Global asymptotic coherence in discrete dynamical systems

Abstract

Spatial synchrony (coherence) in dynamical systems is of both theoretical and applied importance. We address this problem for a generalization of coupled map lattices (CMLs). In the systems we study, which we term “meta-CMLs,” the map at each lattice point may be multidimensional (corresponding, for example, to multispecies ecological systems in which all species have the same dispersal pattern). Most previous work on coherence of CMLs has focused on local stability. Here, we prove a global theorem that provides a useful sufficient condition guaranteeing decay of incoherence in meta-CMLs regardless of initial conditions and regardless of the nature of the attractors of the system. This result facilitates useful analyses of a variety of applied problems, including conservation of endangered species and eradication of pests or infectious diseases.

The simplest possible solutions of dynamical systems with spatial degrees of freedom are coherent (spatially homogeneous), so investigation of coherent solutions is usually the starting point of any mathematically rigorous study. Real-world applications of such analyses abound. For example, in ecological settings, if a species has low population density everywhere simultaneously then it is at risk of global extinction; spatially coherent population dynamics may therefore be associated with extinction risk (1). In epidemiology, coherence may facilitate eradication of pathogens (2). In neurobiology, coherence is associated with neuropathologies such as epilepsy (3–5). In economics, coherence may be stimulated by globalization of markets. The list goes on, including diverse areas of science, engineering, and social science.

Coupled map lattices (CMLs) provide a simple framework for investigation of spatial dynamical processes (6). CMLs are built from iterated discrete maps (7–10),

such as the (normalized) logistic map,

In general, given a fundamental state space X and a fundamental map F : X → X, a CML is defined via

where x_i^t ∈ X is the state at lattice point i at time t, M = (m_ij) is the “connectivity matrix” and n is the number of lattice points. Letting x⃗ = (x₁, …, x_n)^⊤ ∈ Xⁿ, we define the “full map” F : Xⁿ → Xⁿ via

For convenience, we define the abbreviated notation F(x⃗) for the vector (F(x₁), …, F(x_n))^⊤ ∈ Xⁿ, so the full map can be written

Thus, the trajectories of the CML (Eq. 3) are the sequences of points x⃗^t = F⃗^t(x⃗⁰), where F⃗^t(x⃗) = F⃗(F⃗^t−1(x⃗) for t ≥ 1 and F⃗⁰(x⃗) = x⃗.

A CML is in a “coherent state” at time t if x_i^t = x_j^t for all i and j. A coherent solution is a sequence of coherent states that satisfies Eq. 3 or, equivalently, Eq. 5. We are thinking of a CML as a spatial generalization of the fundamental map, so we want to retain the trajectories of the fundamental map as coherent solutions of the corresponding CML. This occurs if and only if

i.e., each row of M sums to 1; equivalently, if we define the n-dimensional vector e = (1, 1, …, 1)^⊤ then

i.e., e is an eigenvector of M with eigenvalue 1. Eqs. 6 and 7 are equivalent to the statement that if x^t is a solution of Eq. 1 then x⃗^t = x^te is a coherent trajectory of the full map (Eq. 5).

In an ecological setting, a CML is known as a “metapopulation model” (11), because it represents a system of isolated patches, within which individuals reproduce and between which they disperse: x_i^t is the population density in patch i at time t and M is the “dispersal matrix.” In this case, the fundamental map F is called the “reproduction function.”

Dispersal matrices must be nonnegative, so we restrict attention to nonnegative connectivity matrices M that satisfy Eq. 6; such matrices are said to be “stochastic” because each row of M can be considered to be a discrete probability distribution on a sample space with n points (12). Eq. 6 may seem to preclude the possibility of death during migration in the ecological context, but this effect can always be included implicitly by absorbing an overall factor into the reproduction function F.

Research on coherence in CMLs has traditionally begun with the assumption that the state variables x_i and the fundamental map F are one-dimensional. A single dimension is certainly sufficient to generate rich dynamical structure, but is too restrictive to cover many important application areas. In the ecological setting, if the population is classified according to species, age, or other factors, then x_i becomes a vector of densities in the various classes, and F becomes a multidimensional map. Similarly, in epidemiology, x_i is a vector of densities of individuals in each of a set of epidemiologic compartments (e.g., susceptible, infectious, recovered). With such models in mind, we define a “meta-CML” via Eq. 3 or 5 but with X a multidimensional state space.

