By Robbin Bastiaansen
Many ecosystems are currently threatened by climatic changes that result from global climate change or local human activities like agriculture, infrastructure, and deforestation. Examples of endangered ecosystems include coral reefs, tropical forests, boreal forests, and peatlands. One of the most iconic and well-studied instances of ecosystem degradation is the process of desertification, during which increased aridity causes drylands to lose their vegetation.
The amount of change that an ecosystem can withstand before its functioning changes is called resilience. Dominant classic resilience theory—developed in the 1970s by scientists such as Crawford Stanley Holling [5], Imanuel Noy-Meir [8] and Robert May [7]—is based on the mathematical theory of ordinary differential equations (ODEs). In short, this theory states that ecosystems can adapt to change only up to a critical value (i.e., a tipping or bifurcation point), after which the system undergoes a critical transition wherein the former ecosystem completely disappears (see Figure 1a). In reality, ecosystems do not necessarily degrade completely at once. Instead, ecosystems that are facing resource droughts (e.g., water deficiency) can adapt by forming spatial patterns. Consider drylands, where localized bands or spots of vegetation emerge in otherwise bare soil and enable the ecosystem to persist and keep its functioning (see Figure 2).
Figure 1. Sketches of ecosystem resilience. 1a. Classic view of a critical shift at a tipping point. 1b. The more gradual degradation pathways of patterned systems (green lines are one-dimensional representations of manifolds \(M_j\)) for a fast (red) and slow (blue) change of rainfall \(a\). 1c. Zoom-in on a green line from 1b. The red pathways jump off the green lines before their end because the pattern is irregular and therefore destabilizes early. Figure courtesy of [3].
Because spatial effects are vital for the emergence of these spatial patterns, one cannot understand their resilience through classic resilience theory, which is based on ODEs that do not account for spatial processes. However, it is possible to capture these spatial effects with partial differential equations that explicitly consider space. The easiest of these equations are reaction-diffusion equations, which model spatial effects—such as the spatial transport of nutrients—as diffusion. For instance, a very simple (non-dimensionalized) model of drylands by Christopher Klausmeier [6] accounts for dynamics of vegetation \(v\) and water \(w\) as
\[\left\{ \begin{array}{l} \frac{{\partial w}}{{\partial t}} = \Delta w + a(t) - w - w{v^2};\\ \frac{{\partial v}}{{\partial t}} = D\Delta v - mv + w{v^2}. \end{array} \right.\tag1\]
Here, the reaction terms model rainfall over time (\(+a(t)\)), evaporation (\(-w\)), plant mortality (\(-mv\)), and plant water uptake and ensuing growth (\(wv_2\)). The latter takes this form because the presence of vegetation promotes local water uptake.
Figure 2. Google Earth satellite images of several different vegetation patterns in drylands. 2a. Banded vegetation in Somalia (8°5’N; 47°27’E). 2b. Gapped vegetation in Niger (12°22’N; 2°24’E). 2c. Spotted vegetation patterns in Zambia (15°38’S; 22°46’E). 2d. Maze pattern in Sudan (11°8’N; 27°50’E). Figure courtesy of Google Earth.
Reaction-diffusion systems can exhibit many spatial patterns. They also display behavior that is similar to real ecosystems, including spatial patterns that emerge via a so-called Turing bifurcation [9] as parameters change. When the different components of such a model spread at very different rates, the system is singularly perturbed. This is often the case for ecosystems; in the dryland example, water spreads much faster than vegetation spreads via seed dispersal.
Figure 3. Sketch of patterns in the dryland reaction-diffusion model \((1)\) with vegetation patches located at \(P_1\), \(P_2\), and \(P_3\). Blue lines indicate water density and red lines indicate vegetation density. Figure courtesy of the author.
Using perturbation techniques, it is possible to reduce the dynamics of patterns in these models to two separate processes. First, patterns slowly rearrange themselves to optimize resource uptake; this is called
pattern adaptation. Second, a system experiencing resource deficiency restructures itself into a new pattern on a fast time scale; this is called
pattern degradation. Both of these processes are typical for modeled pattern dynamics and have also been observed in real-life dryland vegetation patterns [4].
To illustrate these processes, consider the dryland model \((1)\) in one spatial dimension [1-3]. Patterns in this model consist of patches of vegetation (see Figure 3); a pattern is fully characterized by the locations \(P_1(t);...;P_N(t)\) of all \(N\) patches. An ODE then captures pattern adaptation and stipulates the way in which these locations change over time [1-2]. Specifically,
\[\frac{{d{P_j}}}{{dt}} = \frac{{D{a^2}}}{{m\sqrt m }}\left[ {{w_x}{{\left( {P_j^ + } \right)}^2} - {w_x}{{\left( {P_j^ - } \right)}^2}} \right],\ \ \ \ \ \ \left( {j = 1, \ldots ,N} \right),\tag2\]
where \(P_j^{\pm}\) refers to taking the upper and lower limits, respectively. This equation shows that any patch moves in the direction from which the greatest amount of water is coming, thus optimizing its own water uptake and ultimately leading to a configuration wherein patches are equally distributed over the available space; this result optimizes the global water uptake. Furthermore, one can visualize \((2)\) as an \(N\)-dimensional manifold \(M_N\) on which pattern adaptation occurs (see Figure 4).
