18 Thurs Oct 29-Tues Nov 3: Midterm

DUE DATE: Fri Nov 6 at 5pm

18.1 Chapter 1 in Vandermeer and Goldberg

Vandermeer, J.H., Goldberg, D.E., 2013. Population Ecology: First Principles (Second Edition). Princeton University Press, Princeton, United States. Chapter 1. Link

Complete the table below by providing a response in all locations where there is a number in a box. Some of the table has already been completed to give you an idea of how to answer the questions. [20 marks]

If this table is difficult to read, please ask Dr. Hurford for clarification

Figure 18.1: If this table is difficult to read, please ask Dr. Hurford for clarification

18.2 Chapter 6 in Sherratt and Wikinson

Chapter 6. Is Nature Chaotic of Big problems in ecology and evolution by Sherratt and Wilkinson. Link

  1. What is chaos? [1 mark]

  2. Write down a population model that produces chaos? Define all the parameters and variables in the model. [4 marks]

  3. For the population model from your previous answer, for what values of the parameters does chaos occur? [1 mark]

  4. What is the butterfly effect? [1 mark]

  5. Give one example of a laboratory experiment that showed evidence of chaos. List one limitation of this experiment. Provide the full citation. [3 marks]

  6. Give one natural (non-laboratory) example of a population that may show chaotic dynamics. List one limitation of this study. Provide the full citation. [3 marks]

  7. Overall, based on Sherratt and Wilkinson, is there evidence that biological populations exhibit chaotic dynamics? [1 mark]

18.3 p30-36 in Vandermeer and Goldberg

Vandermeer, J.H., Goldberg, D.E., 2013. Population Ecology: First Principles (Second Edition). Princeton University Press, Princeton, United States. p30-36. Link

  1. What is the stable age distribution? [1 mark]

  2. What type of growth do the matrix population models described on p30-36 have? [1 mark]

  3. How can the finite rate of increase of the population be calcuated when the population is described by the matrix population models described on p30-36? [1 mark]

  4. For a 2 \(\times\) 2 projection matrix,

\[\begin{equation} \mathbb{P} = \left[ \begin{array}{cc} p_{11} & p_{12} \\ p_{21} & p_{22} \\ \end{array} \right], \end{equation}\] the eigenvalues are calculated by solving, \[\begin{equation} 0 = (p_{11} - \lambda)(p_{22}-\lambda) - p_{12}p_{21}. \end{equation}\]

If you expand the parenthesis (i.e. using the FOIL method), you will see that the equation can be solved using the quadratic formula.

Consider the projection matrix:

\[\begin{equation} \mathbb{A} = \left[ \begin{array}{cc} 0.1 & 2 \\ 0.7 & 0.5 \\ \end{array} \right]. \end{equation}\]

Calculate the two eigenvalues for this projection matrix, \(\mathbb{A}\). [3 marks]

  1. The dominant eigenvalue value is the one that has largest absolute value (i.e., if \(\lambda_1 < 0\) then \(|\lambda_1| = -\lambda_1\) and if \(\lambda_1 >0\) then \(|\lambda_1| = \lambda_1\)).

What is the dominant eigenvalue for the projection matrix \(\mathbb{A}\)? [1 mark]

  1. If the absolute value of the dominant eigenvalue of a projection matrix is greater than 1, then the population will increase over time. Will the population described by the projection matrix, \(\mathbb{A}\), increase over time? [1 mark]

18.4 Kendall et al. 2019

Kendall et al. 2019. Persistent problems in the construction of matrix population models. Ecological Modelling Link

  1. What are the 3 errors commonly encountered in matrix population models? [1 mark]

  2. What is a post-breeding census? [1 mark]

  3. Give a precise definition of \(b_x, b_i, \sigma_x\) and \(\sigma_i\). [2 marks]

  4. In figure 2d of Kendall et al. 2019, juveniles are shown as reproducing. Is this an error? How is this related to census timing? [2 marks]

  5. If you wanted to aggregrate across ages to make a stage-structured version of an age-structured model what section of this article would you consult? What is meant by age-distribution within a stage? [2 marks]

18.5 Numerical solutions of an ordinary differential equation.

Consider the following equation which is similar to logistic growth except that density dependence is assumed in mortality only and is assumed to have an exponential rather than a linear form:

\[\begin{equation} \frac{dN(t)}{dt} = bN(t) - dN(t)e^{\delta N(t)}. \end{equation}\]

Here, \(N(t)\) is the population density, \(b>0\) is a per captia birth rate, \(d > 0\) is the per capita mortality rate when \(N(t)\) is small, and \(\delta >0\) is a coefficient affecting the strength of density dependence.

  1. Solve this ordinary differential equation in R. You are to hand in your R Script and a figure of the population density (y-axis) versus time (x-axis) with a figure caption. See 1.7.1 and 1.7.2 for expectations for R scripts, figures and captions. You may want to refer to Chapter 12 for a template for how this problem could be solved. [7 marks]