The four considered watersheds (Gorouol, 23 200 km2; Dargol, 7350 km2; Sirba, 40 300 km2; Nakanbé, 21 800 km2), located in central Sahel (Fig. ), are selected on the basis of the availability of long-term observations from the 1950s to the mid-2010s. The first three are tributaries of the Niger River, and the last one feeds the Volta River. The climate of this semi-arid region is governed by the West African monsoon . Seasonal precipitation is provided by mesoscale convective systems between June and September. Intermittent rivers are supplied by surface runoff generated on hillslope during rainstorms. The study area is essentially rural: the north is dominated by natural pastures grazed by cattle, while the south contains a mosaic of savannahs, rainfed agricultural plots and fallows, where scattered shrub and tree species persist .

Annual precipitation over each watershed are computed based on First, for each gauge station of the watershed, annual precipitation series are obtained by summing daily precipitation for each year. Then, annual precipitation means and anomalies are both interpolated using two kriging methods. Finally, annual precipitation of the watershed is computed as the sum of these two interpolated fields (means and anomalies) averaged spatially over the watershed.

Figure a presents annual precipitation over four watersheds for the period 1956–2014. Past annual precipitation can be grouped in three main periods: wet during the 1950s–1960s, followed by a prolonged period with some severe droughts which ended up in the mid-1990s, and since then a period with a strong inter-annual variability where annual precipitation roughly equals the mean of the whole period .

The annual runoff coefficient K (–), ranging between 0 and 1, is equal to the annual watershed outflow V (m3) per unit of watershed area A (m2), normalized by the annual precipitation P (mm): K=VA×P. Annual watershed outflow V (m3) is calculated from the mean daily discharge Q (m3s-1), over the hydrological year, starting on the first of March. For the Nakanbé watershed, we use discharges corrected for the effect of dams built in recent decades . For the three other watersheds, discharges are merged from two databases: SIEREM and ADHI . We exclude years with more than 13 d of missing consecutive daily discharge values during the wet season (June–September). Otherwise, missing values are set to zero in the dry season, and linearly interpolated in the wet season. A sensitivity study (not shown) confirms that this interpolation has a limited effect on the mean annual discharge (bias<10 %).

Figure b presents the annual runoff coefficient series for each watershed. All watersheds have experienced a similar evolution toward higher annual runoff coefficients although the Northern watersheds display higher coefficients.

In this article, variables are denoted with uppercase letters, whereas parameters are written with lowercase letters. Furthermore, in this study, most Sahelian watersheds have sub-domains that do not contribute to runoff at the outlet due to their endorheic nature. In order to account for this partial contribution, our model only focuses on the contributing areas. Therefore, all physical variables are defined at annual time step and at the scale of the watershed areas contributing to runoff. For instance, the runoff coefficient K refers to the annual runoff coefficient for the areas of the watershed contributing to runoff.

The proposed dynamical model is an extension of the hillslope-scale model proposed by , which reproduced a hydrological shift. Our lumped dynamical model simulates the runoff coefficient K. This parsimonious model has a single external forcing, the precipitation P (mm), and a single state variable, the water holding strength S (–). Intuitively, S represents all physical mechanisms that drive water retention within the watershed including soil infiltration properties associated with vegetation dynamics, seepage losses or conversely runoff connectivity. S ranges from 0 to 1. The higher is S, the lower is K. In the model, the variations in S are assumed to be similar to those of a vegetation cover: S increases and decreases as a function of I (mm), an indicator of wetness representing the potential water for the vegetation, i.e. precipitation minus runoff.

Changes in the water holding strength S are modulated by a feedback loop (Fig. ). The value of S impacts directly the proportion of outflow water K, that indirectly affects the indicator of wetness I, which finally drives the growth and decays of S. These assumptions are based on feedback mechanisms frequently observed in drylands worldwide . Wet conditions favor vegetation development, which in turn favors infiltration and thus vegetation growth (i.e. high S values). Conversely, dry conditions are detrimental to vegetation, favoring bare soil and consequently low infiltration (i.e. low S values).

Mathematically, changes in S are prescribed using a differential equation, where the time derivative dSdt is denoted as S˙. The computation of S˙ can be decomposed into three steps (Eqs. –), illustrated in Fig. , and detailed below.

First, we define the runoff coefficient K as a function of the water holding strength S and the precipitation P: 1K=kmax⋅PaPa+Cabwith C=cmin+S×(cmax-cmin) where C (mm) is the water holding capacity; a (–), b (–), kmax (–), cmin (mm), cmax (mm) are positive parameters (Table ). Equation () is a fully derivable variant (well adapted to ODE solving) of the popular SCS model , and of the equation used by to simulate runoff in the Sahel. It is also very similar to the formalism used by Eq. 20 to represent the dependence of runoff to precipitation and to the watershed-scale water retention capacity, which is analogous to our variable S. Equation () is an S-shape function that controls the precipitation-runoff relationship: no runoff is produced below a certain precipitation amount, the runoff ratio varies only little for heavy precipitation, and it increases roughly linearly for intermediate precipitation. The runoff coefficient K is maximized when precipitation is substantially higher than the water holding capacity (P≫C), i.e. when S is close to 0. Inversely, the runoff coefficient K is close to zero if C≫P, i.e. when S is close to 1. The parameters a and b are shape factors allowing the model to adapt to a wide range of watersheds. This equation, although not properly physically-based, proposes a representation consistent with known hydrological processes.

