The classical approach is to consider, as in , first, the static homogeneous equilibrium. To this end, we define the equilibrium B¯=B(x,t) where B¯ is a constant in time and space – likewise with the other variables, which satisfy the relationships: 20=cgmaxW¯B¯W¯+k1-dB¯,30=αOB¯+k2w0B¯+k2-gmaxW¯B¯W¯+k1-rwW¯,40=R-αO¯B¯+k2w0B¯+k2. Two solutions exist depending on the value of R. One solution with vegetation is 5B¯=cRd-k1rwcgmax-d,6W¯=dk1cgmax-d,7O¯=R((cgmax-d)(cR+dk2)-cdk1rw)α((cgmax-d)(cR+dk2w0)-cdk1rw). Physical equilibria for positive parameters must be positive and hence satisfy the relations cgmax>d and R>k1rwd/(cgmax-d). The homogeneous equilibria without vegetation are, by definition, B¯0=0, which implies 8W¯0=Rrw,9O¯0=Rw0α, which are, again, valid for positive parameters. For the parameters chosen here, the homogeneous vegetated equilibrium exists for R>1 and the non-vegetated equilibrium always exists.

The universal existence of the non-vegetated equilibrium does not imply its stability for all values of R.

The analysis in the previous section showed that homogeneous vegetation is linearly unstable against spatially periodic perturbations in the rain range 1mmd-1<R<1.25mmd-1. This linear analysis suggested the existence of periodic inhomogeneous equilibria in the nonlinear system within that rainfall range. Now, our focus shifts towards explicitly computing these nonlinear equilibria. To achieve this, numerical methods are employed to solve the equations involved.

For the time being we assume a periodic domain with a finite size of L=100 m. The choice of the size of the domain influences the number of zero modes exhibited on the homogeneous equilibrium branch. This choice together with periodic boundary conditions effectively discretizes the set of zero modes (Fig. ). Indeed, setting κ=n2πL, with n the wavenumber, we find two pairs (κ,R) corresponding to zero modes, with the range κ=[0:0.6] corresponding to perturbing the homogeneous equilibrium S¯(R) with a perturbation δS(x)=δScos⁡(κx).

For each pair of (κ,R) in the set of zero modes, we identify the corresponding homogeneous equilibrium (S¯) and perturb it with the corresponding perturbation δScos⁡(κx). The periodic boundary conditions are enforced by setting κ=n2πL, with n the wavenumber. The perturbed homogeneous solution 18B0(x,R)=B¯+δBcos⁡(κx),19W0(x,R)=W¯+δWcos⁡(κx),20O0(x,R)=O¯+δOcos⁡(κx), is then taken as the first-guess input in a Newton–Raphson iteration used for finding the corresponding nonlinear inhomogeneous equilibrium.

At this point we need to discretize the spatial domain into, say, N points. We used N=100. Specifically, the discretized first guess reads u0(R)=[B0(R),W0(R),O0(R)]∈R3N and the algorithm 21ui+1(R)=ui(R)-ϵJ-1(ui(R))f(ui(R)), converges towards the equilibrium uS(R) with J(ui(R)) being the Jacobian matrix of the system and f(ui(R)) being the right-hand side as in Eq. (). The Laplacian is discretized with a periodic pseudospectral method. This equilibrium, uS(R), is then used as the first guess to solve for the equilibrium with rainfall R+δR, 22u0(R+δR)=uS(R)+δu, with δu≪ϵ. This iterative procedure leads to the construction of the full branch of equilibria.

A new code has been developed to implement this continuation method. This code is available online in the repository mentioned in the “Code availability” section.

Each branch obtained by the above technique is denoted by n between 1 and 9, with n the wavenumber associated with the perturbation. The periodic boundary conditions, together with a finite domain size, fixed the number of zero modes present within the Turing zone and, therefore, the number of inhomogeneous branches that exist. For the periodic L=100 m domain, only nine zero modes fit within the Turing zone (see Fig. ). Higher-wavenumber modes correspond to linearly stable perturbations that decline exponentially.

All the branches obtained with this approach are shown in Fig. . We now see that the range of rainfall where vegetation is sustained is much wider than expected from the linear analysis. The linear analysis in the previous section predicted the growth of vegetation patterns in the Turing zone: between 1.0 and 1.25mmd-1. The analysis of the full nonlinear system, presented in Fig. , actually reveals time-invariant pattern equilibria on branches attached to zero modes in the range 0.5–1.3 mmd-1. Although the nonlinear equilibrium may differ in shape from the linear perturbation, it is found that wavenumber n used to perturb the homogeneous equilibrium describes reasonably well the shape of the pattern along the branch obtained from that perturbation. One such equilibrium is displayed for illustration in the top panels of Fig. . For this particular equilibrium, with wavenumber n=2, the biomass accumulates in two places, and soil water peaks there due to enhanced infiltration rate. Surface water accumulates around those areas.

