(a) Show that there are two possible equilibrium values for p in [0,1] (which you should calculate) and determine their stability. Give the equation that must be solved to find the possible equilibrium values for p in terms of c. (Type an equation using c and p as the variables.)

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Chapter2: Second-order Linear Odes
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8.2-2a)

In the Levins model, a fixed number of sites, N, are available in the
habitat and can be occupied by a subpopulation or empty, with the fraction of
occupied sites represented by p, the rate at which occupied sites send
propagules to other sites in order to colonize them is represented by c, and
constant fraction m of subpopulations dying out. The Levins model leads to a
(a) Show that there are two possible equilibrium values for p in [0,1] (which you
should calculate) and determine their stability.
Give the equation that must be solved to find the possible equilibrium values for p
in terms of c.
differential equation = cp(1 - p) - mp, where the term cp(1- p) gives the
rate of new colony formation and the term - mp represents the rate of loss by
mortality. This equation is a stable equilibrium at p= 1- m/c if m<c. (If m > c,
the equation still has a stable solution equilibrium, but 1- m/c < 0, so is not a
possible value for p(t)). Often, however, the survival of the population at a site
depends on the number of subpopulations in the surrounding sites. If different
subpopulations compete for limited resources, then the per site mortality rate
may not be a constant, but may increase with p because, as p increases,
competition increases. In this model, called the patch model, the same
colonization rate is used, but now p describes the density-dependent mortality
(Type an equation using c and p as the variables.)
rate.
dp
= cp(1- p) - p?, c>0
dt
Transcribed Image Text:In the Levins model, a fixed number of sites, N, are available in the habitat and can be occupied by a subpopulation or empty, with the fraction of occupied sites represented by p, the rate at which occupied sites send propagules to other sites in order to colonize them is represented by c, and constant fraction m of subpopulations dying out. The Levins model leads to a (a) Show that there are two possible equilibrium values for p in [0,1] (which you should calculate) and determine their stability. Give the equation that must be solved to find the possible equilibrium values for p in terms of c. differential equation = cp(1 - p) - mp, where the term cp(1- p) gives the rate of new colony formation and the term - mp represents the rate of loss by mortality. This equation is a stable equilibrium at p= 1- m/c if m<c. (If m > c, the equation still has a stable solution equilibrium, but 1- m/c < 0, so is not a possible value for p(t)). Often, however, the survival of the population at a site depends on the number of subpopulations in the surrounding sites. If different subpopulations compete for limited resources, then the per site mortality rate may not be a constant, but may increase with p because, as p increases, competition increases. In this model, called the patch model, the same colonization rate is used, but now p describes the density-dependent mortality (Type an equation using c and p as the variables.) rate. dp = cp(1- p) - p?, c>0 dt
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