Lab 5:  Finding equilibria and assessing local stability using Mathematica

In this lab, we will develop a model describing the number of occupied sites over time for a species that has a patchy spatial distribution (see Problem 5.12).  Such population structure is often called a metapopulation.

Consider a large number of suitable territories or patches.  At time t, a fraction, p(t), of these patches are occupied.  Of the 1 - p(t) unoccupied sites, a fraction m p(t) are recolonized from occupied patches.  This colonization rate reflects the idea that the more occupied patches there are, the more migrants are likely to arrive at the unoccupied patch.

Subsequently, each occupied site suffers a risk of local extinction, e, through catastrophic events such as fire or disease.  These assumptions are consistent with the following discrete-time recursion equation for the fraction of occupied sites:

	p(t+1) = (p(t) + fraction of sites colonized)*
			(probability of not going extinct)
		= (p(t) + m p(t) (1 - p(t)))*(1 - e)

The extinction risk, e, equals the proportion of occupied sites that go extinct and so must lie between zero and one (0 ≤ e ≤ 1).  Similarly, m p(t) is the probability that an empty patch is colonized.  Thus, m is the probability that an empty patch is colonized when all the other patches were full (p(t) near one), indicating that m must lie between zero and one (0 ≤ m ≤ 1).

Question # 1

What are all the equilibria of this equation?

First do this by hand and then check it with Mathematica using the Solve[f(x) == 0,x] command.

When is the non-zero equilibrium biologically valid (0 ≤ p ≤ 1)?  [You can use this condition to help interpret the answers to the next two questions.]

Question # 2

When would you predict that the metapopulation will go extinct?

Determine the local stability for the zero equilibrium found in Question #1.

Use Mathematica to calculate the derivative of the recursion equation using the derivative command,  D[f(x),x].  Determine when this derivative is between -1 and 1 (implying stability for a recursion equation).

Question # 3

When is the metapopulation likely to persist?

When this equilibrium is biologically relevant, determine its local stability.  Use Mathematica to calculate the derivative of the recursion equation using the derivative command,  D[f(x),x].  Determine when this derivative is between -1 and 1 (implying stability for a recursion equation)

Question # 4

Think about your results from the previous questions.  Do they make sense?  Do they tell you anything about what must be true for a species to persist in a spatially patchy environment?

For a metapopulation in which e and m have been estimated, you find that the observed fraction of occupied sites matches the predicted equilibrium, leading you to suggest that the metapopulation might be at equilibrium.  Somebody challenges your result by saying that they surveyed one particular patch, which was occupied last year but not occupied this year, so obviously the system is not at equilibrium.  How do you respond?