Lake Erie, the shallowest and warmest of the Great Lakes, is incredibly productive but also heavily impacted by human activity. Decades of stressors, including nutrient pollution and habitat loss, have taken their toll. One significant stressor? Dams. Thousands of dams fragment the rivers and streams flowing into the lake, blocking fish from reaching crucial historical spawning and nursery grounds.
Removing dams can be a powerful tool for river restoration. It can reopen habitat for native fish like the economically important walleye. However, dam removal isn't simple. It costs money, might benefit invasive species like the parasitic sea lamprey, and removing one dam can affect the potential benefits of removing others upstream or downstream. With limited resources, how do managers decide *which* dams to remove to get the biggest ecological bang for their buck?
This is where the research by Zheng, Hobbs, and Koonce (2009) comes in. They developed a mathematical model to help tackle this complex "portfolio problem": selecting the best set of dams to remove across multiple Lake Erie watersheds, considering multiple competing objectives.
Imagine a simplified river system flowing into Lake Erie. Dams act as barriers.
Click on the dams (blue rectangles) in the river below to simulate removing them. Notice how removing a downstream dam makes the habitat upstream potentially accessible to fish migrating from the lake (represented by the green areas).
Restoring access seems good, right? But there are complications:
How do you compare different dam removal plans when they affect so many different things? The researchers used a technique called Multicriteria Value Analysis (MVA). The basic idea is to:
Imagine we have just three (simplified) criteria. Adjust the sliders to see how changes in individual criteria and their assigned importance (weights) affect the overall score. (Note: This is illustrative; the actual study used 8 criteria derived from a complex Lake Erie ecosystem model).
This Ecosystem Health Index ($z_1$ in the paper) becomes one major objective. The other major objective is minimizing the total Cost ($x_9$), which includes dam removal and potential lamprey control.
The core of the paper is a Mixed Integer Linear Program (MILP). Think of it like trying to fill a shopping basket (the portfolio of dams to remove) to get the maximum "value" (Ecosystem Health) without exceeding your budget (Total Cost), while following certain rules.
The key decision for each potential dam $j$ is whether to remove it or not. This is represented by a binary variable $d_j$: $$ d_j = \begin{cases} 1 & \text{if dam } j \text{ is removed} \\\\ 0 & \text{if dam } j \text{ is not removed} \end{cases} $$
The model aims to maximize the Ecosystem Health Index ($z_1$), which is a weighted sum of individual criteria values ($x_i$), each influenced by the dam removal decisions:
Where $W_i$ are the importance weights and $V_i(x_i)$ are the value functions for each ecological criterion $x_i$. The values $x_i$ themselves change based on the predicted increase in walleye recruitment ($\Delta yoy_{walleye}$) resulting from the selected dams ($d_j$).
This maximization is subject to several constraints:
Let's try a miniature version. Below is a river with 5 dams. Each has a removal cost and provides a certain 'Ecosystem Benefit' if removed (a simplified score). Your budget is limited. Try to select dams to maximize the total benefit without exceeding the budget. Remember the rule: you can only select an upstream dam if the one directly downstream is also selected. Click dams to select/deselect.
By running the MILP model with different budget levels ($B$), the researchers traced out an "efficiency frontier". This curve shows the maximum possible Ecosystem Health score achievable for any given level of spending.
The chart below shows the results from the paper (similar to Figure 4a). Each point represents an optimal portfolio of dams found by the MILP for a specific budget. Hover over points to see details. Notice that initial spending yields large health gains, but returns diminish as the budget increases (the curve flattens).
The initial model focused on maximizing the combined health score. But what if decision-makers are particularly worried about increasing sea lamprey populations? The researchers explored this by adding a penalty to the objective function based on the amount of potential lamprey habitat opened up.
They introduced a weight, $WL$, representing the importance placed on *avoiding* lamprey habitat increase. $WL=0$ means no special concern (original model), while a higher $WL$ means lamprey avoidance is prioritized.
Use the slider to adjust the weight given to avoiding lamprey habitat ($WL$). Observe how the optimal solution for a fixed budget (e.g., $15M) changes. A higher $WL$ leads to lower overall ecosystem health scores (as potentially beneficial dams that also open lamprey habitat are avoided) and selects a different set of dams.
(Visualizing the exact dam locations is complex, but note how the number of dams and outcomes change significantly with WL)
The research demonstrates how optimization modeling can be a powerful tool for complex environmental decisions like dam removal. Key takeaways include:
By linking habitat changes in tributaries to their effects on the broader Lake Erie ecosystem and explicitly considering costs and multiple objectives, this work provides a valuable framework for making more informed, transparent, and potentially cost-effective decisions in river restoration.
Based on: 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.