In FredHasselman/nlRtsa: nonlineaR time seRies analysis.

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Assignment: Time-varying parameters

In this assignment we will build a more sophisticated growth model in Excel and look at its properties. The model will be the growth model by Van Geert (1991 etc.) as discussed in the book chapter you read. If your are experienced in R or Matlab you can try to code the models following the instructions at the end of this assignment.

Before you begin, be sure to check the following settings (same as first asignment):

• Open a Microsoft Excel worksheet or a Google sheet
• Check whether the spreadsheet uses a 'decimal-comma' ($0,05$) or 'decimal-point' ($0.05$) notation.
  • The numbers given in the assignments of this course all use a 'decimal-point' notation.
• Check if the $ symbol fixes rows and columns when it used in a formula in your preferred spreadsheet program.
  • This is the default setting in Microsoft Excel and Google Sheets. If you use one of those programs you are all set.

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The growth model by Van Geert (1991)

The growth model by Van Geert has the following form:

$$ L_{t+1} = L_t * (1 + r – r·\frac{L_t}{K}) $$

To build it repeat some of the steps you performed in last week’s assignment on a new worksheet.

• Define the appropriate constants ($r$ in A5, $L_0$ in A6) and prepare the necessary things you need for building an iterative process.
• In particular, add the other parameter that appears in Van Geert’s model:
  • Type $K$ in cell A7. This is the carrying capacity. It receives the value $1$ in cell B7.
• Start with the following values:
  • $r = 1.2$
  • $L_0 = 0.01$

Take good notice of what is constant (parameters $r$ and $K$), for which the $ must be used, and what must change on every iterative step (variable $L_t$). Take about $100$ steps.

• Create the graphs

• You can start playing with the values for the parameters and the initial values in cells B5, B6 and B7. To study this model’s behavior, be sure to try the following growth parameters:

  • $r = 1.2$
  • $r = 2.2$
  • $r = 2.5$
  • $r = 2.7$
  • $r = 2.9$

• For the carrying capacity $K$ (cell B7) you can try the following values:

  • $K = 1.5$
  • $K = 0.5$. (Leave $r = 2.9$. Mind the value on the vertical axis!)

• Changes in the values of $K$ have an effect on the height of the graph. The pattern itself also changes a bit. Can you explain why this is so?

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Conditional growth: Jumps and Stages

Auto-conditional jumps

Suppose we want to model that the growth rate $r$ increases after a certain amount has been learned. In general, this is a very common phenomenon, for instance: when becoming proficient at a skill, growth (in proficiency) is at first slow, but then all of a sudden there can be a jump to the appropriate (and sustained) level of proficiency.

• Take the model you just built as a starting point with $r = 0.1$ (B5)

  • Type $0.5$ in C5. This will be the new parameter value for $r$.
  • Build your new model in column B (leave the original in A).

• Suppose we want our parameter to change when a growth level of $0.2$ is reached. We’ll need an IF statement which looks something like this: IF $L > 0.2$ then use the parameter value in C5, otherwise use the parameter value in B5.

  • Excel has a built in IF function (may be ALS in Dutch).
  • In the first cell in which calculations should start, press $=$ and then from the formula list choose the IF function, or just type it.
  • Try to figure out what you have to do. In the Logical_test box you should state something which expresses $L > 0.2$.
  • The other fields tell Excel what to do when this statement is TRUE (then use parameter value in C5) or when it is FALSE (then use paramter value in B5).
  • Note: the word value might be misleading; you can also input new statements.
• Make a graph in which the old and the new conditional models are represented by lines.
  • Try changing the value of $r$ in C5 into: $1, 2, 2.8, 2.9, 3$.

Auto-conditional stages

Another conditional change we might want to explore is that when a certain growth level is reached the carrying capacity K increases, reflecting that new resources have become available to support further growth.

• Now we want $K$ to change, so type $1$ in B7, $2$ in C7.

• Build your model in column C. Follow the same steps as above, but now make sure that when $L > 0.99$, $K$ changes to the value in C7. Keep $r = 0.2$ (B5).

• If all goes well you should see two stages when you create a graph of the timeseries in column C. Change $K$ in C7 to other values.

