Stabilizing a homoclinic stripe

Abstract

For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.

This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

1. Introduction

Consider a typical reaction–diffusion model, such as the Schnakenberg model, which admits a one-dimensional spike solution corresponding to a homoclinic orbit of the underlying slow-diffusing variable. When such a one-dimensional spike is extended trivially to the second dimension, the result is what we shall call a ‘homoclinic stripe’. In common situations, it is well established that a ‘homoclinic stripe’ is unstable and breaks up into spots, unless a domain is very ‘thin’ [1–4]. This break-up is illustrated in figure 1 row B and figure 2 row 1. In this paper, we examine two situations that can potentially stabilize a homoclinic stripe: adding anisotripy or drift to the fast-diffusing variable. We show that a sufficient amount of anisotropy or drift can stabilize a homoclinic stripe of any length.

Figure 1.

Top: Boundaries for zigzag and break-up instabilities in the d–b parameter plane with ε = 0.025, (x, y)∈ [ − 1, 1] × [0, 1], A = 1. Note a good agreement between asymptotics and numerics. Rows A–C show snapshots of full numerical simulations of (1.1) in different parameter regions. Parameters A are in the stable region. Parameters B: stripe exhibits a break-up instability. Parameter C: The stripe is stable with respect to break-up instability, but unstable with respect to a zigzag instability. As a result, the stripe starts to bend (note the slow time scale). Eventually, this is followed by a break-up of a bended stripe. (Online version in colour.)

Figure 2.

Simulation of (5.1) with ε = 0.05, A = L = 1, c = 10, and with d as follows: (a) d = 1.8ε; (b) d = 1.3ε; (c) d = 1.2ε; (d) d = 1.2ε; (e) d = 1.3ε. Initial conditions are close to those shown in the first column.

For concreteness, we concentrate on the well-studied Schnakenberg model [5–7] and its variants, but we anticipate these techniques can be extended to other settings. We first examine the anisotropic version of Schankenberg model, using the following scaling in two dimensions:

1.1

We will assume Neumann boundary conditions in the x-direction, with the stripe extending to infinity in the y-direction (), which is the most potentially destabilizing case anyway, although our results can be generalized easily to the case of finite stripe.

We further assume that ε≪O(1) and , whereas A and L are O(1). In this case, u and v become weakly coupled and the one-dimensional spike solution to (1.1) is asymptotically close to a sech²-type profile.

Equations (1.1) model the following process: a fast-diffusing substrate v is consumed by a slowly diffusing activator u, which decays with time. The substrate is being pumped into the system at rate A. The reaction kinetics for u and v occur at different scales: u reacts much slower than v, so that v is effectively slave to u. Note that we scaled the system so that the activator u diffuses isotropically; the anisotropy is only present in the substrate (the case of anisotropic diffusion in u can be reduced to (1.1) by scaling the y coordinate appropriately).

Figure 1 illustrates our results. For sufficiently small b, the stripe is fully stable (row A). As b is increased, there are two types of instabilities that appear, depending on how large d is. These instabilities are triggered by either large (O(1)) or small (O(ε²)) eigenvalues, and are illustrated in rows B and C, respectively. We will refer to them as break-up and zigzag instabilities, respectively. As shown in figure 1, the two instability boundaries (indicated by red and blue curves) cross each-other. The crossing point is shown in green. The main result of this paper is to characterize this crossing analytically.

We now summarize the main result as follows. Consider the stripe solution, where u(x, y, t) is localized along a vertical line. Such a solution is shown row A (last snapshot) and has the asymptotic profile u(x, y, t)∼(LA/2) sech²(x/2ε) (see below for derivation). The stripe is stable provided that where b = b₁ and b = b₂ are thresholds for break-up and zigzag instabilities, respectively. The stripe is unstable when . Refer to figure 1 (top).

Suppose further that

1.2

Then the asymptotics for stability boundaries b₁ and b₂ have explicit asymptoics

1.3

and

1.4

Break-up instabilities occur when b > b₁. Zigzag instabilities happen when b > b₂ (the formula (1.4) is actually valid for any whereas formula (1.3) requires further restriction ).

