Water-limited vegetated ecosystems driven by stochastic rainfall: feedbacks and bimodality

Abstract

In arid or semi-arid ecosystems, water availability is one of the primary controls on vegetation growth. When subsurface water resources are unavailable, the vegetation growth is dictated by the rainfall, and the random nature of the rainfall arrivals and quantities induces a probability distribution of soil moisture and vegetation biomass via the coupled dynamic equations of biomass balance and water balance. We have previously obtained an exact solution for these distributions under certain conditions, and shown that the mapping of rainfall variability to observed biomass variability can be successfully applied to a field site. Here, we expand upon our earlier theoretical work to show how the dynamics can give rise to more complicated, bimodal (and multimodal) structures in the biomass distribution when positive feedbacks between growth and water availability are included. We also derive some new analytical results for the crossing properties of this system, which enable us to determine on what time scale the effects of these feedbacks will be felt, and, relatedly, how long the system will take to cross between different modes.

1. Introduction

The ability of plants to grow is dependent on the presence of sufficient water to meet their transpirative needs. In a dryland vegetated ecosystem, it is typically assumed that the dominant control on growth is water availability via precipitation; moreover, the precipitation is characterized not only by its total or average amount, but also by its strongly intermittent and random nature [1]. Vegetation growth is not driven directly by precipitation, but rather indirectly through the soil, which mediates the transpiration demands and the infiltration supply. The probability distribution of soil moisture which results from these processes forms the backbone of the ecohydrological framework developed by Rodriguez-Iturbe & Porporato [2] and Laio et al. [3]. Because the soil moisture typically evolves on a time scale which is fast relative to changes in vegetation [4,5], it is possible to model the soil moisture dynamics with vegetation parameters held constant. Predictions about what those vegetation parameters will or should be (i.e. modelling of vegetation biomass process) can then be inferred post hoc, e.g. based on optimality principles such as water stress avoidance [6,7].

There have been various efforts (many of them recent) to analyse the joint dynamics of biomass and soil moisture (some examples are given shortly), and, broadly speaking, there are two primary difficulties in the formulations of these models. (i) Of the many feedbacks through which plants interact with and modify their (hydrological) environment [8–11], it is not always clear which mechanisms are primary and (ii) there are a wide range of spatial [12] and temporal [13] scales which are relevant to these dynamics, and much work remains to link them [14]. Consequently, the modelling of such systems admits a variety of distinct approaches. Some models focus on the (lateral) spatial interactions between vegetation through a diffusive term [15] which allows water and biomass to drift into adjacent sites. Some introduce auxiliary dynamic variables, such as the surface water [16–18], multiple soil moisture values for distinct depth layers [19] or multiple biomass values corresponding either to multiple species [19,20] or to different biomass functional types, such as wilted biomass [21]. All of these models make compromises. The spatial diffusion models often do not incorporate rainfall stochasticity (though [22] does so), almost all use the vertically integrated soil moisture ([23] discusses a vertically resolved, stochastically driven soil moisture, but with constant vegetation) and high-resolution models such as [24] which do fully resolve the soil and hydrological forcing tend to focus on single plants and shorter time scales (as this degree of resolution at the landscape scale would be prohibitive).

By contrast, our recent work [5] belongs to the class of minimalistic point models, other examples of which are found in [25–28]. All of these works allow for an interacting soil–vegetation dynamics, though our result was novel in the sense that we were able derive an exact steady-state probability distribution for the joint state space corresponding to those dynamics for realistic hydrological forcing without abstracting the stochastic forcing on response as a Wiener (i.e. Brownian motion) process as in [26] or integrating out short time scales (e.g. sub-annual, as in [27]). Our derivation permitted various functional choices to exactly specify the dynamics, but we considered only a specific form which captured the primary water inhibition on growth (whereby growth is limited by the absence of sufficient water to meet the transpirative demands of more biomass), but did not explore additional feedbacks arising from the other ways in which vegetation might modify its hydrological environment, especially through canopy interception, infiltration promotion and evaporation suppression [9,10].

Our intention here is to incorporate these mechanisms and to understand and characterize their effect on the probabilistic behaviour of the system. Our goals are threefold, as follows.

1. To explore the influence of different functional and parameter choices in our model with regards to qualitative regime transitions in the probability density functions, particularly the emergence of multiple modes in the biomass distribution.

2. To introduce a new characterization of the system via the level crossing properties, which can be applied to understand the time scales associated with mode switching and extreme excursions.

3. To provide an alternative understanding of the positive feedbacks focused on temporal variability at the point rather than spatial variability; in the spatial diffusion models, (spatial) variation in biomass between points creates differentials in water availability, driving the dynamics, whereas here it is the randomness of rainfall which creates (temporal) variation of the biomass.

We are of course well aware that such a simplified model as ours has limits to its explanatory power, and that a steady-state analysis neglects the importance of transient behaviours, such as those arising due to seasonal forcing [29]. However, we believe this model helps to distil the effects of rainfall intermittency, and that the simplicity and transparency of the derived analytical results will shed some light on the role of soil moisture–vegetation feedbacks.

2. The soil moisture–vegetation interaction model

(a). Model introduction

Following our previous work (namely [5] but also [4,20]), we describe the coupled dynamics at the daily time scale according to

$$\frac{\mathrm{d}B}{\mathrm{d}t}=g(B)(\alpha \eta (S)-\beta )$$ 2.1

and

$$\frac{\mathrm{d}S}{\mathrm{d}t}=-h(B)\eta (S)+I(t),$$ 2.2

where B (kg m⁻²) is the biomass per unit area and S (dimensionless) is the relative soil moisture in the root zone (i.e. the fraction of the pore space occupied by water), described according to a ‘bucket’ model of soil moisture, vertically integrated over the root zone (e.g. [2,3,30]); the total volume of soil pore space per unit area in the root zone available to store water is the bucket depth nZ_r, the product of porosity n (dimensionless) and root zone depth Z_r (mm). The various terms appearing on the right-hand side of equations (2.1) and (2.2) can be understood as follows. I (d⁻¹) is the infiltration rate (i.e. rainfall post interception) normalized by the bucket depth; it is taken to be a marked Poisson process with arrival rate λ (d⁻¹) and exponential depths with mean size 1/θ (dimensionless). The function η(S) gives the soil moisture dependence of evapotranspiration; it is increasing and convex, and goes to unity at the point where S is no longer limiting. The function h(B) (d⁻¹) captures the biomass dependence of the evapotranspiration rate. The function g(B) (kg m⁻²) gives the scaling of the net biomass gain rate and loss rate with the existing biomass. In light of these definitions, the parameters α (d⁻¹) and β (d⁻¹) may be understood as, respectively, the non-moisture-limited assimilation rate and the biomass loss rate per unit of biomass. Leakage is neglected here, in keeping with the observations that evapotranspiration typically¹ dominates the water balance in arid and semi-arid regions [32,33].

