ-variable systems the definition is applied separately on each stoichiometric compatibility class.
3. An illustrative example.
An illustrative two-species κ-variable system is analyzed geometrically by constructing a convex forward-invariant polygon. Combining reaction-specific flow constraints yields persistence and motivates the extension to reversible and endotactic networks.
- Invariant polygon construction: A convex polygon containing the initial condition is constructed so that the aggregate flow satisfies Nagumo’s sub-tangentiality condition and remains forward invariant.The polygon is formed from a sufficiently large positive-quadrant rectangle with corners cut along reaction-specific regions while preserving convexity.
- Reaction-specific flow: The flow direction for each reaction component is known outside shaded regions, while bounded rate-constant ratios determine the uncertain regions between comparison curves.For the first reaction, the comparison curve is y = (k1(t)/k−1(t))x2, with k1(t)/k−1(t) constrained to (η2, 1/η2).
- Invariant polygon construction: The invariant polygon combines the constraints from three reversible reaction pairs, whose aggregate effects push trajectories toward the polygon’s interior.For each pair, the reaction pointing outward is offset by its reverse reaction when the trajectory lies in the corresponding region.
- Conclusions from the example: The example system is persistent, and the same geometric strategy extends to any two-species reversible κ-variable mass-action system.
- Conclusions from the example: The broader extension targets endotactic networks, a geometric class larger than weakly reversible networks, using source-monomial comparisons up to multiplicative constants.
4. Endotactic networks.
Endotactic networks are defined through directional source-support conditions that generalize weak reversibility and can be checked by finitely many parallel-sweep tests. This class includes weakly reversible networks and motivates persistence and permanence conjectures for variable-rate systems.
- Definition and characterization: Endotacticity requires that reactions originating on a directional essential support do not point toward the swept region.The condition is expressed using essential supports and reaction vectors, and equivalently as a parallel sweep test.
- Definition and characterization: Proposition 4.1 reduces the endotacticity check to inward normals of the source convex hull plus the coordinate directions ±i and ±j.This finite test also handles the case where the source convex hull is a line segment.
- Relation to weak reversibility: Weakly reversible reaction networks form a subclass of endotactic networks because every source-support reaction points into the positive essential-support region.This inclusion connects the new network class to a standard assumption in Chemical Reaction Network Theory.
- Scope and conjectures: The paper extends the target class from weakly reversible to endotactic networks and formulates persistence and permanence conjectures for endotactic κ-variable systems in arbitrary dimensions.The proved results later concern two-species systems, while the all-dimensional statements are presented as conjectures.
5. Construction of a forward-invariant polygon.
The proof constructs a one-parameter family of convex polygons whose convex hull contains the initial condition and is forward invariant. Polygon sides are aligned with source-complex geometry so dominant reaction monomials direct trajectories inward.
- Forward invariance: The construction generalizes the reversible-case argument by identifying a reaction whose source monomial dominates the others up to a constant and whose direction keeps the aggregate flow inward.Forward invariance follows by ensuring the aggregate vector field satisfies the sub-tangentiality condition on the polygon boundary.
- Source-complex geometry: The construction compares source monomials pairwise and uses their comparison curves to organize the polygon’s sides and shaded regions.For source complexes P and P′, the comparison produces power-law curves with exponents determined by source-edge normal slopes.
- Polygon construction: Conditions (P1)–(P5) place the initial condition inside the polygon, control curve intersections, and ensure the vertices occupy regions suitable for invariance.These conditions also make source-monomial comparisons straightforward in the boundary regions.
- Polygon construction: The polygon P is built vertex by vertex on fractional-index curves, with sides orthogonal to the relevant normal vectors and prescribed horizontal or vertical boundary sides.The resulting polygon is uniquely determined by these geometric constraints.
- Parameterized invariant family: The family P(α) varies continuously with α, while conv(P(α)) decreases as α increases and covers the relevant positive state space.This parameterized family supports the later argument that trajectories eventually enter smaller invariant polygons.