When describing computations with meta-CMLs, it is often convenient to use notation that would be appropriate if X were one-dimensional, with the understanding that each component of a vector x⃗ ∈ Xⁿ is itself a vector x_i ∈ X. For example, writing the n-dimensional coherent unit vector ê = e/∥e∥ ∈ ℝⁿ, we use the notation x⃗ · ê to denote the “coherent component” of the state vector x⃗ ∈ Xⁿ, i.e.,

Since each x_i is a vector in X, the result of the dot product is also a vector. The “mean field” of x⃗ is

Again, like each x_i, 〈x⃗〉 ∈ X so it is a vector if dim X > 1. Note also that

Thus, at any time t, the coherent component of x⃗^t corresponds to the state with the mean field in each patch. We shall say that a trajectory {x⃗^t} “relaxes to coherence” or that it is “asymptotically coherent” if ∥x⃗^t − 〈x⃗^t〉 e∥ → 0 as t → ∞ (and “asymptotically incoherent” otherwise). We refer to x⃗^t − 〈x⃗^t〉 e as the “incoherent component” of x⃗. Note that relaxation to coherence does not imply approach to an equilibrium; a coherent attractor can be periodic, quasiperiodic, or chaotic.

In the next section we briefly comment on local analyses of coherent trajectories in CMLs, which relate to the question of possibility or impossibility of coherence in practice. In ecological applications it is particularly important if a metapopulation relaxes to coherence regardless of initial conditions, because substantial extinction risk may then be unavoidable (1). It is this situation of inevitable synchrony or “global asymptotic coherence” that is our primary concern here. A slightly restricted version of the main result we present here was stated informally without proof in ref. 1, in which applications to conservation biology were emphasized.

Local Stability of Coherence

In recent years, numerous studies have given local stability conditions for coherence in CMLs with specific classes of connectivity matrices (13–17). All of these results are subsumed by a proposition that we stated informally in ref. 1. A rigorous statement and proof of this proposition requires careful consideration of all ergodic invariant measures associated with the dynamical system (18, 19) and we defer a rigorous presentation of the local theory to another work (D.J.D.E., S.A.L., and D. A. Rand, unpublished results). Here, we briefly outline some key elements of the local results in order to provide context to the global analysis that we present below.

Consider an attractor 𝒜 of the fundamental map F and define the coherent set 𝒜e = {xe : x ∈ 𝒜}. The set 𝒜e is invariant under the full map F⃗ and, restricted the coherent manifold, 𝒜e is an attractor. The local theory establishes conditions under which 𝒜e is also an attractor of F⃗ in a neighborhood of 𝒜e in Xⁿ. The details of this theory are complicated (19), but the key points can be expressed in rough terms through the local stability condition that we described in ref. 1.

The local stability condition depends primarily on two quantities. The first is the magnitude of (any) subdominant eigenvalue of the connectivity matrix M (an eigenvalue is subdominant if its magnitude is second largest among all eigenvalues). The second is the maximum Lyapunov characteristic exponent of the fundamental map F at a point x ∈ X, which can be defined as

The exponent χ_x characterizes the dynamical nature of the trajectory of F that starts at x. If we restrict attention to “typical” points on an attractor 𝒜 of F, then, if we ignore subtleties, we can drop the dependence of χ_x on x and characterize typical trajectories on 𝒜 by a single number χ (this is an oversimplification, but it will allow us to get across the essence of the local stability condition without getting bogged down in details).

In addition to requiring that the connectivity matrix M be stochastic (and hence that 1 is an eigenvalue of M), we require that the eigenvalue 1 be simple (i.e., has multiplicity 1) and strictly dominant (all other eigenvalues have magnitude strictly less than 1). Suppose that λ is a subdominant eigenvalue of M and that χ is the maximum Lyapunov characteristic exponent for typical trajectories on an attractor 𝒜 of the fundamental map F. If

then coherence is (in practice) impossible: almost all nearly coherent points x⃗ near 𝒜e lie on asymptotically incoherent trajectories. If instead

then coherence is possible: most nearly coherent points that are sufficiently close to 𝒜e lie on asymptotically coherent trajectories (making “most” precise here is complicated).

