Project 4, MATH 481, Spring 2008
Population dynamics: Single species

Assignment of 2008–03–03

Read sections 5.1 through 5.3 in textbook.

Correction of 2008–03–15: Oops, that's for the next project. I should have said: Read sections 3.1 through 3.5 in textbook.

Added 2008–03–07

The current (March 6) issue of Nature has a report on a mathematical model of time-dependent fluctuations of midge population in Lake Midge, Iceland: High-amplitude fluctuations and alternative dynamical states of midges in Lake Myvatn.

The article describes a predator-pray model where the predator is the midge and the pray is the algae that the midge feed on. The model's coefficients are determined from real data gathered over many years. The model exhibits an almost-cyclic variations in the midge population, consistent with observations.

Added 2008–03–14

Project 4 is on dynamics of a single species population. Write an informative article about it. Here are ideas on what to write about but you are not limited to these.

• Population models according to Malthus and Verhulst. You may find information about this in the course's textbook. Additional information is available on the web, for instance:
  • Thomas Malthus and his theory.
  • Pierre François Verhulst and his theory.
  and much more.
• How to plot the solution curves of a differential equation of the form dp/dt = f(p) without solving it.
• What are the equilibria of this differential equation? How are they related to f? What are stable and unstable equilibria?
• The Verhulst model with harvesting: dp/dt = r (1 − p/K) p − h.
• The effect of varying the harvesting rate h on equilibria. Implications on survivability of the population under harvesting.
• There is a critical value h_c such that if h > h_c then there is no chance for any of the population to survive. How do you calculate h_c?

Added 2008–03–17

An article titled Chinook Salmon Vanish Without a Trace in today's New York Times reports on sudden disappearance of Chinook salmon from the Sacramento River. No definitive cause is given but it appears that mismanagement of the regional fisheries is to blame.

Due date

This assignment is due on Friday March 28.