In [5] (with the exception of electronic supplementary material, appendix B), evaporation was neglected, and only the case of a well-vegetated landscape, where transpiration dominated, was considered. Here, the partition between evaporation and transpiration is of primary interest, and so we explicitly write

$$h(b)=E(b)+T(b),$$ 2.3

which are, respectively, the maximum (i.e. when moisture is not limiting) evaporation rate and transpiration rate at a given biomass B. Also, when previously modelling precipitation interception, we used a threshold (censoring) description, in which the first Δ units (in this case, expressed as a fraction of the soil moisture bucket) of rainfall are intercepted and evaporated before infiltrating the soil. It has been derived in numerous places (e.g. [3,30]) that such a censoring of a Poisson process with rate λ_p and exponential marks with mean 1/θ produces a new Poisson process with the same mark distribution (exponential), but with a new arrival rate λ = λ_p e^−θΔ. We took the interception depth Δ to be a constant, but our derived distributions permit it to vary with B, and we allow for that dependence here,

$$\mathrm{\lambda }=\mathrm{\lambda }(B)={\mathrm{\lambda }}_p\hspace{0.17em}{\mathrm{e}}^{-\theta \mathrm{\Delta }(B)}⟺R_{\mathrm{\lambda }}\equiv \frac{\mathrm{\lambda }}{{\mathrm{\lambda }}_p}={\mathrm{e}}^{-\theta \mathrm{\Delta }(B)},$$ 2.4

which will enable us to account for the role of canopy interception and infiltration. Thus, modelling our various soil moisture–vegetation feedbacks and closing the model requires specification of the functions g(B), E(B), T(B), λ(B) and η(S). The detailed physical basis for these selections, as well as some common forms used in the literature, is deferred to electronic supplementary material, appendix A, in the interest of brevity. The essence of the system though is that if the positive feedback induced by the tendency of increased vegetation biomass to promote infiltration and suppress evaporation (i.e. decreasing Δ_s and decreasing E, respectively) is sufficiently strong, the system can qualitatively change to acquire multiple modes in the biomass probability distribution. In §5, we introduce a minimalistic composite feedback function—related to the physical functions previously enumerated—to make this notion precise. The functional possibilities of §5 and electronic supplementary material, appendix A, are summarized in table 1.

Table 1.

Summary of function choices in specifying the soil moisture–vegetation model. Refer to equations (2.1)–(2.4) and (5.1)–(5.7) and equations (A.1)–(A.6) in the electronic supplementary material, appendix A. The ‘—’ indicates that the function is not independently specified in the reduced formulation of §5. Note that specification of λ(B) and E(B) will be of primary importance, with the other functions of B playing a lesser role. Owing to the separable structure demonstrated in §3, the specific choice of function η does not change the biomass distribution, though it can affect the crossing rate, as shown in §5.

function	(typical) physical form	reduced form (see §5)	description
h(B) =	E(B) + T(B)	—	ET when not limited by moisture; the sum of evaporation and transpiration components. Equation (2.3)
T(B) =	γg(B)	same	Transpiration, here taken directly proportional to assimilation, so that the water use efficiency is constant. Equation (A 4) in the electronic supplementary material, appendix A
E(B) =	$\{\begin{array}{ll}\text{various}& \\ E_0{e}^{-a_{\mathrm{c}}\hspace{0.17em}B}& \end{array}$	$\begin{array}{c}1-(R_{\lambda }-\stackrel{\equiv E/\gamma }{\overbrace{b_E}})\\ =F(B)=\\ \{\begin{array}{cc}{F}_0& B<b_F\\ 0& B\ge b_F\end{array}\end{array}$	Direct evaporation from the soil, decreasing with biomass from shading. Equation (A 3) in the electronic supplementary material, appendix A
λ(B) =	${\lambda }_p\underset{\equiv R_{\lambda }(B)}{\underbrace{{\text{e}}^{-\theta ({\mathrm{\Delta }}_{\text{c}}(B)+{\mathrm{\Delta }}_s(B))}}}$		Post-interception arrival rate of rainfall, increasing at low B due to infiltration promotion equation (2.4) and equation (A 2) in the electronic supplementary material, appendix A. This effect is combined with the above evaporation suppression in a new minimalist model (§5) in terms of the feedback function F. Equation (5.7)
Δc(B) =	dc B	—	Interception depth of rainfall at leaf surface, increasing with biomass. Equation (A 1) in the electronic supplementary material, appendix A
Δs(B) =	$\{\begin{array}{l}\text{various}\\ {\mathrm{\Delta }}_s=\frac{1}{\theta }\ln (\frac{B+c_I}{B+c_I{I}_0})\end{array}$	—	Interception depth of rainfall at soil surface, decreasing with biomass. Equations (2.4)
g(B) =	$\{\begin{array}{ll}B& B<b_g\\ m\cdot (B-b_g)+b_g& B>b_g\end{array}$	same	Growth feedback controlling the total dynamical rate and partitioning; some biomass above Bg is partitioned into inert types. Equations (5.3)–(5.5)
η(x) =	$\{\begin{array}{ll}{m}_{\eta }X& X<\frac{1}{m_{\eta }}\\ 1& X>\frac{1}{m_{\eta }}\end{array}$	same	Soil moisture inhibition on growth. Equation (A 6) in the electronic supplementary material, appendix A

3. Characterizing the system: the steady-state description

(a). The meaning of ‘steady state’

The (probabilistic) steady state is formally obtained by allowing the dynamics of equations (2.1) and (2.2) to operate for an infinite amount of time. This is theoretically useful for distilling the effects of the intermittent rainfall which drives the system, but in practice these dynamics have characteristic adjustment time scales over which they realize the features of the infinite time limit, and the significance of the steady state has to be understood with respect to the relationship between these time scales and the others on which the system might operate and respond. There are some processes, such as the seasonal variation of rainfall, which we expect to have comparable time scales to the steady-state adjustment, and these we are simply choosing not to incorporate. Meanwhile, there are infrequent events such as species invasion, drought, fire or perhaps anthropogenic disruption which we are not neglecting out of hand, but which we posit occur on relatively long time scales well separated from the steady-state adjustment, and so it makes sense to use the steady state to characterize the system behaviour in the interim. However, there are cases where the steady-state behaviour of our dynamics only makes sense in light of the longer term processes in which they are embedded, as discussed in §4. The rest of this section focuses on the probability distributions of the state variables (§3b), and on their crossing properties (§3c) corresponding to the steady state, provided it exists.