Figure 4. Sketch of manifold \(M_N\). Each point on this manifold represents a different patch configuration. The blue arrows on \(M_N\) indicate the slow pattern adaptation that redistributes patches, leading to the fixed point (blue circle) that is the regularly distributed configuration. The green part of \(M_N\) is the feasible region, containing all configurations that can be sustained under current conditions. The red double arrows indicate the fast pattern degradation, which moves unfeasible configurations from one manifold to another. Figure courtesy of [3].
The pattern degradation process comes into play because not all patch configurations are feasible, which follows from a linear stability analysis. For example, there is an insufficient local water supply to maintain patches that are located too close to each other. Therefore, only part of \(M_N\) consists of configurations that can actually emerge (the green portion of \(M_N\) in Figure 4). The size of this feasible region depends on the amount of rainfall \(a\) and number of patches \(N\). This region becomes smaller for lower \(a\) or larger \(N\), and the regular configuration on \(M_N\)—the fixed point of \((2)\)—is the last configuration on \(M_N\) to exit the feasible region as rainfall \(a\) decreases.
A pattern degradation occurs when an \(N\)-patch pattern exits the feasible region. Some of the patches then disappear, and an \(M\)-patch pattern (with \(M<N\)) remains that is within the feasible region of \(M_M\). But which patches die down? One can determine this answer from the unfeasible pattern’s configuration. If patches are distributed regularly, half of them disappear; if their distribution is irregular, only the smallest patch is lost [2].
Now that we understand the dynamics of patterns in terms of pattern adaptation and degradation, we can determine a pattern's response to climatic change — i.e., a change in rainfall parameter \(a(t)\) in the dryland model \((1)\). Essentially, there are two competing effects. First, pattern adaptation slowly rearranges the pattern into a regular configuration. Second, the feasible region shrinks with decreased rainfall \(a\). Depending on the rate of climatic change, one of the two processes happens faster than the other; this leads to very different degradation pathways [3] (see Figure 1 and Animation 1).
If climatic change occurs slowly, the feasible region shrinks slowly and pattern adaptation fully takes place, leading to a regular distribution of patches over the domain. As rainfall keeps decreasing, the feasible region continues to shrink until the regular pattern degrades and loses half of its patches. The process then restarts. Overall, this leads to sporadic but large pattern degradation events.
Animation 1. Illustration of the different ecosystem degradation pathways, depending on the rate of change. The upper panel depicts rapid change and the lower panel depicts slow change. The animation has been normalized to show the configuration in both situations for the same (changing) environmental conditions (rainfall \(a\)), meaning that more (simulation) time passes per frame in the bottom panel.
But if climatic change occurs quickly, the feasible region shrinks so rapidly that pattern adaptation hardly takes place. Patterns are thus still quite irregular when they exit the feasible region. This phenomenon leads to a form of pattern degradation wherein only one patch dies down, after which the process restarts. Overall, this gives rise to many minor pattern degradation events.
From this it is clear that the behavior of patterned states is different compared to the classic theory of resilience. Instead of a big, sudden collapse, patterns enhance resilience by providing much more gradual degradation pathways. These pathways still contain state transitions (pattern degradations), but the part of the vegetation that withers away allows the remaining vegetation to keep living. The overall ecosystem is therefore able to keep most of its functioning despite climatic change. However, it can still fully degrade if climatic changes are not stopped in time.
A more elaborate discussion on the ecological implications of these results can be found in [3]; more details on the formal mathematical perturbation analysis explained here can be found in [2].
Robbin Bastiaansen presented this research during a minisymposium at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually in May 2021.
References
[1] Bastiaansen, R., Chirilus-Bruckner, M., & Doelman, A. (2020). Pulse solutions for an extended klausmeier model with spatially varying coefficients. SIAM J. Appl. Dynam. Syst., 19(1), 1-57.
[2] Bastiaansen, R., & Doelman, A. (2019). The dynamics of disappearing pulses in a singularly perturbed reaction-diffusion system with parameters that vary in time and space. Physica D: Nonlin. Phenom., 388, 45-72.
[3] Bastiaansen, R., Doelman, A., Eppinga, M.B., & Rietkerk, M. (2020). The effect of climate change on the resilience of ecosystems with adaptive spatial pattern formation. Ecol. Lett., 23(3), 414-429.
[4] Bastiaansen, R., Jaïbi, O., Deblauwe, V., Eppinga, M.B., Siteur, K., Siero, E., ..., Rietkerk, M. (2018). Multistability of model and real dryland ecosystems through spatial self-organization. Proc. Nat. Acad. Sci., 115(44), 11256-11261.
[5] Holling, C.S. (1973). Resilience and stability of ecological systems. Ann. Rev. Ecol. System., 4(1), 1-23.
[6] Klausmeier, C.A. (1999). Regular and irregular patterns in semiarid vegetation. Science, 284(5421), 1826-1828.
[7] May, R.M. (1977). Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature, 269(5628), 471-477.
[8] Noy-Meir, I. (1975). Stability of grazing systems: An application of predator-prey graphs. J. Ecol., 63(2), 459-481.
[9] Turing, A.M. (1990). The chemical basis of morphogenesis. Bullet. Math. Biol., 52(1), 153-197.
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Robbin Bastiaansen is a postdoctoral researcher at the Institute for Marine and Atmospheric Research Utrecht at Utrecht University. He received his Ph.D. in applied mathematics from Leiden University in 2019. Bastiaansen’s research focuses on applications of dynamical systems, particularly in the climate system and including subsystems such as ecosystems. |