# Modeling the Influence of Outflow and Community Structure on an Endangered Fish Population in the Upper San Francisco Estuary

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Study Area

^{2}when X2 is positioned at 74 km, 50 km

^{2}when X2 is positioned at 81 km, and 40 km

^{2}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

_{i}), the change in the equilibrium level of species n

_{i}

^{*}is

_{i}represent species abundances or abiotic factors. The c

_{i}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 a

_{ij}of a qualitatively defined community matrix (A

^{o}) of order n denotes the net effect 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 effects to variable i from variable j were assigned to each matrix element: a

_{i,j}= 1, a

_{i,j}= −1 and a

_{i,j}= 0, which in the neighborhood of any equilibrium point respectively denote positive, negative, and no effect 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:

^{o}− λI| = a

_{0}λ

^{n}+ a

_{1}λ

^{n}

^{− 1}+ a

_{2}λ

^{n}

^{− 2}+ … a

_{n}= 0

^{o}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].

#### 2.3. Community Structure and Outflow Scenarios

- (1)
- Compilation of long-term data and studies in the upper SF Estuary, including: monitoring data for delta smelt and other species [53], outflow and X2 data [51], interpretation of the spatio-temporal distribution of the salinity field in the LSZ based on the three-dimensional UnTRIM hydrodynamic model for the upper SF Estuary [24,75], and baseline knowledge of the modeled ecosystem (e.g., [6,10,39,40,54,55,63,76,77]).
- (2)
- Consideration of dynamic and stationary factors for developing conceptual models of estuarine communities [78], where dynamic factors include physico-chemical and biological characteristics of the low salinity habitat corresponding to the LSZ each X2 position, and stationary factors include geographically fixed habitat features at each X2 position such as substrate, erodible sediment, and bathymetry in the low salinity habitat [24,38].
- (3)
- Refinement of conceptual models into different subsystems describing the essential community variables influencing subadult delta smelt based on ecological syntheses of long-term field data and studies [24,38]. Community variables selected for each of the three modeled subsystems were based on functional groups (e.g., [74,79]), and their predominant spatio-temporal overlap with dynamic and stationary abiotic and biotic factors. Except for delta smelt and species with significant ecological impacts (the cyanobacterium M. aeruginosa [80], the clams P. amurensis and C. fluminea [56], and the macrophyte E. densa [58], Table 1), other functional groups included trophic levels to minimize redundant species interactions (e.g., [74,81]), (Table 1).
- (4)
- Reformulation of conceptual models into signed digraphs based on qualitative model guidelines (e.g., [71,79,93]). Negative self-effect (self-damping) was assumed to arise for each variable from density-dependent growth rate or a limited source, as in the case of nutrients [71]. Each community variable was then implicitly connected to other variables or abiotic factors through negative feedback [71,74]. Reported community interactions considered for the modeled subsystem included: predation (+, −); interference competition (−, −); and amensalism (0, −), (Appendix A).
- (5)
- Estimation of the direction of change of community variables (+, 0, −) in response to increased outflow (Table 1). Four outflow input scenarios were modeled, with the first scenario accounting for the effect of outflow on the previously referred species having significant ecological impacts. The outflow inputs for the three subsequent scenarios were used to evaluate whether cumulative outflow inputs in each subsystem could reinforce or reverse potential responses on delta smelt and other community variables. These scenarios included: scenario 1 + phytoplankton (scenario 2), scenario 2 + zooplankton (scenario 3), and scenario 3 + delta smelt (scenario 4).

#### 2.4. Qualitative Analyses

_{h}), on the rate of change of a hth variable corresponding to the jth variable in the community matrix was predicted assuming an ecosystem in moving equilibrium [29,71]. Such input could result in positive, negative, or no effect on the equilibrium magnitude or carrying capacity (N

_{i}

^{*}) of a species. The inverse of the negative community matrix (−A

_{ij}

^{−1}) provides an estimate of change in the equilibrium level of variable N

_{i}

^{*}carrying capacity due to change in parameter C

_{h}[95].

