Imagine the vast watershed of Lake Erie, crisscrossed by rivers and streams. For over a century, humans built thousands of dams on these tributaries. While many served useful purposes like power generation or water supply, a large number are now old, deteriorating, and potentially harmful to the ecosystem. They block fish migration, alter flow patterns, and can even pose safety risks.
Removing these dams seems like a good idea for restoring river health. But it's not simple. Removing a dam costs money, and the ecological effects aren't always straightforward. Some removals might benefit desirable native fish like Walleye, crucial for both the ecosystem and recreational fishing. However, the same removal might also help invasive species like the parasitic Sea Lamprey gain access to new breeding grounds. Furthermore, removing one dam can affect the value of removing another upstream or downstream.
With thousands of dams and limited resources, how do managers decide *which* dams to remove to get the biggest ecological bang for their buck, while considering these complex trade-offs? This is the challenge addressed by the paper "Optimizing multiple dam removals under multiple objectives: Linking tributary habitat and the Lake Erie ecosystem" by Zheng, Hobbs, and Koonce (2009).
This post explores the core ideas of their approach, using interactive visualizations to help build intuition for how they model this complex decision-making problem.
The core difficulty lies in the scale and complexity:
To make informed decisions, we first need to quantify the potential effects of removing a dam. The paper focuses on how removal changes fish habitat, particularly for spawning and nursery areas.
One way to measure habitat quality is the Habitat Suitability Index (HSI). It combines factors like water depth, velocity, and substrate type (e.g., gravel for Walleye spawning, fine sediment for Lamprey larvae) into a score from 0 (unsuitable) to 1 (perfectly suitable).
Imagine a river reach. How suitable is it for Walleye spawning based on flow velocity?
Walleye prefer moderate velocities for spawning (roughly 0.3-0.9 m/s). Too slow or too fast, and the habitat becomes less suitable. This is a simplified example; real HSI models incorporate multiple factors.
The paper calculates potential HSI gains for both Walleye and Sea Lamprey for the river segments upstream of each candidate dam.
Changes in tributary habitat, especially for Walleye spawning, don't just affect the river; they impact the entire Lake Erie ecosystem. The researchers used the Lake Erie Ecological Model (LEEM), a complex simulation model, to translate the estimated increase in Walleye Young-of-Year (YOY) resulting from habitat improvements into changes across various lake-wide indicators. These indicators reflect different aspects of "ecosystem health":
Essentially, LEEM provides response functions: how much does indicator X change if we add Y number of Walleye YOY?
We have multiple ecosystem indicators (let's call them criteria $x_1, x_2, ..., x_8$) and the economic cost ($x_9$). How do we combine the ecological criteria into a single measure of "health" to trade off against cost?
The paper uses Multi-Criteria Value Analysis (MVA), specifically an additive value function. The idea is simple: assign a weight ($W_i$) to each ecological criterion reflecting its perceived importance, and define a value function ($V_i(x_i)$) that scales the raw criterion value $x_i$ from 0 (worst outcome) to 1 (best outcome). The overall ecological health score ($z_1$) is the weighted sum:
Where $\sum W_i = 1$. The weights are subjective and represent the priorities of decision-makers or stakeholders.
Let's imagine a simplified scenario with 3 ecological criteria resulting from a hypothetical dam removal plan. Adjust the weights to see how the overall "Ecosystem Health Score" changes based on your priorities.
Notice how changing the weights alters the final score, even though the underlying outcomes (Values 0.8, 0.6, 0.4) remain the same. This highlights the importance of defining priorities in multi-objective problems. The sliders automatically adjust to ensure weights sum to 1.
Now we have the pieces: ways to estimate ecological benefits (via HSI, LEEM, and MVA) and costs (dam removal + lamprey control). The final step is to find the best *set* of dams to remove.
The paper uses a Mixed Integer Linear Program (MILP). This is a mathematical optimization technique.
The Goal: Maximize the overall ecosystem health score ($z_1$) subject to a limit on the total cost ($x_9$).
The Decision: For each candidate dam $j$, decide whether to remove it ($d_j = 1$) or not ($d_j = 0$). These $d_j$ are the decision variables.
The basic structure looks like this (simplified):
Let's break this down:
Imagine a small river network with 10 potential dams. Each has a removal cost and provides some 'Walleye Benefit' but also creates 'Lamprey Habitat'. Use the budget slider to see which portfolio of dams an optimization model might select to maximize total Walleye Benefit within the budget.
Selected Dams: 0
Total Cost: $0.0 M
Total Walleye Benefit: 0 units
Total Lamprey Habitat: 0 units
This is a simplified greedy simulation (prioritizing best benefit/cost ratio respecting upstream constraints) to illustrate the concept. Real MILP solvers find the truly optimal solution. Notice how the selected portfolio changes with the budget. Dams with high benefit but high cost might only be selected at higher budgets.
By running the MILP model for many different budget levels ($B$), the researchers could trace out the efficient frontier (also known as the Pareto frontier). This curve shows the maximum possible ecosystem health ($z_1$) achievable for any given level of cost ($x_9$).
Each point represents an optimal portfolio of dam removals for a specific budget. Moving along the curve reveals the trade-off: gaining more ecosystem health requires spending more money, often with diminishing returns (the curve flattens).
Hover over points to see details. This curve helps decision-makers understand the consequences of different budget allocations. Points below the curve are inefficient (you could get more health for the same cost, or the same health for less cost).
The MVA weights and the very structure of the optimization reflect certain priorities. What if we were particularly concerned about the negative impact of increasing Sea Lamprey habitat?
The paper explored this by adding the amount of new lamprey habitat as a *negative* component in the objective function (or as a separate objective to be minimized). They assigned a weight (WL) to this "lamprey aversion".
Let's revisit the simplified portfolio selection, but now add a slider for "Lamprey Aversion Weight" (WL). This weight penalizes portfolios that create more lamprey habitat. The budget is fixed at $15M for this demo.
Selected Dams: 0
Total Cost: $0.0 M
Total Walleye Benefit: 0 units
Total Lamprey Habitat: 0 units
As you increase the Lamprey Aversion Weight, the model may choose a different set of dams. It might avoid removing dams that offer good Walleye benefits if they *also* create significant Lamprey habitat, potentially selecting less beneficial but "safer" dams, or simply fewer dams overall if the penalty is high enough.
The work by Zheng, Hobbs, and Koonce provides a powerful framework for thinking about complex environmental decisions like dam removal. By:
They created a valuable tool. It doesn't give the "one true answer" – the models have uncertainties, and the weights are subjective. However, it acts as a screening model, identifying *potentially* cost-effective portfolios of dam removals that warrant further, detailed, site-specific investigation before any real action is taken.
This approach highlights how mathematical modeling and optimization can help navigate the complexities and trade-offs inherent in managing our natural resources, making the decision-making process more transparent, rational, and effective.
Reference: Zheng, P. Q., B. F. Hobbs, and J. F. Koonce (2009), Optimizing multiple dam removals under multiple objectives: Linking tributary habitat and the Lake Erie ecosystem, Water Resour. Res., 45, W12417, doi:10.1029/2008WR007589.
Interactive visualizations created for illustrative purposes based on the concepts in the paper. Data used in visualizations is synthetic unless otherwise noted.