As figure 1 shows, the two stability boundaries intersect. Let (d_c, b_c) be the point in (d, b) space at which b₁ = b₂. From (1.3), (1.4), this point is asymptotically given by

1.5

If d > d_c then break-up instabilities occur as b is increased above b₁. If ε^1/2≪d < d_c, then zigzag instabilities occur as b is increased above b₂.

Note that there is a relatively large error in (d_c, b_c) when ε = 0.05. This is due to O(ε^1/3) scaling in (1.5). Table 1 shows the error behaviour as is halfed. It shows that the relative error does indeed decrease as

Table 1.

Error in d_c and b_c as a function of ε.

	numerics		asymptotcs		rel. err
ε	dc	bc	dc	bc
0.05	0.2201	0.1375	0.2720	0.1861	19.1%	26%
0.025	0.18648	0.09255	0.2159	0.1172	13.6%	21%
0.0125	0.1536	0.06102	0.1714	0.0738	10.3%	17%

2. Stripe steady state

We begin by constructing a stripe steady state. Such a solution independent of y or t, so that (1.1) reduce to

2.1

with Neumann boundary conditions u_x( ± L) = v_x( ± L) = 0. The asymptotic construction of a spike profile to (2.1) is by now very standard (e.g. [7,8]) but we briefly review it here for completeness and to settle the notation. We assume the spike has a maximum at x = 0. In the inner region, we scale u(x) = U(z), v(x) = V (z), which yields

2.2

Assume that

2.3

Then to leading order we have V_zz∼0 so that V is a constant: V ∼v₀. Then to leading order, U satisfies U∼w/v₀ where w is the ground state satisfying

2.4

with a well-known explicit solution given by . It remains to determine v₀. To do so, we integrate the second equation in (2.1) which yields

2.5

where we used the fact that In conclusion, we obtained

2.6

3. Stability: large eigenvalues

We linearize the stripe using

which results in the following one-dimensional eigenvalue problem,

3.1a

and

3.1b

with Neumann boundary conditions ϕ^′( ± L) = ψ^′( ± L) = 0.

In the inner region x = εz, to leading order, (3.1a) can be written as

3.2

where

3.3

is the local linear operator of the associated ground state w(z), and ψ₀ = ψ(0). To determine ψ₀, we approximate so that

3.4

where G satisfies

3.5

The solution to (3.5) is given by

and, in particular,

Upon plugging in x = 0 into (3.4) we obtain

Now consider the critical scaling

so that (3.2) becomes

3.6

where we used Problem (3.6) is equivalent to solving

3.7

where

3.8

The function f(μ) can be expressed in terms of hypergeometric functions (e.g. [9]). However, the resulting expression is unwieldy and at the end still needs to be evaluated numerically. Here, we compute f(μ) directly by solving numerically the associated BVP (L₀ − μ)ϕ = w² and integrating numerically. Figure 3 shows the graph of f(μ).

Figure 3.

f(μ) versus μ.

The vertical asymptote corresponds to the eigenvalue of the local operator L₀, and asymptotic expansion near shows that f blows up to the left of this asymptote, in agreement with the graph above. Moreover, .

Simple sketching show that there is always an instability band when is sufficiently large, and there is full stability when is sufficiently small. The instability threshold happens when there is a double root for the equation

3.9

Table 2 gives the double-root boundary for selected values of β.

Table 2.

Instability thresholds for large eigenvalues.

0.05	30.416	1.5208	608.319
0.1	15.2277	1.52277	152.277
0.2	7.65315	1.53063	38.2658
0.5	3.16896	1.58448	6.33792
1	1.76	1.76	1.76
1.4	1.40648	1.96907	1.00463
2	1.17	2.35	0.588
4	0.998	4	0.249
8	0.975	7.8	0.121

Of particular interest are the following limits:

• —d small: In this limit, so that tanh in (3.9) can be replaced by 1, and the instability threshold corresponds to the double-root of the equation which numerically yields α∼0.975 as In the original variables, this yields the threshold (1.3).
• —
  d large: then and the instability threshold corresponds to the double-root of the equation which numerically yields αβ∼1.5208 as In the original variables, this yields the threshold

  3.10

  We will come back to (3.10) in the context of incorporating large drift, see below.