(b). The steady-state probability distribution

The evolution of the joint biomass and soil moisture probability density function is governed by the Kolmogorov equation,

$$\begin{array}{rl}{\mathrm{\partial }}_t{f}_{B,X}(b,x,t)& ={\mathrm{\partial }}_x[h(b)\eta (x)f_{B,X}(b,x)]-\mathrm{\lambda }{f}_{B,X}(b,x)+\mathrm{\lambda }\theta {\int }_0^x{\mathrm{e}}^{-\theta (x-z)}{f}_{B,X}(b,z)\hspace{0.17em}\mathrm{d}z\\ & -{\mathrm{\partial }}_b[g(b)(\beta -\alpha \eta (x))f_{B,X}(b,x)].\end{array}$$ 3.1

We discuss the derivation and meaning of this equation in [5]. To summarize, the left-hand side represents the net rate of accumulation of probability at an arbitrary point in the state space, and the right-hand side gives the transport rate (due to jumps and advection) to that point; the equation thus represents the conservation of total probability. The steady-state distribution arises when the transient influence of the initial conditions has abated, so the solution is no longer changing in time, f_B,X(b, x, t) = f_B,X(b, x). The specific form of this solution is given in [5], and a slight generalization is presented in the electronic supplementary material, appendix B, following the same procedure. The solution turns out to be separable,² f_B,X(b, x) = f_B(b)f_X(x), and each of the marginal distributions is generated by an ODE, which may be integrated:

This solution covers both zero and non-zero biomass steady states, according to

$$\frac{⟨\mathrm{\lambda }⟩}{⟨h⟩}=\hspace{0.17em}\{\begin{array}{lll}\frac{\beta \theta }{\alpha }& ⟨B⟩\ne 0& \\ \frac{\mathrm{\lambda }(0)}{h(0)}\equiv \frac{{\mathrm{\lambda }}_0}{E_0}& ⟨B⟩=0,& \text{i}\text{.e}\text{. }{f}_B(b)=\delta (b).\end{array}\hspace{0.17em}$$ 3.4

Here, 〈 · 〉 denotes an expectation with respect to the biomass distribution, which may be either that given in equation (3.3) or the Dirac delta function δ(b) when the system either starts with no vegetation or cannot maintain vegetation at steady state (see §4a). In the latter case, the expectation of the (biomass dependent) quantities λ(B), h(B) simply becomes their value at B = 0, namely λ(0)≡λ₀ and h(0)≡E₀, as per table 1.

(c). Crossing properties

The steady-state distribution tells us the fraction of time the system will spend in a current biomass–soil moisture state (or equivalently the fraction of systems in an ensemble which can be found in that state at a single time), but it does not tell us how frequently such a state is accessed or for how long the system dwells there. These are important features of dynamics, as there may be critical levels which lead to major disruptions of the system, and moreover we will see a posteriori that under some conditions the steady-state distribution may have multiple distinct modes, the frequency of switching between which we will want to know.

Consider an arbitrary region Ω as shown in figure 1a. The continuous probability mass-flux field (the product of the probability density function and the continuous advection terms in the governing dynamics) f_B(b)f_X(x)[(αη(X) − β),  − h(b)η(X)] will advect probability into the region over those parts of the boundary ∂Ω where the flux field points inward (in figure 1b, this part of the boundary is shown in solid, in contradistinction to the dashed portion where advection is outward). Note that the general flux is always downward, i.e. decreasing with respect to soil moisture, but the direction of the biomass advection changes at the critical value x_c, where the soil moisture where assimilation just balances losses 0 = αη(x_c) − β. Meanwhile Ψ is defined as the set of points outside of Ω but beneath at least one point in Ω, so that, equivalently, Ψ is the region from which soil moisture jumps can enter Ω. The mean rate of entry is given by the sum of these contributions:³

$$\begin{array}{rl}{\nu }_{\mathit{\Omega }}& ={\int }_{l\in \mathrm{\partial }\mathit{\Omega }}{f}_B(b)f_X(x){[(-h(b(l))\eta (x(l))\mathbf{x}+g(b(l))(\alpha \eta (x(l))-\beta )\mathbf{b})\cdot (-\mathbf{n})]}_{+}\hspace{0.17em}\mathrm{d}l\\ & +{\int }_{\mathit{\Psi }}{f}_B(b)f_X(x)\mathrm{\lambda }(b)({\int }_{x+y\in \{\mathit{\Omega }\cap b\}}{f}_Y(y)\hspace{0.17em}\mathrm{d}y)\mathrm{d}b\hspace{0.17em}\mathrm{d}x,\end{array}$$ 3.5

where the boundary curve is parametrized by l, n is the outward pointing normal vector, and f_Y(y) = θ e^−θy is the distribution of infiltration pulses. When the region is a simple rectangle (figure 1b), this can be written more explicitly as⁴

$$\begin{array}{rl}{\nu }_{\mathit{\Omega }}& ={\int }_{x_1}^{x_2}{f}_X(x)(f_B(b_1)g(b_1){[\alpha \eta (x)-\beta ]}_{+}-f_B(b_2)g(b_2){[\alpha \eta (x)-\beta ]}_{-})\hspace{0.17em}\mathrm{d}s\\ & +{\int }_{b_1}^{b_2}{f}_B(b)\mathrm{\lambda }(b)({\int }_0^{x_1}{f}_X(x)({\mathrm{e}}^{-\theta (x_2-x)}-{\mathrm{e}}^{-\theta (x_1-x)})\hspace{0.17em}\mathrm{d}x)\mathrm{d}b.\end{array}$$ 3.6

As we are primarily interested in the biomass distribution, we may take the upper boundary to infinity and the lower boundary to zero, thereby eliminating advection on those segments (since the flux field vanishes) and eliminating the jump-access region Ψ. If the left boundary is taken to zero, or the right boundary to infinity, we have the mean rate of up or down crossings of the biomass level, which upon integration over the X region [0, x_c] (for downcrossings) or [x_c, ∞] (for upcrossings) gives the much reduced expression⁵

$$\nu (\xi )=[\left(\frac{\alpha }{\theta }\right)\eta (x_{\mathrm{c}})f_X(x_{\mathrm{c}})]g(\xi )f_B(\xi )=[\left(\frac{\beta }{\theta }\right)f_X(x_{\mathrm{c}})]g(\xi )f_B(\xi ).$$ 3.7