The stability condition (Eq. 13) is local not only in requiring that an initial point x⃗ be nearly coherent (∥x⃗ − 〈x⃗〉 e∥ must be small) but also in requiring that the coherent component of x⃗ be close to an attractor 𝒜 of F (〈x⃗〉 must be close to 𝒜). The theorem we present in the next section is extreme in the opposite sense: it applies to all initial states and makes no reference to particular attractors of the system. Intermediate to these two extremes is the theory of normally hyperbolic invariant manifolds (20, 21), which could be used to develop stability conditions that apply to all coherent states (i.e., this would address the local stability of the entire coherent manifold as opposed to local stability of a coherent invariant set that is an attractor for the system restricted to the coherent manifold).

Global Asymptotic Coherence (GAC)

Consider first the simplest possible case: a one-dimensional fundamental state space X (an interval) and only two lattice points (“patches”). The most general 2 × 2 stochastic matrix has the form

where m₁, m₂ ∈ [0, 1]. Using a prime to denote the next iteration of the map, we have

and hence

We shall assume that the fundamental map F is differentiable, so the mean value theorem implies that there exists ξ between x₁ and x₂ such that F(x₂) − F(x₁) = F′(ξ)(x₂ − x₁). If, moreover, F′ is bounded (say r = sup_x∈X |F′(x)|) then for any x₁, x₂ ∈ X, we have

If |1 − (m₁ + m₂)| r < 1 then the difference between the states at each of the lattice points will decrease to zero asymptotically, regardless of the initial conditions. Thus, |1 − (m₁ + m₂)| r < 1 is a sufficient condition for GAC in this simple model.

Now note that the eigenvalues of M are 1 and λ = 1 − (m₁ + m₂), and |λ| ≤ 1. In fact, unless m₁ = m₂ = 0 (in which case M is the identity matrix) or m₁ = m₂ = 1 (in which case the state at each of the two lattice points is simply swapped at each iteration), |λ| < 1. In these two special cases in which |λ| = 1 (so λ is not subdominant) there is actually no connectivity between the two lattice points. If we exclude these cases, we can express the GAC condition as |λ| r < 1, where λ is the subdominant eigenvalue of M. This observation motivates our formulation of Theorem 1 below.

Before stating the theorem, we recall a few definitions. The segment joining two points x and y in a vector space is the set S_x,y = {αx + (1 − α)y : 0 ≤ α ≤ 1}. A subset X of a vector space is convex if for any two points x, y ∈ X the segment S_x,y is contained in X. A Banach space is a complete, normed vector space. Given the norm ∥·∥ on a Banach space ℬ, the norm of a function F : ℬ → ℬ is

Where it exists, the derivative of F at x ∈ ℬ is the unique continuous linear transformation D_x F : ℬ → ℬ with the property that

where the limit is taken over all nonzero sequences that converge to zero.

Theorem 1.

Let X be a convex subset of a Banach space ℬ and suppose the fundamental map F : X → X is differentiable at each x ∈ X, that ∥D_x F∥ is bounded in X, and that r = sup_x∈X ∥D_x F∥. Suppose M is a stochastic n × n matrix, the eigenvalue 1 is simple and strictly dominant, and λ is a subdominant eigenvalue of M. If r|λ| < 1 then the full map F⃗ : Xⁿ → Xⁿ, defined by F⃗(x⃗) = M · F(x⃗), is globally asymptotically coherent, i.e., every initial state x⃗₀ ∈ Xⁿ asymptotically approaches a coherent trajectory. If r < 1 then F⃗ has a globally asymptotically stable fixed point.

Usually the fundamental state space X is a subset of ℝ^k, but the theorem is valid for any X that is a convex subset of any Banach space. This extension to arbitrary dimensionality is a potentially useful generalization of the result as stated in ref. 1, because X can now represent, for example, a continuum of ages or continuous space, or both.

Note that from the definition of the maximum Lyapunov exponent (Eq. 11), we must have χ_x ≤ log r for all x ∈ X (where r is defined in the statement of Theorem 1). Given the local coherence condition (Eq. 13) discussed in the previous section, this observation increases the plausibility of Theorem 1. Our proof, however, does not rely in any way on the local theory.

