In dynamic-R/RTM: Reaction Transport Modelling: Course Tutorials and Exercises

Questions

Problem formulation

In the previous lecture you made a food-web model of salmon, bears, scavengers and salmon carcasses. You first created the conceptual model (flow diagram), then wrote the mass balance equations.

Now, you will create a rate expression for every flow that entered the mass balance equations. Once you have done that, you can see how this model is implemented in R.

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Mass balance equations

Suitable mass balances for the state variable SALMON, BEAR, CARCASS and SCAVENGERS are:

$$\frac{dSALMON}{dt} = Migration - BearFeeding - SalmonLoss$$ $$\frac{dBEAR}{dt} = BearFeeding-CarcassProd-BearLossIngest-BearLoss$$ $$\frac{dCARCASS}{dt} = CarcassProd -ScavFeeding - CarcassDecay$$ $$\frac{dSCAVENGERS}{dt} = ScavFeeding -ScavLossIngest - ScavLoss$$

Assumptions

The assumptions are the following:

1. The units of the state variables are $kg~C~m^{-2}$; the time unit is days.

2. Question: What are the units of the flows?

3. The model domain (river + surrounding area) is $1000~m\times 1000~m$ large.
4. The input of the salmon in the river is defined by a parameter called migrate. Per day, $10000~kg$ of salmon carbon migrate up the river.
5. The mortality + spawning + basal respiration of salmon, the hunting and basal respiration of the bear, and the losses of the scavengers are expressed by a first-order rate.
6. The parameters that you should use, and their values are:

| parameter | units | description | |---------------------------|--------------- | ---------------------------------------------------- | | area = 1000000 | [$m^2$] | | | migrate = 10000 | [$kg~C~d^{-1}$] | Amount of salmon entering the river system | | rSalmonLoss = 0.05 | [$d^{-1}$] | Salmon loss rate constant (Spawning+death+basalResp) | | rBearFeeding = 0.02 | [$d^{-1}$] | Bear feeding rate constant | | ksSalmon = 0.01 | [$kg~C~m^{-2}$] | Half saturation ct for ingestion bear | | rBearLoss = 0.01/365 | [$d^{-1}$] | Death rate constant (hunting and other) for bear | | pLossToCarcass = 0.5 | [$-$] | Losses of salmon to carcass (fraction) | | pBearLossIngest = 0.4 | [$-$] | Ingested fraction that is lost (respiration+faeces) | | rCarcassDecay = 0.001 | [$d^{-1}$] | Decay rate constant of carcasses | | rScavFeeding = 0.02 | [$d^{-1}$] | Feeding rate constant of scavengers | | ksCarcass = 0.002 | [$kg~C~m^{-2}$] | Half saturation ct for ingestion scavenger | | pScavLossIngest = 0.7 | [$-$] | Growth efficiency of scavengers | | rScavLoss = 1/365 | [$d^{-1}$] | Mortality+basal respiration rate constant scavengers |

Tasks

The above information is all you get. Write the mathematical equations for each flow on a piece of paper.

Note: to make the equations (and the model code) easy to follow, we use the following formalism:

• State variables are written in CAPITAL LETTERS,
• Flows start with a capital letter, and
• parameters start with a lower case letter.

If you have finished with this exercise, then you can try to read, understand and then run the implementation of this model in R, using R markdown.

It is in the file called RiverRun_Rcode.Rmd. The model is run twice, with different rate constants characterising ingestion by the bears.

• Try to explain the output of this model.
• Assuming the mean weight of a bear at the start of the simulation is 300 $kg$, how many bears are there in the area? What is their final weight?

dynamic-R/RTM documentation built on Feb. 28, 2025, 1:23 p.m.