Following the same argument as in [2], the mean excursion times below or above the level are, respectively (with F≡F_B the cumulative distribution function of biomass):

$${\overline{T}}_{-}(\xi )=\frac{F(\xi )}{\nu (\xi )}$$ 3.8

and

$${\overline{T}}_{+}(\xi )=\frac{1-F(\xi )}{\nu (\xi )}.$$ 3.9

Note that while the probability density function for biomass was independent of the soil moisture inhibition function η, the crossing frequency and crossing times are not; they depend on it through the value of f_X(x_c). However, this dependence is quite structured, and has a physical significance. The soil moisture state X = x_c represents the boundary between increasing and decreasing biomass trajectories, and if the system spends a lot of time there, then the biomass trajectories will more frequently switch sign, contributing to a relatively higher rate of crossing any level. However, it can be shown that the soil moisture factor f_X(x_c)/θ is less than 1, so the overall influence is to buffer the hydrological process which drives the biomass dynamics, resulting in less randomness, fewer crossings and longer excursion times than in the no-storage limit which we discuss in [5]. In brief, this is the limit mathematically corresponding to taking h(b) → ϕh(b), α → ϕα, and then ϕ → ∞, so that the relative strength of evaporation to transpiration is preserved, as is the water use efficiency, but water is used immediately upon arrival. It is clear from equation (3.3) that the biomass distribution is unchanged (since ϕ will cancel out), and it can be shown (see electronic supplementary material, appendix B.3) that f(x_c) → θ. In this limit, the crossing rate becomes

$$\nu (\xi )=\beta g(\xi )f_B(\xi )\text{[no-storage limit crossing rate]}.$$ 3.10

If the no-storage limit is thought not to apply, then f_X(x_c) must be computed explicitly for a given choice of parameters and function η; some examples are given in §5, and the explicit formulae for the distributions are provided in the electronic supplementary material, appendix B.2.

Figure 1.

An example of the probability flux fields generated by the steady-state distribution and the system dynamics. Let us think of many instances of the random process as being particles moving in the space (b, x). In between rainfall arrivals, particles flow smoothly like a fluid, and the light blue arrows represent advective transport by this continuous part of the dynamics, in which soil moisture is decreasing as the vegetation transpires, and biomass is decreasing below x_c and increasing above it, depending on whether there is enough water for assimilation to compensate for the losses. Wherever the arrows point into the region Ω, a contribution to the total entry rate is being made. Particles in this system can also enter Ω by jumping from below, i.e. by an instantaneous increase in soil moisture when rainfall arrives. The region from which it is possible to make such a jump is indicated as Ψ. The total rate at which particles enter Ω is the sum of the advective transport rate—obtained by integrating over all parts of the boundary ∂Ω with inward directed arrows—and jump rate—obtained by integrating over Ψ. This is given algebraically in equation (3.5) for the case of arbitrary Ω, as depicted in (a). The expression can be simplified for a simple rectangular region Ω = [b₁, b₂] × [x₁, x₂], as shown in equation (3.6), corresponding to (b), in which the points on the boundary where advection is into the system (and over which the advection integral is taken) are shown explicitly with solid lines, while dotted lines indicate advection outward. (a) An arbitrary region Ω and (b) a simple rectangular region.

(d). Mean first passage times

The mean first passage time (MFPT) generalizes the mean excursion time, with the ‘mean’ in question applied to trajectories with arbitrary start and end conditions rather than just the return time to the initial state. For the case above (with a single biomass level of interest), the MFPT would be denoted ${\overline{T}}_{\xi }(b,x)$, i.e. the mean time to go from an initial point (b, x) to the biomass level ξ. This is related to the mean excursion time by

$${\overline{T}}_{+/-}(\xi )=\mathbb{E}\left[\frac{{\overline{T}}_{\xi }(b,x)\hspace{0.17em}|\hspace{0.17em}\{x>x_{\mathrm{c}}\}}{\{x<x_{\mathrm{c}}\}}\right],$$ 3.11

which is to say that the mean excursion time above/below ξ is the average MFPT over all points which are initially at b = ξ and increasing/decreasing in b, i.e. with {x > x_c}/{x < x_c}. Unlike the excursion time, the MFPT does not have a simple closed form solution; it may be obtained from a PDE as given in the electronic supplementary material, appendix B.4. Nor does the MFPT inherently refer to the steady state; as the process is Markovian, conditioning on the initial state (b, x) obviates the history of the process up to that point. However, there is a particular useful MFPT which is associated with the steady state, namely the mean time of a random member of the steady-state ensemble to reach the level (from above/below). Because this quantity is the average of a mean, we denote it with a double bar,

$${\overline{\overline{T}}}_{+/-}(\xi )={\int }_{\xi /0}^{\mathrm{\infty }/\xi }{\overline{T}}_{\xi }(b)f_B(b)\hspace{0.17em}\mathrm{d}b.$$ 3.12

Unlike the general MFPT, this quantity can be analytically related to the mean excursion time as follows. Define n(t) as the density of systems undergoing an excursion of time t (either above or below the level ξ; the proof is identical), so that n(t) dt is the number of systems in the ensemble currently undergoing an excursion of time t∈[t, t + dt]. Define g(t) as the excursion time density function. Then the rate at which new systems begin excursions in the time range [t, t + dt] is the rate of level crossing ν(ξ) multiplied by the probability g(t) dt of a system entering into such a trajectory,

$$r_{\mathrm{in}}(t)=\nu (\xi )g(t)\hspace{0.17em}\mathrm{d}t.$$ 3.13

The rate at which systems exit from an excursion in this range is the exit frequency per system 1/t multiplied by the number of systems n(t) dt currently in the range,

$$r_{\mathrm{out}}(t)=\frac{n(t)\hspace{0.17em}\mathrm{d}t}{t}.$$ 3.14

At steady state, these two quantities must be equal, so that

$$n(t)=tg(t)\nu (\xi ).$$ 3.15

Among all systems currently undergoing such an excursion, the average excursion time $\overline{\tau }$ in which they are engaged is