Proof of Theorem 1

Our proof of Theorem 1 depends on two propositions that we state as lemmas. The first is a special case of a standard result of analysis, while the second expresses a useful algebraic property of stochastic matrices.

Lemma 1 (The Mean Value Inequality).

Suppose ℬ is a Banach space and S_x,y is the segment joining x, y ∈ ℬ. If F is a continuous mapping of a neighborhood of S_x,y into ℬ, and F is differentiable at every point of S_x,y, then

where r_x,y = sup_z∈Sx,y ∥D_z F∥.

Proof:

See Dieudonné (22), proposition 8.5.4, p. 160.

Lemma 2.

Let M = (m_ij) be an n × n stochastic matrix and suppose the eigenvalues of M, including multiplicities, are (λ₁, λ₂, …, λ_n−1, 1). Let M̃ = (m̃_ij) be the (n − 1) × (n − 1) matrix defined via m̃_ij = m_ij − m_nj for i, j = 1, …, n − 1. Then the eigenvalues of M̃ are (λ₁, λ₂, …, λ_n−1).

Proof:

Let A be the n × n matrix with 1s on the diagonal and along the last column, and zeros elsewhere. It is easily verified that A⁻¹ is the matrix with ones on the diagonal, −1s above the diagonal in the last column, and zeros elsewhere. If we define B = A⁻¹MA then B and M are similar matrices so they have exactly the same spectrum (12). But the submatrix consisting of the first n − 1 rows and columns of B is M̃, and (using stochasticity of M) the last column of B is zero except in its last entry, which is 1. Consequently, the eigenvalues of B (and hence of M) are the eigenvalues of M̃ together with 1.

Proof of Theorem 1:

It is convenient to express the map (Eq. 3) as follows (where prime denotes one iteration of the map):

Now, since Σ_j=1ⁿ m_ij = Σ_j=1ⁿ m_nj = 1, we have Σ_j=1ⁿ (m_ij − m_nj) = 0, so we can rewrite Eq. 21a above as

With M̃ defined as in Lemma 2, we can write Eq. 22

Given the norm on ℬ, we can define a norm on ℬⁿ in many ways, for example via ∥x⃗∥ = max_i=1ⁿ ∥x_i∥ (we use ∥·∥ to denote norms on both ℬ and ℬⁿ with meaning hopefully clear from context). Adopting this particular norm on ℬⁿ for convenience, Lemma 1 implies (since r_x,y ≤ r for all x, y ∈ X) that

Now, since λ is a subdominant eigenvalue of M, Lemma 2 implies that it is a dominant eigenvalue of M̃, so

where ∥·∥ here denotes the matrix norm of M̃, i.e., ∥M̃∥ = sup_{v∈ℝⁿ⁻¹,∥v∥=1} ∥M̃v∥. Therefore, taking norms of Eq. 23, we have

Thus, if |λ| r < 1 then ∥x_i − x_n∥ → 0 for all i, i.e., the system is asymptotically coherent.

To obtain the last part of Theorem 1, note that for general x⃗, y⃗ ∈ Xⁿ we have ∥F⃗(x⃗) − F⃗(y⃗)∥ ≤ r ∥x⃗ − y⃗∥(where we are using a norm on ℬⁿ induced by the norm on ℬ and applying Lemma 1 as above). Thus, if r < 1 then the full map F⃗ is a contraction mapping and therefore has a globally attracting fixed point.

Discussion

Theorem 1 gives a sufficient condition for GAC that can be applied to a large class of discrete dynamical systems (meta-CMLs) arising in a variety of different fields. Although this condition is sufficient for GAC, we emphasize that it is certainly not necessary. For specific systems, i.e., specific fundamental maps F, it is possible to show that the region of parameter space in which the system is globally asymptotically coherent is larger than the region identified by Theorem 1. The value of Theorem 1 is in providing a relatively easy way to identify a parameter region in which synchrony is inevitable.

As we have defined them, meta-CMLs are by no means the most general class of dynamical systems for which synchrony is important (for a comprehensive review see ref. 23). For example, we have considered systems with a single connectivity matrix so in the biological context the possibility that different species disperse in different ways is excluded. Local analyses of systems with different patterns of connectivity between different components have been conducted (24, 25), but global analyses of such systems have yet to be explored.

Abbreviations

CML

coupled map lattice

GAC

global asymptotic coherence.