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
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?
What does dN/dt mean?
Assume that N<K. For what values of r will N increase over time?
Assume that r>0 and K>N. Will N increase or decrease in size over time?
Assume that r,K≠0. For what values of N is the population size constant (i.e., not changing over time)?
What is the main difference between exponential and logistic growth?
Sketch a graph of the logistic growth equation, dN(t)dt=rN(t)(1−N(t)K), with time, t, on the horizontal axis (x-axis), and population size N(t), on the vertical (y-axis).
- Add to your graph a dashed line corresponding to the carrying capacity, N(t)=K.
- Label on your graph, N(0): the population size at t=0.
- As drawn in your graph, is N(0)<K? i.e. is the population size at t=0 less than the carrying capacity, K?
- As drawn, is r>0? i.e. is the net reproductive rate when the population size is small, positive?
- 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?
- 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.
Draw a graph of a. exponential growth, N(t)=N(0)ert or dN(t)dt=rN(t) (both are the same equation), and b. logistic growth dN(t)dt=rN(t)(1−N(t)K), where the value of r is the same for both a. and b.
Draw a graph of logistic growth, where the population size is decreasing 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, dN(t)dt<0, but positive, N(t)>0.