$$\overline{\tau }=\frac{{\int }_0^{\mathrm{\infty }}tn(t)\hspace{0.17em}\mathrm{d}t}{{\int }_0^{\mathrm{\infty }}n(t)\hspace{0.17em}\mathrm{d}t}=\frac{{\int }_0^{\mathrm{\infty }}{t}^{2}g(t)\hspace{0.17em}\mathrm{d}t}{{\int }_0^{\mathrm{\infty }}tg(t)\hspace{0.17em}\mathrm{d}t}=\frac{{\overline{T}}^2}{\overline{T}}>\frac{{\overline{T}}^2}{\overline{T}}=\overline{T},$$ 3.16

where $\overline{T}$ is an abbreviation for the mean excursion time as previously defined ($\overline{T}\equiv {\overline{T}}_{+/-}(\xi )$), and the inequality above follows from Jensen's inequality. As the completed fraction of an excursion for a randomly chosen system is uniformly distributed, the average time remaining in the excursion will be half of the excursion length. Putting these results together gives

$${\overline{\overline{T}}}_{+/-}(\xi )=\frac{\overline{\tau }}{2}>\frac{{\overline{T}}_{+/-}(\xi )}{2}.$$ 3.17

Thus, one half of the mean excursion time, e.g. above the level, forms a lower bound on the mean time for a system at steady state starting above the level to downcross it for the first time. This is an important result for the case discussed in §4c, in which the latter quantity $\overline{\overline{T}}$ may have a physical interpretation but the former quantity $\overline{T}$ does not.

4. Interpreting the (non-)existence of the steady state

Beyond the conditions inherent in the structure of the dynamics (equations (2.1) and (2.2)), we have thus far not discussed when the steady state described by equations (3.2) and (3.3) will prevail, or equivalently under what circumstances it will fail. These conditions could have been laid out a priori, as strictly mathematical conditions for the above to hold, but their meaning is more readily apparent in light of the structure of the distributions. Below we will consider the requisite conditions of the low biomass limit, those of the high biomass limit, and finally what can be gleaned from the steady-state distribution even when the steady-state distribution ‘fails’ to exist.

(a). Low biomass conditions

The steady-state distributions describe the system state after an arbitrarily long time, and if vegetation is to persist all the while, then the probability of reaching a zero biomass state in finite time must vanish, as the dynamics here do not allow the system to recover from such a state (i.e. there is no external source of new biomass). This requires⁶ lim_b → 0g(b)/b < ∞, and we will assume that g∼b for b small. The system may nevertheless decay towards a non-vegetated state if the vegetation is not well adapted. Note that normalizability of the biomass density function requires that lim_b → 0bf_B(b) = 0 (this is a general feature of density functions; if f_B(b) grows more quickly than 1/b as b → 0, its integral in that neighbourhood will be unbounded), and so

$$\begin{array}{rl}0=\underset{b\to 0}{lim}b{f}_B(b)=\underset{b\to 0}{lim}g(b)f_B(b)& =\underset{b\to 0}{lim}\hspace{0.17em}{\mathrm{e}}^{-(\theta /\alpha )\int ((h(b^{\prime })-(\alpha /\beta \theta )\mathrm{\lambda }(b^{\prime }))g(b^{\prime }))\hspace{0.17em}\mathrm{d}{b}^{\prime }}\\ & ⟹\hspace{0.17em}h(0)=E_0<\frac{\alpha }{\beta \theta }{\mathrm{\lambda }}_0.\end{array}$$ 4.1

In other words, if the water uptake from evaporation is too strong relative to the infiltration rate in the limit of no biomass, system trajectories nearing that state will be unable to recover, leading to a zero-biomass steady state; we made such an observation in Schaffer et al. [5]. However, we can give some more meaning to the definition of ‘too strong’ evaporation. Consider the steady-state soil moisture distribution for a zero-biomass system, for which⁷

$${⟨\eta (X)⟩}_0\equiv ⟨\eta \hspace{0.17em}|\hspace{0.17em}B\equiv 0⟩=\frac{{\mathrm{\lambda }}_0}{\theta E_0}.$$ 4.2

Combining equations (4.1) and (4.2) gives the condition for non-zero biomass,

$${⟨\eta (X)⟩}_0>\frac{\beta }{\alpha }.$$ 4.3

However, this turns out to be the same as the condition for invasibility of the zero-biomass landscape. Suppose a small amount of biomass ϵ were introduced into such a landscape and followed the dynamics of equation (2.1). Then demanding that its relative growth rate be positive (in the sense of expectation, with respect to the extant soil moisture distribution) yields

$$0<{⟨\frac{\dot{ϵ}}{g(ϵ)}⟩}_0={⟨\alpha \eta (X)-\beta ⟩}_0\hspace{0.17em}⟹\hspace{0.17em}{⟨\eta (X)⟩}_0>\frac{\beta }{\alpha },$$ 4.4

and precisely the same condition is obtained, implying that a landscape too harsh to permit biomass to persist is equivalently one too harsh to permit it to invade. The caveat to this observation is that there may be tissue types not resolved (or resolvable) by the dynamics given here, upon which the various quantities above depend. For instance, if an underlying woody structure of branches and coarse roots were established and assumed static, this might be sufficient—e.g. via the mechanism of root infiltration enhancement—to ensure that the normalizability condition be met for the varying types of biomass, such as fine roots and leaves. Moreover, this might be true even if the balance between evaporation and infiltration were unfavourable to vegetation establishment in the completely unvegetated landscape. In short, we must be careful not to improperly elide ‘invasion’ as defined above with vegetation establishment in general, which might hinge on extra-dynamical processes.

In this same vein, we might ask in general: how sensitive are the steady-state results to physical processes omitted from our mathematical description? As an example, suppose that the system fails the normalizability condition, i.e. h(b) − (α/βθ)λ(b) < 0 at b = 0, so that formally the steady-state biomass distribution would collapse to a delta function. But now suppose also that this inequality is satisfied above some small level δ_B. Then a comparably small intervention, say an influx rate of new biomass greater than βδ_B, would be sufficient to ensure that B(t) > δ_B, ∀t, the prohibitive conditions near zero would never be experienced and the biomass distribution might have finite support. Thus, we need some way of understanding what, if anything, the steady-state distribution tells us in such cases, which is the problem considered in §4c.

(b). High biomass conditions

Just as we considered the possibility that the system would collapse towards zero, we also must ensure that it does not escape towards infinity. It is intuitively clear that this is less of a concern, as a water-limited system hardly lends itself to runaway growth, but it is worth being thorough, and there is one special case of interest in this regime.