4. Small eigenvalues

Small eigenvalues arise from the translation invariance of the inner problem for the spike [8]. They correspond to an odd solution of the linearized equation (3.1). To leading order in ε, the eigenvalue is zero and a higher-order expansion is necessary to resolve the stability. Proceeding as in [3], we, therefore, expand both the steady state and the eigenfunction in the inner region using the stretched variable z = x/ε:

The resulting equations are

4.1

4.2

4.3

To leading order in ε, the solution to (4.2) is given by Φ∼U_y, λ = 0. To resolve the next order, we start by multiplying (4.2) by U_z and integrating to obtain

4.4

Estimating Φ∼U_y and integrating the right-hand side by parts yields

4.5

and using (4.1) we further estimate

Finally, from (4.3) we estimate

so that

4.6

and (4.6) becomes

4.7

Next we compute the outer solution to and match Ψ_z(∞). In the outer variables, we get

Recall that ψ satisfies (3.1b). The last two terms decay exponentially and are taken are zero, so that

To get C, multiply by x and integrate, assuming that Φ, Ψ is odd to get

so that

and finally

4.8

Upon substituting (4.8) into (4.7) we obtain, after some algebra,

4.9

where we used

Rewrite (4.9) as

4.10

where

4.11

From (4.10), there is a double root when α = 0.23716 corresponding to Instability exists when α > 0.23716, or b > b₂, where b₂ is given by (1.4).

This concludes the derivation of (1.3) and (1.4).

5. Vegetation patterns on a sloping ground

Finally, we extend this analysis to a variant of the Schnakenberg model which incorporates the water flowing down the hill. The equations are

5.1a

and

5.1b

(as in (1.1) and for the same reasons, we have replaced v_t by zero in (5.1b) to simplify the exposition). For simplicity, we will assume periodic boundary conditions in the x-direction:

with the stripe extending to infinity in the y-direction (). Mathematical models of vegetation patterns have a long history, starting with the work by Lefever and Lejeune on Tiger Bush patterns, [10]. The link between localized vegetation spots and localized structures was established in [11]. Vegetation spots can also be unstable through self-replication [12]. The particular model (5.1) that we chose is a variant of the Klausmeier vegetation model [13]. There, v represents water concentration in the soil whereas u is the plant density. In addition to the usual diffusion (assumed to be isotropic here for simplicity), the water flows down the slope. This is represented by the cv_x term (positive c assumes the water flows from right to left). In fact equations (5.1) are special case of Klausmeier model when water evaporation is assumed small (water evaporation is modelled by adding −ev term to the right-hand side of (5.1b)).

There are numerous papers studying the transition between homogeneous vegetation, vegetation spots and vegetation stripes, as well as self-replication. For a very incomplete list of references, see for example [4,14–22] and references therein. Generally speaking, spots transition to stripes, and stripes to uniform state, when precipitation rate A is increased.

Here, we show the transition of spots to stripes analytically in the regime of large slope (represented by parameter c) and small diffusion d. The stripes extend in the y-direction, so that they are perpendicular to the slope. The existence of stable stripes for large c was observed numerically in [4,20], matching observations in nature [23,24]. By contrast, in [4] the authors show analytically that a stripe is unstable (in particular when the slope is sufficiently small) and breaks up into spots.

Generally speaking, we find a spot-to-stripe transition as d is sufficiently decreased (and with c large). We restrict the analysis of (5.1) to the regime

5.2

With this restriction, the vegetation density has a sech²-type profile in the x-direction, which we extend into a two-dimensional stripe. This makes it possible to determine the stability thresholds of the stripe analytically.