_{ji}

^{T}is the adjoint of the negative community matrix (Adj − A

_{ij}), and |−A

_{ij}| is the determinant of the negative community matrix. Since |−A

_{ij}| is constant and positive for each element of −A

^{−1}in stable ecosystems, the predicted response under local equilibrium was inferred from the Adj − A:

_{k}) due to a disturbance k acting directly upon two or more variables, the combined effect of P

_{k}on the direction of change for community variable i and their corresponding uncertainty level was computed from the corresponding matrix element Ws

_{ik}(Equation (5)) and these were derived from a modified community matrix specifying press inputs (${A}^{P}$):

_{11}to a

_{nn}represent the interaction elements a

_{ij}of community matrix A

^{o}, and P

_{ik}denotes the press input k on community variable i, which in a qualitatively defined ecosystem is specified as having positive (1), negative (−1) or no effect (0). The last row of matrix ${A}^{P}$ includes zero elements to denote no effect of community variables on P

_{k}, and a negative element (−1) to preserve the stability of matrix ${A}^{P}$ provided the community matrix A

^{o}is stable. Qualitative analyses were conducted in Maple 18 [47].

#### 2.5. Quantitative Simulations

_{ik})* and f(−Adj − A

_{ik})* respectively denote the absolute frequencies of positive and negative Adj − A

_{ik}in stable matrices, and N is the total number of simulations for each perturbation scenario, including stable and non-stable matrices (N = 10,000). As in the case of Ws, the value of Δ$\hat{p}$ provided both the overall direction of change and its uncertainty for each community variable, with extreme potential values (Δ$\hat{p}$ = −1, negative response and Δ$\hat{p}$ = 1, positive response) both indicating no uncertainty in the direction of change among simulations, Δ$\hat{p}$ ≥ |0.5| and < |1|indicating low uncertainty, and Δ$\hat{p}$ = 0 denoting total uncertainty. Quantitative simulations were performed in a program written in C# version 3.0 [96].

#### 2.6. Statistical Analyses

## 3. Results

#### 3.1. Community Stability

#### 3.2. Community Response to Outflow

#### 3.3. Delta Smelt Abundance and X2

## 4. Discussion

#### 4.1. Community Stability

#### 4.2. Community Model Predictions

#### 4.3. Delta Smelt Abundance under Different Ranges of X2

#### 4.4. Community Models and Prediction Metrics

#### 4.5. Management Implications

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Qualitative interactions for modeled community variables defined in Table 1 at the high-, mid-, and low-X2 positions in the upper San Francisco Estuary. (References shown in parentheses).