The system will escape towards infinity in the long time limit if and only if the probability density function decays too slowly to be normalized.⁸ It is sufficient for the following to hold:

$$\underset{b\to \mathrm{\infty }}{lim}(h(b)-\frac{\alpha }{\beta \theta }\mathrm{\lambda }(b))>0,$$ 4.5

which ensures that the distribution will decay at least exponentially for large values of b. This equation is related to the water balance requirement. As 〈I|b〉 = λ(b)/θ (i.e. the arrival rate at biomass level b multiplied by the mean infiltration depth) and 〈h(b)η|b〉 = h(b)〈η〉 = h(b)(β/α) by the separability property (see electronic supplementary material, appendix B, [5]), the above condition can be rewritten as

$$\underset{b\to \mathrm{\infty }}{lim}(⟨h(b)\eta \hspace{0.17em}|\hspace{0.17em}b⟩-⟨I\hspace{0.17em}|\hspace{0.17em}b⟩)>0,$$ 4.6

which reads that the conditional expectation of the evapotranspiration must always exceed the conditional expectation of the infiltration rate for b sufficiently large. If it did not, then high biomass trajectories would tend to produce wetter soils, creating a positive feedback. In practice, this is an easy condition to meet. The post-interception arrival rate λ is bounded (in fact decreasing due to increasing canopy interception at large b values), as is the evaporative component of h, so the condition will be met provided g is strictly increasing. A corner case arises in the case of threshold partitioning (table 1). If g is constant above some value b*, the water balance condition might fail. This situation corresponds to vegetation which grows roots and leaves to a level where infiltration is high and evaporation low, but then switches to storing all newly assimilated biomass (e.g. as non-structural carbohydrate) before the transpiration is sufficient to use all incoming water.

This specific outcome is of course somewhat contrived, as it depends on the possibility of storing an arbitrary amount of inert biomass with no losses, but it reminds us of an important fact about the steady state: depending on the parameter and functional choices, the features resolved by the steady-state probability distribution may arise on time scales that can only be interpreted meaningfully by comparison with processes exogenous to the dynamics posed here, e.g. those of reseeding, mortality or competition.

(c). The cultivated steady state

In §4a, we raised the question of what can be gleaned from the steady-state distribution if some of the feedback functions are not well resolved on certain biomass domains, or if they collapse the steady state entirely. The answer is that the derived distributions may still be useful, provided that the problematic regions in the state space are only rarely visited by system trajectories.

In order to make this notion more precise, we proceed in the vein of Maxwell's demon. Suppose that at the interface between a biomass region where the system is well behaved and one where it is not there sits an agent capable of tweaking system trajectories which reach him. To be concrete (and anticipating some of the examples in the next section), this interface might be at a trough in the biomass density function separating a central biomass mode from a low biomass mode. We would like to use the steady-state density to characterize the central mode without dependence on the low mode, for it might be that the low mode is a trapping state (failing the normalizability condition previously discussed), or it might be that vegetation and soil moisture behaviour are not well understood at low biomass.

To do this, we consider a system cultivated by a demon at level ξ_d separating the modes, who acts to keep system trajectories from leaving the central mode. The demon does this by adding water to all trajectories incoming from the right—which have decreasing biomass and soil moisture in the range [0, x_c]—such that, after his intervention, they have soil moisture greater than x_c, and move back to the right (towards higher biomass), whence they came. If he performs his intervention just so, he can maintain the soil moisture distribution for the outgoing trajectories in keeping with the steady-state distribution. The results of equations (3.2) and (3.3) will hold exactly, but with biomass restricted to the domain b∈(ξ_d, ∞), resulting in the cultivated distribution, which is renormalized,

$$f_{B|{\xi }_d}(b)=\frac{C_{B|{\xi }_d}}{g(b)}\hspace{0.17em}{\mathrm{e}}^{-(\theta /\alpha )\int ((h(b^{\prime })-(\alpha /\beta \theta )\mathrm{\lambda }(b^{\prime }))/g(b^{\prime }))\hspace{0.17em}\mathrm{d}{b}^{\prime }}\text{such that}\hspace{0.17em}{\int }_{{\xi }_d}^{\mathrm{\infty }}{f}_{B|{\xi }_d}(b)\hspace{0.17em}\mathrm{d}b=1.$$ 4.7

We now have the cultivated downcrossing rate and mean excursion time, defined analogously to before,

$${\nu }_{{\xi }_d}(\xi )=\left(\frac{\beta }{\theta }\right)\hspace{0.17em}{f}_X(x_{\mathrm{c}})g(\xi )f_{B|\xi }(\xi )$$ 4.8

and

$${\overline{T}}_{+|{\xi }_d}(\xi )=\frac{1-F_{B|{\xi }_d}(\xi )}{{\nu }_{{\xi }_d}(\xi )}.$$ 4.9

Taking ξ_d = ξ above gives the rate and mean excursion time for the positive mode, when the boundary is cultivated by the demon as described. Note that the new normalization cancels out in the ratio defining ${\overline{T}}_{+|{\xi }_d}(\xi )$, so that it is equal to the previous value ${\overline{T}}_{+}(\xi )$.

This conceptualization gives two useful characterizations of the degree of independence of the central mode from the low. We can describe the size of the intervention required by an agent (perhaps a real one) to keep the system in the central vegetation mode. Specifically, we can compute the average amount of water per unit time Q the demon needed to apply to perform his role as the difference between the average soil moisture of incoming and outgoing trajectories at b = ξ,

$$\begin{array}{rl}Q& =g(\xi )f_{B\hspace{0.17em}|\hspace{0.17em}\xi }(\xi )({\int }_{x_{\mathrm{c}}}^{\mathrm{\infty }}x{f}_X(x)(\alpha \eta (x)-\beta )\hspace{0.17em}\mathrm{d}x-{\int }_0^{x_{\mathrm{c}}}x{f}_X(x)(\beta -\alpha \eta (x))\hspace{0.17em}\mathrm{d}x)\end{array}$$ 4.10

$$\begin{array}{rl}& =g(\xi )f_{B|\xi }(\xi ){\int }_0^{\mathrm{\infty }}x{f}_X(x)(\alpha \eta (x)-\beta )\hspace{0.17em}\mathrm{d}x.\end{array}$$ 4.11