In the second step, the indicator of wetness I (mm) is derived from the precipitation P and the runoff coefficient K: 2I=P⋅(1-K)

Finally, we further assume that the derivative of the water holding strength S˙ can be calculated from S and I: 3S˙=rg⋅II+ig⋅S⋅(1-S)︷(1)-rd⋅idI+id⋅S︷(2)+μ⋅(1-S)︷(3) where rg (yr-1), ig (mm), rd (yr-1), id (mm) and μ (yr-1) are positive parameters. Equation () contains a growth term (1) and a decay term (2), whose rates (rg and rd) are modulated by I. The balance between these terms depends on S and I, and determines the sign of the derivative of S, hence its variation. The third term of Eq. () is small and only prevents the unrealistic convergence to S=0. It can be interpreted as the representation of all the physical processes not included in the model which increase the water holding strength such as sand deposits by wind and vegetation growth due to seeds imported. In general, this third term μ⋅(1-S) remains negligible as compared to the other terms of Eq. (3), because the range of values of μ (Table 1) are 2 to 3 orders of magnitude lower that the growth and decay rates rg and rd. The only exception is when S becomes close to 0: in this case terms 1 and 2 in Eq. (3) are also close to 0, and S˙ is close to μ (always positive), avoiding the trajectory of S to remain stuck in 0.

For each watershed, the selection of best parameterizations rely on a comparison between observed runoff coefficients Kobserved with simulated runoff coefficients Ksimulated at the watershed scale. The watershed-scale simulated runoff coefficient Ksimulated can be computed from K, the runoff coefficient of the areas contributing to runoff (Eq. ), as follows: 4Ksimulated=f×K where f (–) is the fraction of the watershed (between 0 and 1) that actually contributes to the watershed outlet.

Let θ={kmax,a,b,cmin,cmax,rg,rd,ig,id,μ,f} denote a parameterization of the dynamical model, i.e. a set of parameters for the equations (Eqs. –). For each parameter, values (or ranges of values) are shown in Table .

Selecting a single best parameterization θ* can be prone to equifinality, i.e. different parameterizations lead to similar results . To circumvent this issue, we adopt a three-step approach to select an ensemble of relevant parameterizations:

106 parameterizations are sampled from a Latin hypercube sampling, using a uniform distribution over a range of parameter values (Table ). These ranges were adjusted through preliminary tests. Note that a, b and kmax are fixed.

For each parameterization θ of the ensemble Θ*, we compute its bifurcation diagram (Fig. ). This diagram is a graph that associates each forcing value (precipitation P) with its fixed points (attractors, repeller), i.e. values S such that S˙=0.

Here fixed points are calculated using a continuation method that finds the isoline defined by S˙=0, starting from an initial solution. In practice, we rely on the “pycont-lint” Python package, which is dedicated to numerical bifurcation analysis. We begin by setting the state value close to 0 (S=0.01) for a very low precipitation (P=0.1), and stop the continuation method when the isoline exceeds 4000 mm of precipitation. The precipitation range explored (until 4000 mm) is on purpose larger than the actual precipitation range (Sect. ) in order to assess the sensitivity of the continuation method.

We define that a parameterization θ is monostable if it has a single fixed point (an attractor) for every precipitation value between 1 and 4000 mm, and bistable if it has several fixed points for at least one precipitation value. In this case, the continuation algorithm provides both the stable (attractor) and unstable (repeller) fixed points. Figure  illustrates the bifurcation diagram for a monostable parametrization (Fig. a), and a bistable parametrization (Fig. b). In the bistable case, for any precipitation in the range 700 to 1500 mm, three S values (two attractors, one repeller) can be reached asymptotically depending on the initial state value. We denote as lower and upper branch the lines formed by the lower and upper attractors, respectively.

A bifurcation diagram is a useful tool to define regimes. For monostable parameterization (Fig. a), there is a unique regime, and changes in the state variable S or in the runoff coefficient K adapt to changes in precipitation in a reversible way. For bistable parameterizations, regimes and regime shifts require more subtle definitions, that we describe below.

Regimes and regime shift, i.e. the transition from one regime to the alternative regime, are often conceptualized with state values that remain equal to attractors . However, many systems (including ours) have transient dynamics, where state values remain far from attractors . Here we present two definitions of regimes suited for such a transient case. Both definitions split the bifurcation diagram into two regions: the “Low runoff coefficient regime” and the “High runoff coefficient regime”, that correspond to a high water holding strength and low water holding strength, respectively.