High-X2 Variables | DS | ZO | PH | ED | PR | CF | MA |

DS | −1 [71] | 1 [41] | 0 | −1 [64] | −1 [92] | 0 | −1 [70] |

ZO | −1 [41] | −1 [71] | 1 [113] | −1 [114] | 0 | 0 | −1 [69] |

PH | 0 | −1 [113] | −1 [71] | −1 [85] | 0 | −1 [115] | −1 [80] |

ED | 0 | 0 | 0 | −1 [71] | 0 | 1 [116] | −1 [117] |

PR | 1 [92] | 0 | 0 | 0 | −1 [71] | 0 | −1 [80] |

CF | 0 | 0 | 1 [115] | −1 [100] | 0 | −1 [71] | −1 [118] |

MA | 0 | 0 | 0 | −1 [119] | 0 | −1 [120] | −1 [71] |

Mid-X2 Variables | DS | ZO | PH | PA | PR | CF | MA |

DS | −1 [71] | 1 [41] | 0 | 0 | −1 [92] | 0 | −1 [70] |

ZO | −1 [41] | −1 [71] | 1 [113] | −1 [66,88] | 0 | 0 | −1 [69] |

PH | 0 | −1 [113] | −1 [71] | −1 [60] | 0 | −1 [115] | −1 [80] |

PA | 0 | 1 [66,88] | 1 [60] | −1 [71] | 0 | 0 | −1 [118] |

PR | 1 [92] | 0 | 0 | 0 | −1 [71] | 0 | −1 [80] |

CF | 0 | 0 | 1 [115] | 0 | 0 | −1 [71] | −1 [118] |

MA | 0 | 0 | 0 | 0 | 0 | −1 [120] | −1 [71] |

Low-X2 Variables | DS | ZO | PH | PA | PR | ||

DS | −1 [71] | 1 [41] | 0 | 0 | −1 [92] | ||

ZO | −1 [41] | −1 [71] | 1 [113] | −1 [66,88] | 0 | ||

PH | 0 | −1 [113] | −1 [71] | −1 [60] | 0 | ||

PA | 0 | 1 [66,88] | 1 [60] | −1 [71] | 0 | ||

PR | 1 [92] | 0 | 0 | 0 | −1 [71] |

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**Figure 1.**The San Francisco Estuary showing the Suisun Bay and Delta regions in the upper estuary. Dashed lines denote the positions of the 2 psu isohalines (X2) considered in community models (74, 81, and 85 km) and the range of the average X2 position in September–October (64–94 km) from 1967 to 2017. Map adapted from [24].

**Figure 2.**Example of (

**A**) a 3 × 3 community matrix (A

^{o}) 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 (N

_{i}), 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].

**Figure 3.**Alternative fall community models for the Delta smelt subsystem under three scenarios of the salinity field in upper San Francisco Estuary. Dashed arrows show where the low salinity zone overlaps the position of the 2 psu isohaline (X2) at (

**A**) high X2 (85 km), (

**B**) mid X2 (81 km), and (

**C**) low X2 (74 km) at low-, mid- and high-outflow, respectively. Community variables included in signed digraphs: CF = Corbicula fluminea; DS = Delta smelt; ED = Egeria densa; MA = Microcystis aeruginosa; PA = Potamocorbula amurensis; PH = phytoplankton; PR = predators of delta smelt; ZO = zooplankton. Salinity field after [75].

**Figure 4.**Signed digraphs of the modeled delta smelt subsystems at three positions of the 2 psu isohaline (X2) in the upper San Francisco Estuary. Dashed lines indicate the four alternative outflow (FL) input scenarios for community variables. Dashed lines in scenario 1 represent outflow perturbations to Corbicula fluminea (CF), Egeria densa (ED), Microcystis aeruginosa (MA), and Potamocorbula amurensis (PA). Dashed lines in outflow scenarios 2, 3, and 4 cumulatively add outflow perturbations to phytoplankton (PH, scenario 2), zooplankton (ZO, scenario 3), and delta smelt (DS, scenario 4). PR denotes predators of delta smelt. Community variables show the predicted direction of change in abundance (+, −) in response to sustained outflow input. Direction of change without parentheses is unconditional. Direction of change in parentheses denote high-certainty (asterisk) and low-conditional certainly (no asterisk), and ? denotes complete ambiguity.

**Figure 5.**Predicted direction of change and its uncertainty for community variables based on signed weighted predictions (Ws) at low-, mid-, and high-X2 positions under outflow–input scenarios (1–4, Figure 4). Values of |Ws| for each community variable range between unconditional sign determinacy (1) and complete uncertainty (0), with |Ws| between 0.50 (dotted line) and ≤1 indicating high sign determinacy.

**Figure 6.**Predicted direction of change of community variables for low-, mid-, and high-X2 positions based on the difference between the proportions of quantitative simulations with positive and negative Adj − A (Δ$\hat{p}$). Acronyms represent species or trophic groups (Table 1, Figure 3) and legend denotes the four outflow input scenarios (Figure 4). Values of |Δ$\hat{p}$| can range between 1 (unconditional sign determinacy) and 0 (complete uncertainty), with |Δ$\hat{p}$| between 0.5 (dotted lines) and <1 implying low uncertainty in the predicted direction of change.

**Figure 7.**Relations between the net proportion of positive and negative Adj − A in stable quantitative models (Δ$\hat{p}$) at low-, mid-, and high-X2 positions and: (

**A**) the estimated direction of change of corresponding variables in qualitative models (Adj − A); (

**B**) the signed weighted feedback in qualitative models (Ws), as indicated for modeled subsystems in the low salinity zone at low-, mid-, and high-X2 positions.

**Figure 8.**Relative abundance for subadult delta smelt based on the fall midwater trawl (FMWT) index (Y) versus the average position of the 2 psu isohaline (X2) during September and October (X) under four ranges of X2 over the period 1967–2017. Dark and light shaded areas denote 95% confidence intervals for regression lines and predicted values, respectively. Regression coefficients in bold type are significant (P < 0.05).