Moreover, we can consider what would happen if the demon were removed, so that the trajectories started to cross into the low-mode region. This seemingly artificial situation has important applications. Suppose that the system has been ‘cultivated’ by its historical environment, which produces the steady-state distribution at levels above ξ, but is then subjected to a subtle ecological or climatological perturbation. This perturbation might not meaningfully alter the dynamics when the system is in a robust⁹ (high biomass) state, but could have a large influence on the low-biomass states when the system is more sensitive, perhaps jeopardizing its ability to recover from such states. This prompts the question: how long would it take the system to ‘feel’ the effects of whatever feedbacks govern the dynamics in the newly exposed low-biomass states b < ξ? The mean excursion time above level ξ can no longer be applied directly, as for all we know ξ might be/have become an absorbing boundary under the perturbation, so that no future trajectories will return to the central mode upon reaching it. However, the results of §3d show that the typical time for the perturbed steady-state system to reach the level ξ is bounded by the mean excursion time, and so it is sufficient that this time is much greater than the persistence time¹⁰ in order to establish that the space b < ξ is well separated from the central mode,

$${\overline{T}}_{+}(\xi )\gg \frac{1}{\beta }.$$ 4.12

5. Feedback quantification: results and implications

In the preceding sections, we discussed how feedbacks might arise and be described. Here we explore the implications of specific parameter and function choices. It is convenient to work with rescaled units, so that β = 1 and μ≡αλ_p/γβθ = 1; time is described in units of the biomass decay time scale, and the reference level of biomass μ, which is the mean value when there are no feedbacks, is the unit mass. Defining also b_E(b) = E(b)/γ as the biomass equivalent of evaporation, the distributions become

$$f_X(x)=\frac{C_X}{\eta (x)}{\mathrm{e}}^{-\theta x+(\theta /\alpha )\int (\mathrm{d}{x}^{\prime }/\eta (x^{\prime }))}$$ 5.1

and

$$f_B(b)=\frac{C_B}{g(b)}\hspace{0.17em}{\mathrm{e}}^{-{\mathrm{\lambda }}_p\int ((b_E(b^{\prime })+g(b^{\prime })-R_{\mathrm{\lambda }}(b^{\prime }))/g(b^{\prime }))\hspace{0.17em}\mathrm{d}{b}^{\prime }}.$$ 5.2

Additionally, it was suggested in §2 that g(B) could be used to model the partitioning of biomass into active biomass such as leaves and roots and inert biomass such as non-structural carbohydrates (i.e. stored energy). Under this interpretation, g(B) represents the active biomass and B − g(B) the stored remainder. If this is the case, then by the chain rule

$$\dot{g}=\frac{\mathrm{d}g}{\mathrm{d}B}\dot{B}=g^{\prime }(B(g))g(\alpha \eta -\beta ).$$ 5.3

As all of the feedbacks previously discussed pertain to the active biomass (previously synonymous with the total biomass), the new system of equations,

$$\dot{g}=g^{\prime }g(\alpha \eta -\beta )$$ 5.4

and

$$\dot{S}=-h(g)\eta +I(g;t),$$ 5.5

is the same as the old one, with B → g, g → g′ = g′(B(g)). The rest of this section is devoted to examining the range of distributions which result for the variables B, g(B) and X. In order to determine the presence and location of the biomass modes, we need to locate the peaks in the distribution, which corresponded to sign changes in its derivative. It follows from equation (5.2) that

$$\text{sign}[f_B^{\prime }(b)]=\text{sign}[\underset{\equiv 1-F(b)}{\underbrace{R_{\mathrm{\lambda }}(b)-b_E(b)}}-g(b)-\frac{g^{\prime }(b)}{{\mathrm{\lambda }}_p}].$$ 5.6

The first two terms on the right define the effective water availability for the plant: the relative amount of infiltrating water minus the share taken by evaporation. The feedback function F(b) identified above is the inverse quantity, and gives a measure of the water potentially available to the plant which is lost. As F(b) encapsulates various underlying feedback mechanisms as discussed in §2, and these mechanisms are extremely variable, we proceed with a simple, synthetic form of F(b),

$$F(b)=\{\begin{array}{ll}{F}_0& b<b_F\\ 0& b\ge b_F.\end{array}$$ 5.7

This step function has an intensity F₀ which captures the effect of limited infiltration and evaporation competition for water at low biomasses, but these effects rapidly dwindle when the vegetation exceeds a critical feedback threshold of b_F. It follows from the arguments of §4 that for a vegetated steady state to exist F₀ < 1, and F₀ > 0 by definition (because 0 ≤ R_λ(b) ≤ 1 and b_E(b)≥0). Figure 2 illustrates the influence of the feedback function within this range.

Figure 2.

Biomass distributions for a range of feedback biomass thresholds b_F, feedback intensities F₀, and precipitation arrival rates. The mean excursion time above the threshold value (i.e. the trough between the modes) is given for each curve of each plot, following the legend. The excursion time calculation here uses the crossing rate ν of equation (3.5), with the requisite values furnished by the explicit distribution forms given in the electronic supplementary material, appendix B.2, taking θ = 40, x_c = 0.03 and x* = 0.1. All values use the rescaled units of equation (5.1) and equation (5.2).

When the feedback is too weak and the threshold too low (see, for example, F₀ = 0.82, b_F = 0.2), the distribution is dominated by the single mode which represents the balance between transpiration and precipitation alone. On the other hand, for very strong feedbacks with high thresholds (see, for example, F₀ = 0.98, b_F = 0.5), the system can never escape the unfavourable low-biomass conditions, and the distribution is dominated by a mode near zero. Intermediate values, however, can produce a non-trivial bimodal distribution. These modes are graphically distinct, and are also temporally well separated; the mean excursion time above the threshold value b_F (i.e. in the central biomass mode) is typically one to two orders of magnitude greater (and sometimes much more) than the unit biomass persistence time scale 1/β. The sharpness of the modes, and consequently the temporal separation, is governed by λ_p; as λ_p increases, the process looks more and more deterministic, and trajectories rarely make large excursions from the model values.

At high biomass, we can consider the effect of partitioning into inert tissue types, using g(b) as in table 1 with m < 1. As shown in figure 3, this high biomass feedback only slightly affects the structure shown in figure 2 or excursion times above the low mode. However, the distribution of the non-partitioned active biomass (leaves, roots) acquires an additional peak at the partitioning threshold, as shown in figure 4. The mode defined by this peak is of a somewhat different nature from the low-biomass mode, as it is generated by saturating a boundary rather than by competing attractors. As a practical consequence, it can only acquire a meaningful share of probability when it impinges on the central mode, where it is not temporally well separated; the mean excursion times below this high mode given in figure 4 are smaller than the excursion times above the low mode given in figure 3.

