the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Existence and influence of mixed states in a model of vegetation patterns
Lilian Vanderveken
Marina Martínez Montero
Michel Crucifix
Abstract. The Rietkerk vegetation model is a system of partial differential equations, which has been used to understand the formation and dynamics of spatial patterns in vegetation ecosystems, including desertification and biodiversity loss. Here, we provide an in-depth bifurcation analysis of the vegetation patterns produced by Rietkerk's model, based on a linear stability analysis of the homogeneous equilibrium of the system. Specifically, using a continuation method based on the Newton-Raphson algorithm, we obtain all the main heterogeneous solutions for a given size of the domain. We confirm that inhomogeneous vegetated states can exist and be stable, even for a value of rainfall for which no vegetation exists in the non-spatialized system. In addition, we evidence the existence of a new type of solution, which we called "mixed state", in which the equilibria are always unstable and take the form of a mix of two solutions from the main branches. Although these equilibria are unstable, they influence the dynamics of the transitions between distinct stable states, by slowing down the evolution of the system when it passes close to it. Our approach proves to be a helpful way to assess the existence of tipping points in spatially extended systems and disentangle the fate of the system in the Busse balloon. Overall, our findings represent a significant step forward in understanding the behavior of the Rietkerk model and the broader dynamics of vegetation patterns.
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Lilian Vanderveken et al.
Status: final response (author comments only)
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RC1: 'Comment on egusphere-2023-1351', Robbin Bastiaansen, 20 Jul 2023
The manuscript under consideration investigates the spatially explicit vegetation model by Rietkerk. Numerical continuation is used to compute a large part of its bifurcation diagram, which includes the finding of unstable stationary patterned states with peaks on unequal height. It is suggested that these play a crucial role in the dynamics of the system during pattern-to-pattern transitions.
I believe the analysis and results are correct (though some things should be cleared up; see my minor comments below), and the manuscript gives a nice and detailed overview of patterned states in the Rietkerk model. However, many - but not all - things were already studied, observed or remarked before by others, which are not all cited. Thus, the paper in current form seems unable to firmly place itself among the literature, making it feel like it misses some urgency and relevance. But I think this can be solved by better incorporating existing literature, some of which I point out below, and perhaps by slightly expanding some of the new aspects in the manuscript. Hence I cannot recommend the paper in the current form.
MAJOR COMMENTS
1. As the authors point out themselves, the Rietkerk model has been studied before, and bifurcation diagrams were contructed for it before as well. Specifically, the cited paper by Zelnik et al already provides bifurcation diagrams for the Rietkerk model. Therefore, many of the figures (that is, Figures 01 to 05) are not new results (although the authors are very clear about this in the manuscript). The report on the mixed states in this model is new as far as I know. So it is a bit unfortunate that this is only covered on pages 7-8 of the paper. It would help the paper if that section could be expanded a bit, and possible be connected more to ecology or open questions in mathematical pattern formation theory.
For instance, in Figure 08 a n=5 to n=3 transition is followed that visits one intermittent state. But what if n=6 is the starting configuration? Based on e.g. Bastiaansen & Doelman (2018) I expect that to transition to n=3 (i.e. a period doubling), but will that transient visit n=5bis first and then n=4bis before ending on n=3? In general, when there are 2N pulses and they become unstable, I expect N pulses to die off, which I would expect to go in a sort of travelling front of pulse mergings -- that is, thus visiting all sort of mixed state branches. Perhaps authors can shine some light on this using their findings?
2. Section 3.1: here, random initial conditions are used, that are close to the bare soil state. Therefore, this procedure only samples the basin of attraction close to the bare soil state. I don't know what the purpose of that is? For instance, as rainfall decreases this is not the relevant part of the state space, but instead the regions close to the vegetation branches are.
In particular, in a scenario the precipitation might decrease slowly over time (e.g. conform Bastiaansen et al, 2020). So it might be more interesting to see in which basin of attraction a solution is when one of the branches destabilises. I do wonder if it is possible to completely follow a full scenario from uniform vegetation to bare soil including the pattern-to-pattern transitions and see whether they also visit the mixed states.
3. Lines 244-245 & lines 275-276: The importance of unstable states in ecological models has been reported on before. For instance, in Sherrat et al (2020), Morozov et al (2020), Van de Leemput et al (2015), Eigentler & Sherratt (2019), Hastings et al (2018) and Eppinga et al (2021). In the context of spatial patterns in partial differential equations this has also already been observed before, e.g. in Sherratt et al (2009). In these papers, also further references can be found. I suggest to incorporate those papers, including their terminology, to better place the paper into the context of the field.