As shown in appendix A, with restriction (5.2), the cross-section of the stripe has the spike-type profile of the form (2.6), except that x is replaced by x − st, where s is the (slow) speed of motion. This speed is easily computed using solvability condition (as was already done in [25], or see appendix A) with the result that

5.3

Repeating the analysis for large eigenvalue yields the problem (3.6) but with

As a consequence, the analysis leading to (3.10) applies, and it follows that stripe is stable provided d < d₁ where

5.4

This threshold is illustrated in figure 2. In the first two rows, d > d₁ and the stripe breaks up into spots, which then travel together forming a spotted-stripe pattern. In the second row, we took d = 1.3AL^1/2ε. This is very close to the stable regime; the stripe still breaks up but not completely, indicating that the break-up instability is supercritical (i.e. reversible). For d = 1.2AL^1/2ε (so that d = 0.97d₁, the stable side of the threshold), the stripe is seen to be stable. Stripe stability appears to be very robust: even initial conditions consisting of a single spot evolve into a stable stripe or spotted stripe as shown in rows (d,e), depending on whether d < d₁ or d > d₁.

6. Discussion

For the Klausmeier model, we have shown that adding sufficient anisotropy can lead to stable stripes. Unlike the previous works [1–4] which show that only a stripe of very small O(ε) length can be stabilized in the isotropic case, here we show that the stripe of infinite length can become stable under sufficient anisotropy.

We also showed that the boundaries for zigzag and break-up instabilities cross when d = d_c = O(ε^1/3), as b is increased. This analysis has similarities to the analysis of double-Hopf point of a spike in the Gray–Scott model—see for example [26–28]—although the resulting NLEP problem is has a somewhat different form, and it is enough to consider real solutions here (whereas full complex solutions arise when discussing a Hopf bifurcations). In these works, it is shown that a single spike in one dimension can become unstable due to either oscillations in position or in height, when the zero in the left-hand side in (1.1) is replaced by τv_t and as τ is increased. These two instabilities also cross each-other when A = O(ε^1/6) (with d = 1, b = 0). We remark that this hopf-hopf crossing has been recently extended to a two-dimensional spike in [29], where an unusual loglogε scaling for the crossing point is derived.

The observed patterns are closely connected to the concept of dissipative structures in out of equilibrium matter introduced by Ilya Prigogine and his collaborators [30–33]. See also [34] for more recent special issue on dissipative pattern formation in general, and localized structures specifically.

An entirely different mechanism to stabilize a stripe is by adding saturation as described in [35]. Adding sufficient saturation can turn homoclinic (unstable) stripe into a heteroclinic (stable) stripe having a mesa-type profile consisting of back-to-back interfaces.

In the limit of large d (or equivalently, the limit of large c for the drift problem (5.1)), v only depends on y. We may integrate out the x-direction for (1.1) to obtain the following limiting system

6.1

This is analogous to a so-called shadow limit, but in one direction only. In particular, the break-up instability threshold (3.10) can be derived directly from (6.1). It would be interesting to study spike dynamics for this system. The same system also arises in the limit of large drift for (5.1).

There are many open questions. For example, even when the stripe breaks up, the resulting spots can still align, forming a ‘spotted stripe’. As the anisotropy is further decreased, the spotted stripe itself undergoes secondary bifurcation, eventually resulting in a more uniform spot distribution. This is illustrated in figure 4. We plan to address this transition in future work.

Figure 4.

Simulation of (1.1) on a square 2 × 2 domain with ε = 0.025, A = 1 and with d and b as indicated. For each value of (d, b), the eventual steady state is shown (at t = 10⁷). As anisotropy is decreased, first, the stripe bifurates into a spotted stripe, then the spotted stripe breaks up resulting in a more uniform spot distribution.

Appendix A. Speed of propagation

In this appendix, we derive the formula (5.3) for the speed of propagation of a stripe. Similar formula was previously derived in [25]. Assuming stripe moves in the x-direction only, equations (5.1) reduce to

A 1

Assume that c scales like

and expand

Then expand

At the leading order, we obtain

A 2

and the next order equations are

A 3

Equation (A 2) yield

so that (A 3) simplifies to

Multiplying the equation for U₁ by w_z and integrating then yields

As before, to compute V₀ we integrate the equation for v in (A 1) using periodic boundary conditions to obtain

so that

Finally, using we obtain (5.3).

Data accessibility

This article has no additional data.

Competing interests

We declare we have no competing interests.

Funding

T.K. and M.W. are supported by NSERC, Canada.