Characterizing the stability of equilibria will further help us to understand the dynamics of the system inside the Busse balloon. The Busse balloon is the region in the parameter space (κ,R) where at least one stable pattern equilibrium exists. A linear stability analysis gives partial but valuable information about a given solution and its possible evolutions. As we will see in the following, while stable equilibria tend to be the end state of time evolutions, unstable equilibria may still be relevant for the dynamics. The stability of equilibria is classically estimated based on the Jacobian matrix evaluated at each equilibrium, positive real parts of the eigenvalues signalling instability.

As a first example, the equilibrium for rainfall equal to 0.9mmd-1 of the branch n=2, the associated eigenvalues and the first eigenvector are displayed in Fig. . The majority of eigenvalues have a negative real part. One eigenvalue has a very small, positive real part (3×10-10), which suggests quasi-neutral stability. Further inspection of the associated eigenvector shows that this quasi-neutrally stable mode is the spatial derivative of the equilibrium pattern. Therefore, it corresponds to a translation mode, which is indeed expected with periodic boundary conditions. Hence, we have a stable pattern, which may however translate consistently with the periodic character of the boundary conditions. We therefore attribute the small real part to a numerical artefact (it should be zero) and consider equilibria with positive real part of the highest eigenvalue of the order of 10-10 as stable.

This stability analysis is repeated for all the equilibria shown in Fig. . Stable sections of the branches are drawn as solid lines and unstable ones as dashed lines. This gives us a more precise idea about what is happening inside the Busse balloon. We now know what the shape of those stable states is and how and at what rainfall value they lose stability. The existence of multiple stable states for a given value of rainfall is also consistent with previous work based on models and real systems .

Some branches of equilibria, such as n=5, 6, 7, 8, 9, are unstable across all their existence domain. Other branches, such as n=1, 2, 3, 4, change stability along the branch. These branches of equilibria start as unstable from the low-vegetation zero modes on the homogeneous equilibrium. Then, they become stable and eventually become unstable again shortly before joining the highly vegetated homogeneous equilibrium at the corresponding zero mode. The change in stability may take place at the extreme rain values for which the equilibrium exists, as in the n=1 branch, or at a rainfall value in the middle of a branch. As we know, when an equilibrium loses stability, the real part of one of its eigenvalues changes from negative to positive, which indicates that the equilibrium becomes linearly unstable in the direction of the associated eigenvector. In all of the cases here the change of sign of the eigenvalue happens through zero (as opposed to, through infinity), meaning that there are zero modes at the intersection of the stable and unstable branch sections. As we show next, this indicates the branching of another branch of the equilibrium of the full nonlinear system. A simple way to identify those zero modes is proposed in Appendix .

In the previous section we studied the stability of the “main” branches of equilibria shown in Fig. . These branches of equilibria originated from the zero modes present on the homogeneous vegetated branch. We also saw that some of these branches of equilibria had zero modes at the intersection of their stable and unstable parts. These zero modes act as bifurcation points from which new branches of equilibria emerge. The latter can be found by starting from the equilibrium at the bifurcation point, perturbed in the direction of the eigenvector associated with the newly positive eigenvalue (zero mode) corresponding to a new unstable direction. Again, a Newton–Raphson iteration allows us to find the new equilibrium. From there, the continuation method explained in the previous section allows us to trace the full new branch.

The result of this routine applied to the bifurcation point of the branch n=4 is shown in Fig. a and b. At the transition between stability and instability, there are three zero modes. Hence three branches of equilibria start at this transition, and they are shown in Fig. . They all begin at the transition between the stable and unstable part of the n=4 branch, and they reconnect with it for higher rainfall. We call those branches n=4 bis1, n=4 bis2 and n=4 bis3. These three branches are close (with regard to the mean biomass) to that of the n=4 branch, but if we look closely at their profile (Fig. c–f) we see how they differ. For a given value of rainfall, the mixed-state equilibria look like a modulation of the n=4 equilibrium by another wavenumber.

Equilibria branching out of zero modes in the homogeneous equilibrium tend to exhibit a single perturbation mode; see Fig. . By contrast, equilibria branching out of zero modes in those inhomogeneous equilibria exhibit a mixture of modes (Fig. c–f), hence the name “mixed states”. The equilibria along the mixed-state branch n=4 bis3 are unstable, with positive eigenvalues. The branches n=4 bis2 and n=4 bis3 reconnect around 1.14mmd-1. There are numerous mixed-state branches, obtained from the different zero modes at bifurcation points found along the main branches. These can also be found to emerge from zero modes present on the unstable sections of the main branches. For example, in the low vegetated unstable part of the n=3 branch (see Fig. ), we find two zero modes at the same value of rainfall, from which two mixed-state branches emerge: one labelled n=2 bis because it connects to the n=2 branch and one labelled n=3 loc, which connects to the n=4 branch. The n=3 loc refers to the fact that the equilibrium of that branch is localized in space. The n=1 main branch is special in two ways. First, as stated, it is the only stable localized state. Second, it starts from a zero mode on the homogeneous equilibrium, but it ends up connecting to the n=2 branch as if it was a mixed state; see Fig. . The evolution of the shape along this branch goes from one localized vegetation peak to two localized vegetation peaks in the unstable part close to the connection with the n=2 branch.

