Eutrophication Modeling Example

Lecture 05

September 1, 2023

Review and Question

Last Class

Used mass-balances and reaction decay to formulate model for wastewater concentrations downstream of multiple releases.
Key Point: Systems management requires accounting for the full dynamics, not (usually) one process/decision at a time.

Questions?

Text: VSRIKRISH to 22333

URL: https://pollev.com/vsrikrish

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Lake Eutrophication Example

What is Eutrophication?

Eutrophication: common environmental problem in which plants and algae feed on excess nutrients and become overabundant.

Lake Eutrophication Causes

In lakes, eutrophication is often caused by excess input of nutrients (particularly N and P). Excess N and P can come from:

point sources (such as industrial/sewage processes); and/or
non-point sources (such as agricultural runoff).

Excess nutrients are stored in sediment and recycled back into the lake, as well as transported by organisms/consumers.

Effects of Lake Eutrophication

hypoxia (reduction in oxygen from the decomposition of organic matter), leading to “dead zones”;
acidification(from the CO₂ produced by decomposition);
reduced sunlight (from an accumulation of surface algae);
clogged water intakes; and
reduction in recreational value and drinking water quality.

Management of Eutrophication

Once a lake is eutrophied, it can be difficult to restore to oligotrophic state:

Reduce N and P going forward to reduce pressure;
Remove and treat sediment/water;
Biofiltration.

Restoration takes a long time and is not guaranteed!

Shallow Lake Model

Model introduced by Carpenter et al (1999).
(Simplified) lake management problem
Tradeoff between economic benefits and the health of the lake.

Shallow Lake Model: Variables

Variable	Meaning	Units
$X_t$	P level in lake at time $t$	dimensionless
$a_t$	Controllable (point-source) P release	dimensionless
$y_t$	Random (non-point-source) P runoff	dimensionless

Shallow Lake Model: Runoff

Random runoffs $y_t$ are sampled from a LogNormal distribution.

Figure 1: Lognormal Distributions

Shallow Lake Model: P Dynamics

Lake loses P at a linear rate, $bX_t$.
Nutrient cycling reintroduces P from sediment: \[\frac{X_t^q}{1 + X_t^q}.\]

Shallow Lake Model

So the P level (state) $X_{t+1}$ is given by: \[\begin{gather*} X_{t+1} = X_t + a_t + y_t + \frac{X_t^q}{1 + X_t^q} - bX_t, \\ y_t \underset{\underset{\Large\text{\color{red}sample}}{\color{red}\uparrow}}{\sim} \text{LogNormal}(\mu, \sigma^2). \end{gather*} \]

Shallow Lake Model Dynamics

Dynamics Without Inflows

First, let’s look at the dynamics without inflows ($a_t=y_t=0$) to get a sense of baseline behavior.

We’ll focus on the case where $q = 2.5$.

Figure 2: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows

Important: where is the black line (recycled P from sediment) relative to the straight line (outflows)?

This reflects trend in natural P dynamics (red arrows).

Figure 3: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Equilibria

The intersection points are equilibria, where the state of the system is fixed and doesn’t change.

Figure 4: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Equilibria

For $b=0.4$, three equilibria:

$X=0$;
$X = X_o = 0.67$ (oligotrophic)
$X = X_e = 2.2$ (eutrophic)

Figure 5: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Equilibria

These equilibria have different nearby behaviors:

$X=0$ and $X=2.2$ are stable: the state will recover after a perturbation.
$X=0.67$ is unstable.

Figure 6: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Equilibria

This gives rise to a tipping point as we cross $X=0.67$: stable oligotrophic behavior suddenly switches to eutrophication.

Figure 7: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Bifurcation

For different values of $b$, the number of equilibria changes.

$b = 0.2$: Only two equilibria
- stable at $X=0$
- unstable at $X=0.36$
$b=0.6$: Only stable equilibrium at $X=0$.

Figure 8: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Bifurcation

These changes in the type and/or number of equilibria is called a bifurcation, and these are relatively common properties of systems.

Figure 9: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics Without Inflows: Bifurcation

Bifurcations are important, because they can result in unexpected outcomes if we plan for one type of stability but get another.

This is particularly acute when we are uncertain about the values of relevant parameters.

Figure 10: Lake eutrophication dynamics based on the shallow lake modelwithout additional inputs. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics With Inflows

Let’s look at what happens if $a_t + y_t = 0.05$ (constant inflows). The equilibria are shifted.

With $b=0.2$, this eliminates equilibria: the lake will inevitably eutrophy.

Figure 11: Lake eutrophication dynamics based on the shallow lake model with. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Dynamics With Inflows

But this assumes constant inflows: remember that we would treat $y_t$ as random.

What are the implications for management?

Figure 12: Lake eutrophication dynamics based on the shallow lake model with. The black line is the P recycling level (for $q=2.5), which adds P back into the lake, and the dashed lines correspond to differerent rates of P outflow (based on the linear parameter $b$). The lake P level is in equilibrium when the recycling rate equals the outflows. When the outflow is greater than the recycling flux, the lake’s P level decreases, and when the recycling flux is greater than the outflow, the P level naturally increases. The red lines show the direction of this net flux.

Key Takeaways

Nonlinear system dynamics (e.g. bifurcations, but also feedbacks) can complicate management.
Important to bear these in mind: many methods we will see involve finding a solution under assumptions about parameters.
What do we do about random external forcings? Will talk about later…

Upcoming Schedule

Next Class

Wednesday: Simulating Systems; Box Models.

Variable	Meaning	Units
\(X_t\)	P level in lake at time \(t\)	dimensionless
\(a_t\)	Controllable (point-source) P release	dimensionless
\(y_t\)	Random (non-point-source) P runoff	dimensionless