  • Try to also change the growth rate r after reaching $L > 0.99$ by referring to C5. Start with a value of $0.3$ in C5. Set $K$ in C7 to $2$ again.
  • Also try $1, 2.1, 2.5, 2.6, 3$.

Connected growers

You can now easly model coupled growth processes, in which the values in one series serve as the trigger for for parameter changes in the other process. Try to recreate the Figure of the connected growers printed in the chapter by Van Geert.

Demonstrations of dynamic modeling using spreadsheets

See the website by Paul Van Geert, scroll down to see models of:

• Learning and Teaching
• Behaviour Modification
• Connected Growers
• Interaction during Play

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Assignments: Iterating 2D Maps and Flows

In this assignment we will look at a 2D coupled dynamical system: the Predator-Prey model (aka Lotka-Volterra equations). If your are experienced in R or Matlab you can try to code the models following the instructions at the end of this assignment.

Predator-prey model

The dynamical system is given by the following set of first-order differential equations, one represents changes in a population of predators, (e.g., Foxes: $f_F(R_t,F_t)$ ), the other represents changes in a population of prey, (e.g., Rabbits: $f_R(R_t,F_t)$ ).

$$ \begin{align} \frac{dR}{dt}&=(a-bF)R \ \ \frac{dF}{dt}&=(cR-d)F \end{align} $$

This is not a difference equation but a differential equation, which means building this system is not as straightforward as was the case in the previous assignments. Simulation requires a numerical method to 'solve' this differential equation for time, which means we need a method to approach, or estimate continuous time in discrete time. Below you will receive a speed course in one of the simplest numerical procedures for integrating differential equations, the Euler method.

Euler Method

A general differential equation is given by:

$$\frac{dx}{dt} = f(x)$$

Read it as saying: "a change in $x$ over a change in time is a function of $x$ itself". This can be approximated by considering the change to be over some constant, small time step $\Delta$:

$$\frac{(x_{t+1} – x_t)}{\Delta} = f(x_t)$$

After rearranging the terms a more familiar form reveals itself:

$$ \begin{align} x_{t+1} – x_t &= f(x_t) * \Delta \ x_{t+1} &= f(x_t) * \Delta + x_t \end{align} $$

This looks like an ordinary iterative process, $\Delta$ the time constant determines the size of time step taken at every successive iteration. For a 2D system with variables R and F on would write:

$$ \begin{align} R_{t+1} &= f_R(R_t,Ft) * \Delta + R_t \ F_{t+1} &= f_F(R_t,F_t) * \Delta + F_t \end{align} $$

Simulate the Coupled System

Implement the model in a spreadsheet by substituting $f_R(R_t,Ft)$ and $f_F(R_t,F_t)$ by the differential equations for Foxes and Rabbits given above.

• Start with $a = d = 1$ and $b = c = 2$ and the initial conditions $R_0 = 0.1$ and $F_0 = 0.1$. Use a time constant of $0.01$ and make at least $1000$ iterations.
• Visualize the dynamics of the system by plotting:
  • $F$ against $R$ (i.e., the state space)
  • $R$ and $F$ against time (i.e., the timeseries) in one plot.
• Starting from the initial predator and prey population represented by the point $(R, F) = (0.1, 0.1)$, how do the populations evolve over time?
• Try to get a grip on the role of the time constant by increasing and decreasing it slightly (e.g. $\Delta = 0.015$) for fixed parameter values. (You might have to add some more iterations to complete the plot). What happens to the plot?
  • Hint: Remember that $\Delta$ is not a fundamental part of the dynamics, but that it is only introduced by the numerical integration (i.e., the approximation) of the differential equation. It should not change the dynamics of the system, but it has an effect nonetheless.

The Competetive Lottka-Volterra Equations

The coupled predator-prey dynamics in the previous assignment are not a very realistic model of an actual ecological system. Both equations are exponential growth functions, but Rabbits for example, also have to eat! One way to increase realism is to consider coupled logistic growth by introducing a carrying capacity.

• Follow the link to the Wiki page and try to model the system!

This is what interaction dynamics refers to, modeling mutual dependiencies using the if ... then conditional rules isn't really about interaction, or coupling between processes.

FredHasselman/nlRtsa documentation built on May 6, 2019, 5:07 p.m.