Figure 3.

Biomass distributions for a range of partitioning feedback thresholds b_g, and intensities 1 − m (the case m = 1 corresponds to no partitioning). The times associated with each plot are for excursions above the same low mode as in figure 2, and are only slightly affected.

Figure 4.

Distributions for the non-stored biomass (rather than the total biomass of figure 3). A new mode emerges, and the times indicated are the mean excursion times below that mode, i.e. in the central/lower mode.

Note that the high and low biomass portions of the distribution are essentially independent of each other; the distribution below b_g in figures 3 and 4 is the same as in figure 2, up to the overall scaling. This is a graphical illustration of the mathematical result that the steady-state distribution, as given by equations (3.2) and (3.3) or (equations (5.1) and (5.2)) obeys

$$\frac{f_B(b_1)}{f_B(b_2)}=\frac{g(b_2)}{g(b_1)}\hspace{0.17em}{\mathrm{e}}^{-{\int }_{b_1}^{b_2}(\cdots )\hspace{0.17em}\mathrm{d}b},$$ 5.8

and so the relative probability on an interval [b₁, b₂] depends only on the values of the feedbacks inside that interval, and the overall distribution looks like a superposition of the feedback effects when those effects occur over disjoint intervals, as above (regarding intervals b < b_F and b > b_g, respectively). For a single variable Markov process, this property would be guaranteed, as the process is memoryless and behaviour inside an interval depends only on the dynamics in that interval. For the two-variable process here, the system state upon entering an interval [b₁, b₂] might depend on the biomass feedbacks outside that interval through the soil moisture dimension, and it is a feature of the specific form of the dynamics here that it does not, i.e. that f_X turns out to be independent of the behaviour of any biomass feedback functions.

Considering equation (5.6), we see that modal structures emerge at any point where the feedbacks ‘kick in’ quickly enough to change the sign, and if these points are well separated then a superposition-like structure emerges in the sense just described. The system might also have long-range feedbacks which act over the whole biomass range and not only over well-defined subintervals, but these will typically modulate the modal structure rather than qualitatively alter it. Of the feedbacks identified in §2, evaporation suppression and canopy interception will continue to influence the dynamics and thus the statistics at higher biomass (above b_F), but, as these are typically of such a long-range nature (and moreover have opposite signs, tending to cancel each other out), we do not expect them to fundamentally alter the general picture given by figures 2–4.

Finally, it is interesting to note that buffering of the infiltration process by the soil moisture creates a significant effect on the biomass crossing time scales for this parameter regime even though it was shown in [5,31] that its effect on the distribution is mild. In particular, we showed in [31] that the time-dependent biomass distribution in a seasonal climate could be well described by the no-storage limit, and that in turn this gave a good description of a field site; the end-of-season biomass statistics were similar to what they would have been if the water arriving in each rainfall event had been instantly transpired. Moreover, the steady-state distribution analysed here is independent of the no-storage approximation, as it features only the ratio of the evapotranspiration and assimilation coefficients. However, using parameter values θ = 40, x_c = 0.03, x* = 0.1, similar to those in [31], introduces a factor of ∼3 in all of the mean excursion time calculations, and figure 5 shows that this buffering effect persists over a relatively wide value range. By contrast, assuming a no-storage model led to only a slight over-prediction of the time-dependent variance, and so we have the interesting result that the temporal buffering effect is much larger than the variance buffering effect.

Figure 5.

The factor f_X(x_c)η(x_c)α/θ introduced in the crossing rate ν of equation (3.5) by the soil moisture distribution. Note that this factor is significant (i.e. meaningfully different from 1) across a large range of values, even though the biomass distribution is identical across variations in these parameters. The factor introduced to the mean excursion time is the inverse of the above values. (Online version in colour.)

6. Discussion

In the preceding sections, we have shown how the various feedbacks on vegetation biomass growth can create qualitatively new modal structures in the biomass distribution when the system is stochastically forced by a random rainfall process; these distributions stand in contrast to those associated with the simpler functional forms of, for example, [4,5,31], which were dominated by a single mode. We have introduced a new characterization for these modal structures in the form of the derived mean crossing times—given algebraically by equations (3.6) and (3.7), and numerically in figures 2–4—and show that these define a time scale which is distinct from the time scales of soil moisture fluctuations and the biomass persistence identified in [5]. Critically, the modal crossing times do not obviously arise from any single fundamental time scale in the vegetation growth dynamics or in the rainfall forcing alone, but emerge in a non-trivial way from the interaction of the two. Taken together, these time scales will provide a way of situating the randomly, hydrologically driven vegetation growth with respect to other processes, and determine whether these processes should be studied hierarchically or jointly, e.g. whether disruptions from fire or anthropogenic causes are playing out in between water-induced modal switching events (as discussed here), or vice versa.

Looking forward, the most natural and perhaps most important extension of this line of research is to apply the stochastic biomass and soil moisture model to problems of competition and fitness, as we have now established the basic model properties [4,5], shown (at least provisionally) that a simple form of that model could successfully map the rainfall variability onto the biomass variability in the field [31], and here demonstrated how more complicated structures can arise in the distribution of biomass. The next step is to ask why these structures arise; how do competitive and evolutionary pressures force the vegetation to adapt to the stochastic hydrological environment we have modelled? Even in cases where our model provides a good representation of the current state of affairs, it is not yet clear how different species might harness the biomass–soil moisture interaction to invade other species or ensure they are not invaded, and this sort of behaviour is not necessarily well resolved by the current model. For example, two species with similar transpiration curves η might exhibit a total biomass in keeping with the system described here, but with one completely eclipsing the other's share of that biomass over time. Other behaviours, such as partitioning into stored biomass as described above, only make sense in light of a competitive environment. Thus, the challenge going forward will be not only to predict the probabilistic structure of the biomass–soil moisture system but also to establish a normative analysis, and answer: when (and why) is such a structure good or bad for the plant species in question?

Data accessibility

This work does not have any experimental data.

Author's contributions

B.E.S. and I.R.I. jointly developed the mathematical models and developed their ecological interpretation. B.E.S. wrote the original manuscript, which was later developed jointly as well. Both authors gave final approval for publication.

Competing interests

We have no competing interests

Funding

This work was supported by National Science Foundation grant no. 1514606 and the Texas Experimental Engineering Station of Texas A&M University.