10 Jan 24: Density dependent growth

10.1 Required reading

Vandermeer, J.H., Goldberg, D.E., 2013. Population Ecology: First Principles (Second Edition). Princeton University Press, Princeton, United States. p9-17. Link

10.2 Questions

  1. What is the equation for continuous time logistic growth in its classic form? Define all the symbols in the equation by writing their meanings in words. Can \(K\) be negative?

  2. What does dN/dt mean?

  3. Assume that \(N < K\). For what values of \(r\) will \(N\) increase over time?

  4. Assume that \(r > 0\) and \(K > N\). Will \(N\) increase or decrease in size over time?

  5. Assume that \(r,K \neq 0\). For what values of \(N\) is the population size constant (i.e., not changing over time)?

  6. What is the main difference between exponential and logistic growth?

  7. Sketch a graph of the logistic growth equation, \(\frac{dN(t)}{dt} = rN(t)\left(1 - \frac{N(t)}{K}\right)\), with time, \(t\), on the horizontal axis (x-axis), and population size \(N(t)\), on the vertical (y-axis).

  1. Add to your graph a dashed line corresponding to the carrying capacity, \(N(t) = K\).
  2. Label on your graph, \(N(0)\): the population size at \(t=0\).
  3. As drawn in your graph, is \(N(0)<K\)? i.e. is the population size at \(t=0\) less than the carrying capacity, \(K\)?
  4. As drawn, is \(r > 0\)? i.e. is the net reproductive rate when the population size is small, positive?
  5. What does it mean if \(r<0\) in terms of the per capita birth rate when the population size is small, \(b\), relative to the per capita death rate when the population size is small, \(d\)?
  6. If you answered ‘yes’ to c. add another line for \(N(t)\), but when \(N(0)>K\) (assume \(r>0\)). Note that \(N(0)>K\) means that at time \(t=0\) the population size, \(N(0)\), is greater than the carrying capacity, \(K\).

  1. Draw a graph of a. exponential growth, \(N(t) = N(0)e^{rt}\) or \(\frac{dN(t)}{dt} = rN(t)\) (both are the same equation), and b. logistic growth \(\frac{dN(t)}{dt} = rN(t)\left(1 - \frac{N(t)}{K}\right)\), where the value of \(r\) is the same for both a. and b.

  2. Draw a graph of logistic growth, where the population size is decreasing \(\frac{dN(t)}{dt}<0\), but positive \(N(t)>0\). Give a condition on the initial value of \(N(0)\) or the per capita net reproductive rate when the population size is small, \(r\), such that the population size is decreasing, \(\frac{dN(t)}{dt}<0\), but positive, \(N(t)>0\).