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1) A mathematical (differential equation) representation for a metapopulation can be written as
dp/dt = cp(1- p) - ep
where p is the proportion of patches occupied at time t, c is the rate of patch colonization, and e is the rate of patch extinction. The term on the left hand side is the ‘rate of change in the proportion of patches occupied.’ The first term on the right hand side of the equation provides the effective gain in patch occupancy as a function of p due to the colonization process, and the term in parentheses represents the fraction of patches not currently occupied. The second right hand side term provides the effective loss of patch occupancy due to the (local) extinction process.
a) Pick a positive value for c and draw a plot that shows the colonization process (first right hand side term) as a function of p.
b) Pick a positive value for e and draw a plot that shows the extinction process (second right hand side term) as a function of p.
c) What proportion of patches is occupied “at equilibrium” ? In this case, ‘at equilibrium’ means that the ‘rate of change in the proportion of patches occupied’ is zero. Another way of saying this is that equilibrium occurs when occupancy is neither increasing nor decreasing. Yet another way of saying this is that equilibrium occurs when the colonization process exactly balances the extinction process.
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