MINOR COMMENTS
1. Lines 16-17: Turing's article is not about vegetation patterns as is suggested in the text.
2. Line 32: It would be good to have the model parameters in this paper explicitly as well.
3. Lines 42-36: In the (non-spatial) Rietkerk model there is no fold bifurcation; the 'vegetation' branch and the 'bare soil' branch coincide in a transcritical bifurcation instead. So, there you can go smoothly from one to the other, without tipping. Interestingly, the addition of spatial effects here seems to create saddle-node tipping points from patterned vegetation states to other patterned vegetation states or to the bare soil state. Hence, the text in the manuscript seems to misrepresent the findings later in the paper. I suggest to rephrase these sentences to reflect the dynamics in this Rietkerk model.
4. Section 2: The stability analysis including the 'Turing' onset of patterned states has been studied before. This analysis for the here considerd Rietkerk model does appear in e.g. Siero (2020).
5. Lines 98-101: The reasoning here is incomplete; it is not explained how it is clear that two roots are negative (which means they have negative real parts, I suppose?), and how the region in which the other one is positive can be computed. Some more detail on this should be added in the text.
6. Lines 129-136: It reads as if a new code has been written for the continuations instead of using available continuation software. I suggest to say so explicitly in the text. Further, often people use pseudo-arclength continuation to be able to follow the folds in the branches, but the text suggests only natural continuation is used. Is this correct? In any case, I suggest to be explicit about this.
7. Line 136: It would be good to explain why no higher n values are used: are these higher values absent, or were these not considered?
8. Lines 129-136 & lines 155-158: Since a domain with periodic boundary conditions is used, the system is translational invariant. Typically, that leads to problems in the continuation as solutions are never locally unique. How is this handled in the continuation? Further, stability of such periodic solutions could be studied using Floquet theory.
9. Lines 175 & 180: It is suggest by the use of the words 'may' and 'can' that something else could happen besides a bifurcation from which new branches emerge. What would that be?
10. Table 1: I do not understand the contents of this table. I suggest to add a bit more text to the caption to help interpret it.
11. Figure 01: Labels for the different components are missing.
12. Figure 08: The left panel summarizes the infinite-dimensional state space via the mean biomass only. So, theoretically, not necessarily does this show that the passage is close by the mixed state equilibrium. Further, in the right panel, I find it also hard to judge how regular the transient intermittent state is (as the 'n=4bis' equilibrium solution is regularly spaced, but the simulation suggest the intermittent state is not fully regular?). So I suggest to investigate a bit further how close to the unstable equilibrium solution the transient passes in the full state space.
13. Figure 08: The simulations are done for a value for one fixed value of the rainfall for which the n=5 branch is unstable. I wonder if this process is the same for all points on the unstable part of this branch. In particular, in e.g. Siteur et al (2014) and Doelman et al (2012), it is shown that patterned state can destabilise according to different mechanisms (sideband, period doubling, in or out of phase Hopf). Hence, it would be good to check that the reported behaviour is consistent along the unstable part of the branch.EDITORIAL
1. Line 69: P -> B
2. Line 73: R -> R/d
3. Section 2: I suggest to add a sentence saying you have two types of solutions to alert the reader it is not only the first one given, which the the text on line 68 now suggests.
4. Equation (17): in row 2, column 1, I think the 'c' should be deleted, and in row 2, column 3 a minus sign should be added.
5. Line 172: delete either "changes" or "loses".
6. Line 189: "a an other" -> "another"
7. Line 274: "they affect" -> "their effect"
8. Figure 05: "wit ha red" -> "with a red"LITERATURE (not already appearing in manuscript)
Doelman, A., Rademacher, J. D., & van der Stelt, S. (2012). Hopf dances near the tips of Busse balloons. Discrete Contin. Dyn. Syst. Ser. S, 5(1), 61-92.
Eigentler, L., & Sherratt, J. A. (2019). Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bulletin of mathematical biology, 81(7), 2290-2322.
Eppinga, M. B., Siteur, K., Baudena, M., Reader, M. O., van’t Veen, H., Anderies, J. M., & Santos, M. J. (2021). Long-term transients help explain regime shifts in consumer-renewable resource systems. Communications Earth & Environment, 2(1), 42.
Hastings, A., Abbott, K. C., Cuddington, K., Francis, T., Gellner, G., Lai, Y. C., ... & Zeeman, M. L. (2018). Transient phenomena in ecology. Science, 361(6406), eaat6412.
Morozov, A., Abbott, K., Cuddington, K., Francis, T., Gellner, G., Hastings, A., ... & Zeeman, M. L. (2020). Long transients in ecology: Theory and applications. Physics of Life Reviews, 32, 1-40.
Sherratt, J. A., Smith, M. J., & Rademacher, J. D. (2009). Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion. Proceedings of the National Academy of Sciences, 106(27), 10890-10895.