**Table 1.**Variables considered in qualitative community models for the upper San Francisco Estuary. Black circles denote community variables included in the low salinity zone at three positions of the 2 psu isohaline (X2).

Variable Type | Variable Code (Description) | Low X2 74 km | Mid X2 81 km | High X2 85 km | Functional Role | Community Response to Outflow |
---|---|---|---|---|---|---|

Explanatory | FL (Delta outflow, m ^{3} s^{−1}) | 323 [24] | 227 [24] | 142 [24] | Major abiotic forcing factor controlling the overlap between dynamic and stationary factors in the upper San Francisco Estuary, the position and size of the low salinity zone and the low salinity habitat of delta smelt and community interactions [24,38]. | |

Response (Community variables) | CF (Corbicula fluminea, Asian clam) ^{1} | ● [24] | ● [24] | Filter feeder exerting grazing pressure on phytoplankton [82,83]. | (+) [56] | |

DS (Hypomesus transpacificus, delta smelt) | ● [38,45] | ● [38,45] | ● [38,45] | Zooplanktivorous preying primarily on copepods, mysids, and cladocerans [40,41]. | (+) [10,54] | |

ED (Egeria densa, Brazilian waterweed) ^{1} | ● [58,59] | Primary producer of dense submersed vegetation beds which reduce open water habitat, alter the habitat for other aquatic plants [58,65] and provide habitat for centrarchids [84]. | (−) [59,85] | |||

MA (Microcystis aeruginosa, cyanobacterium) ^{2} | ● [24,86] | ● [24,86] | Biological contaminant due to its poor nutritional value and transfer of toxic microcystins into the aquatic food web [24,69,70,80]. | (−) [86] | ||

PA (Potamocorbula amurensis, overbite clam) ^{1} | ● [24] | ● [24] | Filter feeder exerting grazing pressure on phytoplankton and zooplankton [87,88]. | (−) [56] | ||

PH (Phytoplankton) | ● [38,89,90] | ● [38,89,90] | ● [38,89,90] | Primary producer fueling higher trophic levels of the food web in the open waters of the Delta and Suisun Bay [89,90]. | (+) [38,91] | |

PR (Predators of delta smelt) | ● [24,38] | ● [24,38] | ● [24,38] | Piscivores deemed to exert predation pressure on delta smelt throughout the Delta and Suisun Bay [24], including two introduced species, striped bass Morone saxatilis [62,92] and largemouth bass Micropterus salmoides [64,84]. | (0) ^{3} [38] | |

ZO (Zooplankton) | ● [24,38] | ● [24,38] | ● [24,38] | Primary and secondary consumer supporting pelagic and benthic food webs [41,88]. | (+) [55] |

**Table 2.**Percent of quantitative community models meeting zero, one and both Routh–Hurwitz (R–H) stability criteria at three X2 positions in the low salinity zone of the upper San Francisco Estuary (based on 10,000 simulations at each X2 position).

R–H Criteria | Low-X2 (74 km) | Mid-X2 (81 km) | High-X2 (85 km) |
---|---|---|---|

None | 2.7 | 39.9 | 48.2 |

Only I | 0.3 | 3.5 | 10.6 |

Only II | 0.0 | 0.0 | 0.0 |

Both | 97 | 56.6 | 41.2 |

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**MDPI and ACS Style**

Castillo, G.C. Modeling the Influence of Outflow and Community Structure on an Endangered Fish Population in the Upper San Francisco Estuary. *Water* **2019**, *11*, 1162.
https://doi.org/10.3390/w11061162

**AMA Style**

Castillo GC. Modeling the Influence of Outflow and Community Structure on an Endangered Fish Population in the Upper San Francisco Estuary. *Water*. 2019; 11(6):1162.
https://doi.org/10.3390/w11061162

**Chicago/Turabian Style**

Castillo, Gonzalo C. 2019. "Modeling the Influence of Outflow and Community Structure on an Endangered Fish Population in the Upper San Francisco Estuary" *Water* 11, no. 6: 1162.
https://doi.org/10.3390/w11061162