6. Endotactic two-species κ-variable mass-action systems: persistence and permanence.
For two-species endotactic κ-variable mass-action systems, the invariant-polygon argument establishes persistence and boundedness, then permanence. The proof also yields persistence for bounded lower-endotactic trajectories and supports a positive-equilibrium consequence in an autonomous setting.
- 6.1. Persistence: The persistence proof controls boundary behavior by showing that a suitably dominant source reaction produces an inward directional effect at polygon-boundary points.Source monomial comparisons are organized according to the relative position of each source complex to the selected reaction source.
- 6.1. Persistence: Any two-species endotactic κ-variable mass-action system is persistent and has bounded trajectories.The proof uses directional reactions guaranteed by endotacticity and the forward-invariant polygon constructed earlier.
- 6.1. Persistence: Bounded trajectories of two-species lower-endotactic κ-variable systems are persistent, although full endotacticity is not required for this conclusion.Lower-endotacticity only requires the parallel sweep condition for vectors pointing into the closed positive quadrant.
- 6.2. Permanence: Any two-species endotactic κ-variable mass-action system is permanent.This is the paper’s main permanence theorem for time-varying reaction rates constrained to a compact positive interval.
- 6.2. Permanence: For rate functions depending on the phase point while remaining uniformly bounded away from zero and infinity, the persistence and permanence conclusions still hold.In this setting, Theorem 6.4 and Brouwer’s Fixed Point Theorem yield a positive equilibrium inside an invariant polygon.
7. The Global Attractor Conjecture for three-species networks.
The paper proves the Global Attractor Conjecture for three-species complex-balanced networks by constructing invariant compact sets that prevent trajectories from approaching the boundary.
- For complex-balanced systems, convergence to the equilibrium set reduces global attraction to excluding boundary ω-limit points.Complex-balanced systems have a unique positive equilibrium in each compatibility class, together with a strict Lyapunov function and convergence to the equilibrium set.
- Theorem 7.2 establishes the Global Attractor Conjecture for three-species networks.Every positive trajectory converges to the unique positive equilibrium in its stoichiometric compatibility class.
- The construction relies on prior boundedness and separation from the origin supplied by a Lyapunov function.These bounds ensure the trajectory stays in a controlled region while boundary ω-limit points are ruled out.
- The proof constructs compact planar invariant polygons for projected two-species systems and combines them into a compact set K in R3.Weak reversibility is preserved under projection, making the projected networks endotactic and allowing invariant polygons to be constructed.
- The trajectory remains inside K because the vector field points inward at every boundary point, using normal-cone inequalities and the planar invariant-polygon construction.The argument considers a first boundary contact and reduces the relevant inequality to the projected polygon whenever one coordinate is bounded below by ϵ.
8. Examples.
The examples apply the permanence theory to Thomas, S-system, and related power-law models, while showing that endotactic structure is essential for κ-variable systems and that the three-species theorem resolves previously open examples.
- 8.1. Thomas-type models: The Thomas model is permanent because its endotactic κ-variable mass-action representation satisfies Theorem 6.4.This holds for continuous nonvanishing interaction functions on a suitable compact region, including the model’s T(u,v).
- 8.2. Power-law systems: The S-system is permanent, so every positive trajectory remains bounded away from both zero and infinity.Its generalized monomials and reaction vectors form an endotactic configuration.
- 8.2. Power-law systems: The same permanence conclusion extends to the displayed power-law family with time-varying rate functions constrained by κ_i(t) ∈(η, 1/η).Theorem 6.4 applies to the corresponding endotactic configuration.
- 8.3. Lotka-Volterra systems: The fixed-parameter Lotka-Volterra system is persistent but not permanent, and its κ-variable version is generally not persistent.The network is neither endotactic nor lower endotactic; fixed-parameter trajectories can be constant or closed orbits.
- 8.4. Examples for the three-species Global Attractor Conjecture: The three-species theorem proves global asymptotic stability for two complex-balanced examples that previous results could not resolve.Both networks have boundary equilibria, including equilibria on a codimension-two face, while their unique positive equilibria are nevertheless globally asymptotically stable.