Sherratt, J. A., Liu, Q. X., & van de Koppel, J. (2021). A comparison of the “reduced losses” and “increased production” models for mussel bed dynamics. Bulletin of mathematical biology, 83(10), 99.
Siero, E. (2020). Resolving soil and surface water flux as drivers of pattern formation in Turing models of dryland vegetation: A unified approach. Physica D: Nonlinear Phenomena, 414, 132695.
van de Leemput, I. A., van Nes, E. H., & Scheffer, M. (2015). Resilience of alternative states in spatially extended ecosystems. PloS one, 10(2), e0116859.Citation: https://doi.org/10.5194/egusphere-2023-1351-RC1 -
AC1: 'Reply on RC1', Lilian Vanderveken, 17 Aug 2023
We are grateful to Robbin Bastiaansen for his valuable feedback and insightful comments on our manuscript. We agree with the request of expanding the section on mixed states and appreciate the suggestion of exploring the effect of different initial conditions on transient orbits. The relevant literature will also be added. All technical comments will be addressed point by point in a revised version.
Citation: https://doi.org/10.5194/egusphere-2023-1351-AC1
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AC1: 'Reply on RC1', Lilian Vanderveken, 17 Aug 2023
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RC2: 'Comment on egusphere-2023-1351', Anonymous Referee #2, 21 Jul 2023
The manuscript presents a numerical analysis of the existence of mixed states in a spatially vegetation model as previously introduced by Rietkerk. The main new result is that unstable stationary patterns with different peajs govern the mixing of trajectories and states. The manuscript is well written and surely fits the scope of NPG. I have three main suggestions to improve the clarity and credits to previous works and some technical/minor remarks as outlined below.
Main comments
1. The manuscript provide a detailed analysis of the patterns of the model, although some results have been previously found in existing literature. I would suggest to give proper credits to previous papers.
2. Figure 8 (right): this is an interesting observation and it is the main result of the paper. I would suggest the authors to add more comments on this and to compare their results in terms of striped spatial patterns with those obtained for energy-balance models (e.g., Adams et al., 2003, Alberti et al., 2015). This can strenghten the importance of the authors' main result in terms of inspecting the role of vegetation states into different models and contexts.
3. Figures 8-9: I would suggest the authors to add more details on the possible bifurcations between different states. What about the crossing starting from a different initial equilibrium n?
Technical comments
- Line 14: missing space between the two references.
- Line 24: check the style of units "mm".
- Line 27: check units "mm.day-1".
- Line 30: additional/missing space before/after parenthesis.
- Line 44 and Line 47: check the style of the reference.
- Eq. (5): should be "B" instead of "P"?
- Line 73: is there a missing "d" after k1rw?
- Line 94: delete duplicated "value".
- Line 119: "to" should be "two"?
- Line 129: "discretization" should "discretize"?
- Line 140: check units "mm.day-1".
- Line 166: missing space after comma.
- Line 167: "was" should be "as".
- Line 210: delete duplicated "which".
- Line 233: also the Daisyworld models show similar features (Adams et al., 2003; Alberti et al., 2015).
- Line 233: missing space before the second "Bastiaansen".
- Line 240: this is not proper true since these findings are also observed in Daisyworld models.
- Figure 06: it is difficult to see the pentagone.
- Figure 07: "R = 1,13" should be "R = 1,1"?
Citation: https://doi.org/10.5194/egusphere-2023-1351-RC2 -
AC2: 'Reply on RC2', Lilian Vanderveken, 17 Aug 2023
We greatly appreciate the reviewer's thorough review and the valuable comments provided on our manuscript, for which we are thankful.
Specifically, we have taken good note of the comments regarding proper crediting of previous work, expanding on our main result, and providing additional details on bifurcations between different states. Technical comments are also duly noted, and will be addressed point by point, such as to enhance the manuscript clarity and balance.
Citation: https://doi.org/10.5194/egusphere-2023-1351-AC2
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RC3: 'Comment on egusphere-2023-1351', Anonymous Referee #3, 27 Jul 2023
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AC3: 'Reply on RC3', Lilian Vanderveken, 17 Aug 2023
We sincerely appreciate the reviewer's time and effort. The thorough feedback and insightful comments have provided us with valuable insights for improving the clarity and quality of our work.
Specifically, the reviewer's observations and recommendations regarding various aspects of the manuscript, including definitions, terminology, figures, notation, and other formal issues will be addressed point by point.
Citation: https://doi.org/10.5194/egusphere-2023-1351-AC3
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AC3: 'Reply on RC3', Lilian Vanderveken, 17 Aug 2023
Lilian Vanderveken et al.
Lilian Vanderveken et al.
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