*2.1. Study Area*

The SF Estuary is the largest estuary in the west coast of the United States (Figure 1), and has been profoundly altered physically, chemically, and biologically [6,8]. Multiple stressors in the upper SF Estuary have been implicated in the pelagic organism decline occurring in the early 2000s [48,49]. Reservoirs upstream of the SF Estuary, and substantial water diversions within an extensive network of interconnected channels surrounding leveed agricultural islands (the Delta) have greatly altered the hydrodynamics of this estuary [13,50]. Two dataset are central to the evaluation of hydrologic conditions in the SF Estuary: (1) Dayflow, a 2-D mass balance model of freshwater outflows from the tributaries of the Upper SF Estuary [51], and (2) the California Data Exchange Center, a query-based portal which maintains, and operates an extensive hydrologic data collection network [52]. In addition, the Interagency Ecological Program (IEP) provides long-term monitoring data of pelagic, planktonic, and benthic organisms in the SF Estuary [53]. The field data generated by the IEP have been collected consistently over time at multiple locations representing the study areas in the upper SF Estuary (e.g., [24,38,54]). The temporal and spatial coverage of these field surveys has enabled to evaluate the role of hydrodynamic forcing on the distribution and relative abundance of planktonic, pelagic, and benthic species (e.g., [24,38,55–57]). Additional monitoring of aquatic macrophytes by the University of California at Davis, has been conducted extensively using a combination of field sampling and remote sensing in the upper SF Estuary (e.g., [58,59]).

The LSZ area varies greatly with outflow, and is about 90 km<sup>2</sup> when X2 is positioned at 74 km, 50 km<sup>2</sup> when X2 is positioned at 81 km, and 40 km<sup>2</sup> when X2 is positioned at 85 km [24]. Altered fresh water outflow in the upper SF Estuary has greatly changed the abiotic habitat for pelagic fishes [9,54], and facilitated the introduction of species [60,61]. A high proportion of pelagic and planktonic organisms in the upper SF Estuary are also lost due to prevailing upstream flows in the south Delta caused by massive water pumps [5,6]. As is the case of the lower SF Estuary, most communities in the upper SF Estuary are dominated by introduced species [61,62]. The introduced macrophyte *Egeria densa* was first reported in the Delta in 1946 [63] and became the most abundant submersed plant in the Delta, where it has greatly altered community structure and functions [58,64,65]. The introduction of the clam *Potamocorbula amurensis* in the late 1980s caused major trophic changes in the benthic and pelagic communities of the SF Estuary [60,66], greatly adding to the filter feeding activity of the clam *Corbicula fluminea*, an earlier introduction in the upper SF Estuary [67]. In addition, the cyanobacterium *Microcystis aeruginosa* was first observed in the Delta in 1999 [68] and produces toxic algal blooms that can be harmful to upper trophic levels [69,70].

### *2.2. Community Matrix Models*

The community structure and interactions in this study were based on qualitative analysis of the community matrix [29,71]. Given a classical dynamical ecosystem of *n* interacting variables ( *Ni*), the change in the equilibrium level of species *ni* \* is

$$\frac{dn\_i^\*}{dt} = f i(n\_1, n\_2, \dots, n\_n; c\_1, c\_2, \dots, c\_n) \quad (i = 1, \ n). \tag{1}$$

where the *ni* represent species abundances or abiotic factors. The *ci* are parameters that govern biological rates (e.g., birth, mortality, growth), and depend on the species and environment [72,73]. Modeled subsystems were represented by a signed digraphs, where element *aij* of a qualitatively defined community matrix ( *Ao*) of order *n* denotes the net e ffect to variable *i* from variable *j* (for *i* and *j* from 1 to *n*). For example, a subsystem composed of three trophic levels can be represented by three variables and their interactions (Figure 2). Three possible qualitative e ffects to variable *i* from variable *j* were assigned to each matrix element: *ai,j* = 1, *ai,j* = −1 and *ai,j* = 0, which in the neighborhood of any equilibrium point respectively denote positive, negative, and no e ffect on the instantaneous growth of variable *i* due to increase in the level of variable *j*. The stability of the community matrix was evaluated using the characteristic polynomial:

$$p(\lambda) = |A^o - \lambda \mathbf{I}| = a\_0 \lambda^n + a\_1 \lambda^{n-1} + a\_2 \lambda^{n-2} + \dots \ a\_n = 0 \tag{2}$$

where, *Ao* is as defined earlier, I is the n-dimensional identity matrix, and λ are the roots or eigenvalues [71]. Following a perturbation, negative real roots means the community can return to equilibrium. Conversely, positive real roots indicate the ecosystem moves away from equilibrium, as in unstable ecosystems. Multiple zero roots can result in unstable equilibrium [73], and zero overall feedback indicates neutral stability [74].

**Figure 2.** Example of (**A**)a3 × 3 community matrix (*Ao*) showing qualitative interaction terms for 3 variables such as species or trophic groups, with diagonal matrix elements −*a*11, −*a*22, and −*a*33 representing negative self-effects of each variable and zero denoting no effect, and (**B**) its corresponding signed-digraph. Open circles denote variables (*Ni*), with lines between variables ending in arrows and bubbles respectively denoting positive and negative effects, and curved lines originating and ending in the same variable denoting self-effects, after [29].

### *2.3. Community Structure and Outflow Scenarios*

Qualitative analysis was used to model three flow-dependent fall community structures in the upper SF Estuary under high-, mid-, and low-net Delta outflows corresponding respectively to X2 positions of 74, 81, and 85 km [24]. The three essential components of qualitative models (community variables, interactions or links, and driving force or perturbation) were based on the following five steps:



*Water* **2019**, *11*,

**Table 1.** Variables considered in qualitative community models for the upper San Francisco Estuary. Black circles denote community variables included in the low

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