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) Look at the term for the colonization process. How would you modify this term if you wanted to capture the idea that habitat loss destroyed (i.e., made uninhabitable) a proportion D from the patches that are available for colonization? d) 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. e) Using the modification in part a , modify Equation 1. What is the new proportion of patches occupied at equilibrium? f) What value of D guarantees complete extinction of the whole metapopulation? g) Do you have to destroy all the patches to cause extinction of the whole metapopulation? h) What does this say about the value of ‘empty’ patches where a species is not currently observed?
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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) Look at the term for the colonization process. How would you modify this term if you wanted to capture the idea that habitat loss destroyed (i.e., made uninhabitable) a proportion D from the patches that are available for colonization?
d) 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.
e) Using the modification in part a , modify Equation 1. What is the new proportion of patches occupied at equilibrium?
f) What value of D guarantees complete extinction of the whole metapopulation?
g) Do you have to destroy all the patches to cause extinction of the whole metapopulation?
h) What does this say about the value of ‘empty’ patches where a species is not currently observed?
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