Question 2. Consider the following figure representing the theoretical relations

Question

Question 2. Consider the following figure representing the theoretical relationships between patch area (size) and isolation (distance from one another) and the equilibrium proportion of occupied habitat patches in a metapopulation: 0.25 0.20 0.15 0.10 0.05b 0 0.2 0.4 06 0.8 1.0 Given the metapopulation equation (dP/dt- cP(1-P) - eP), at equilibrium, what is the predicted change in patch occupancy from one year to the next for a metapopulation with large patches located relatively far from one another? Show your work

Explanation / Answer

The rate dP/dt is given by the difference between colonization rate C and extinction rate E. Colonization rate is the number of successful colonizations of unoccupied patches as a proportion of all available patches, and extinction rate is the proportion of patches becoming empty. Thus,

dP/dt = C - E, or dP/dt = cP(1 - P) - eP,

At equilibrium dP/dt = C - E = 0, or P' = 1 - e/c. For persistence of the metapopulation, it is required that P' >0, which is true if e/c < 1 or c > e.as long as the probability of a patch to be colonized exceeds its probability to go extinct, the metapopulation exists, with a single stable equilibrium incidence P'. The metapopulation is stable at the point in where dP/dt = 0, at the intersection of the C and the E lines. Where P' is depends on the shape of these lines. If actual observed P lies in the area where the difference between the lines is positive, the number of occupied patches P increases. If P is in the area where the difference is negative, P decreases. If colonization exceeds extinction (C > E) then there is an equilibrium value for incidence (0 < P' < 1).

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