All mixed-state branches share similar characteristics, with states appearing as a modulation of a main branch. Even though they are unstable, we expect mixed-state equilibria to influence the dynamics of nearby trajectories, depending on the value of the positive eigenvalue.

In order to assess the relevance of the various branches for the dynamics, we propose the following numerical experiment. The Rietkerk model is run with rainfall changing over time at a rate of dRdt=-5×10-6mmd-2. The starting point is the vegetated homogeneous equilibrium with rainfall R=1.4mmd-1. The final rainfall is 0.5mmd-1. The result is shown in Fig. . Until R∼1.2mmd-1, the system stays in the homogeneous equilibrium. At this point, the system undergoes a Turing bifurcation and jumps to a heterogeneous equilibrium. The patterned branch on which the system lands is the n=2 branch. We can understand this transition by looking at Fig. . Indeed we see there that the first mode to lose stability in a decreasing rainfall scenario is the n=2. So this simple linear analysis can give us a first idea about the way the system will be destabilized in a slow-change scenario. After that, the system tracks the n=2 branch until this equilibrium loses its stability. Finally, the system switches to the only other stable equilibrium, the n=1. For this experiment, the mixed states do not have an influence on the trajectory. But in the following we will see that even if the mixed states are unstable, they can still play a role in the dynamics.

Here we address the question of whether unstable mixed-state equilibria are also able to influence system-transient trajectories. In this case the rainfall is set to rainfall R=1.05mmd-1, for which a mixed-state equilibrium exists.

The initial condition is an unstable equilibrium (n=5) with a small perturbation along the direction of the first eigenvector. Figure a and b summarize the evolution of the biomass pattern over time and the corresponding trajectory in a summary phase space. This summary phase space consists of a two-dimensional phase space where the two dimensions are the mean biomass and the maximum biomass. With that in mind, we observe that the system leaves the n=5 unstable equilibrium by first reorganizing itself: mean biomass remains constant while maximum biomass increases. This means that one or more vegetation bumps are growing at the expense of others. After that initial reorganization, an excursion on higher mean and maximum biomass takes place. It is during this out-of-equilibrium excursion that one of the vegetation bumps is lost. At this point in time the vegetation profile, containing four vegetation bumps, passes close to the three mixed states originating from zero modes on the n=4 branch explained identified in the previous section. Then, the system undergoes another transition to the n=3 equilibrium. To have a better representation of this trajectory, we also show the time evolution in Fig. b. We see that the system quite slowly leaves the n=5 unstable equilibria. The state is indeed unstable, but the associated positive eigenvalues are small, such that the dynamics around the equilibrium are slow. After abruptly departing from the n=5 equilibrium, one biomass bump disappears, leading to a rearranged state with four bumps.

We see that even though the summary phase space may let us think that the system passes close to the mixed-state n=4 bis2, we have to keep in mind that this two-dimensional space is obtained by the projection of the infinite-dimensional phase space on a two-dimension summary space. The biomass profiles at three points in time are represented in Fig. c–e, and their corresponding positions are also shown in panel a. In each of panels c–f, the shape of the n=4 bis3 mixed state is also shown. The shape of the dynamical solution is actually close to the n=4 bis3. That equilibrium is unstable, but the system lingers in its vicinity for a considerable period of time before another bump of biomass vanishes, propelling the system towards the stable equilibrium of n=3.

Switching from one state to another by losing one or more vegetation bumps is a known feature of the vegetation pattern model . This mechanism is called sideband instability . We used the same setup with different rainfall values along the n=5 branch. The fact that the system passes close to a mixed state is consistent along the n=5 branch only for values of rainfall for which mixed-state equilibria exist. For rainfall lower than ∼0.91mmd-1, the system jumps directly to the n=3 equilibrium. Now, if we consider the same rainfall but we start from another unstable branch like n=6 or n=7, the dynamics are different. For n=6, we observe a different type of destabilization called period doubling . With that mechanism the system transitions to the n=3 equilibrium without passing through a mixed state, and the system spends less time around the unstable equilibrium of n=6 compared to the n=5 case. This might be explained by the fact that the positive eigenvalue on the n=6 solution is larger than the one on the n=5 solution. For n=7, the transition is even more rapid, as expected for an even larger positive eigenvalue, but the landing point is still n=3. In that case, we observe a destruction of four bumps similar to a sideband instability.