Classically, a regime is associated with an attraction basin, i.e. the set of initial values converging to the same branch of attractors (Fig. a). One drawback of this classical definition is that a state S cannot be directly classified into one regime, because the attraction basin also depends on the precipitation P. In other words, the same state S can switch regimes following changes in precipitation. For instance, S=0.2 corresponds to the “Low runoff coefficient regime” for P=750 mm, and to the “High runoff coefficient regime” for P=600 mm.

We introduce an alternative definition of regimes based on a norm . Visually, regimes are separated by a threshold (Fig. b). This definition is more practical as any state value S can be directly classified into the low or high regime by comparing it with the threshold. Specifically, S is in the “High runoff coefficient regime” if S<♢S=▾S+▴S2 while it is in the “Low runoff coefficient regime” if S≥♢S=▾S+▴S2. For a given bistable parameterization θ, ▴S and ▾S are the water holding strength for the highest attractor of the lower branch and for the lowest attractor of the upper branch, respectively (Fig. b). The associated precipitation values are denoted as ▴P and ▾P.

For each watershed, we study the ensemble Θ* made of the best 1000 parameterizations (Sect. ). Figure  shows the ensemble of simulated runoff coefficients (Fig. a and b) and water holding strength (Fig. c and d). The starting year of the trajectories differs between each watershed as the initial state value S is computed with the first observed runoff coefficient and Eq. (). The Sirba and the Dargol watersheds are initialized in 1956 and 1957, respectively (Fig. a and c), whereas the Gorouol and the Nakanbé watersheds are initialized in 1961 and 1965, respectively (Fig. b and d).

In Fig. , we observe that until 2014 the four watersheds have increasing trends of runoff coefficients (and decreasing trends of water holding strength, consistently with Eq. ). This confirms that this simple dynamical model can reproduce the first-order dynamics (the trend) of the observed runoff coefficient. We note that the ensemble mean water holding strength reaches a plateau around the year 1990 for the Gorouol watershed, and around the year 2000 for the Dargol watershed. A more detailed quantitative assessment of the fit is presented in Appendix .

A bifurcation diagram is computed for each parameterization of the ensemble. Bistable parameterizations represent more than 90 % of the ensemble for all the watersheds, whereas monostable parameterizations amount to 8.1%, 6.4%, 4.4% and 0% for the Dargol, Nakanbé, Sirba, and Gorouol watershed, respectively. For these bistable parameterizations, Fig.  shows the distribution of ▾=(▾P,▾S) the lowest attractor of the upper branch and ▴=(▴P,▴S) the highest attractor of the lower branch. ▾P and ▴P are bifurcation-induced tipping points, i.e. where small changes in annual precipitation can cause a shift to the alternative regime. In general, we find that the distribution of ▾P, a tipping point for the shift to the high runoff coefficient regime, is less spread than the distribution of ▴P, a tipping point for the inverse transition. This is because this inverse transition (the decrease of runoff coefficient) was not observed over the study period, and thus ▴P can hardly be constrained. Lastly, we observe that the drier the watershed (box plots in Fig. ), the more the distribution of ▾P shifts towards lower precipitation values, and the narrower the range of variation of these values becomes.

We analyze regime shifts between 1965 and 2014 (common simulation period for all watersheds) by computing, each year, the percentage of selected parameterizations (from the ensemble) that are the “High runoff coefficient regime”.

When regimes are defined with attraction basins (Fig. a), this percentage is non-null in 1965 and increases afterwards for all watersheds, even though the four curves are far from smooth. This is most likely because this definition of regimes is sensitive to the variability in precipitation (Sect. ) which results in a lack of robustness for the identification of regime shift.

When regimes are defined with a threshold (Fig. b), all selected parameterizations are in the “Low runoff coefficient regime” in 1965 for the Dargol, Sirba, and Nakanbé watersheds. For the Gorouol watershed, 40 % of the selected parameterizations are already in the “High runoff coefficient regime” in 1965. At the end of the simulation period in 2014, more than 80% of selected parameterizations are in a “High runoff coefficient regime” for all watersheds. Thus, most selected parameterizations underwent a regime shift between 1965 and 2014.

To analyze the timing of the regime shift, we define the year of the shift as the year with the greatest number of regime shifts. Graphically, it corresponds to the year when the slope of the curve “percentage vs. time” is maximized. In practice, we compute it as the year where “% of selected parameterization in the High runoff coefficient regime (year+1) – % of selected parameterization in the High runoff coefficient regime (year-1)” is maximized. Following this definition, the regime shift occurred in 1971, 1972, 1973, 1983 for the Gorouol, Nakanbé, Dargol, and Sirba watershed, respectively (Fig. b). For the Dargol and Nakanbé watersheds, the percentage of selected parameterizations in the “High runoff coefficient regime” increased from 0% to 75% in 15 years, while it took approximately 30 years for the Sirba watershed.