8.2-2a) In the Levins model, a fixed number of sites, N, are available in thehabitat and can be occupied by a subpopulation or empty, with the fraction ofoccupied sites represented by p, the rate at which occupied sites sendpropagules to other sites in order to colonize them is represented by c, andconstant 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 youshould calculate) and determine their stability.Give the equation that must be solved to find the possible equilibrium values for pin terms of c.differential equation = cp(1 - p) - mp, where the term cp(1- p) gives therate of new colony formation and the term - mp represents the rate of loss bymortality. This equation is a stable equilibrium at p= 1- m/c if m c,the equation still has a stable solution equilibrium, but 1- m/c < 0, so is not apossible value for p(t)). Often, however, the survival of the population at a sitedepends on the number of subpopulations in the surrounding sites. If differentsubpopulations compete for limited resources, then the per site mortality ratemay not be a constant, but may increase with p because, as p increases,competition increases. In this model, called the patch model, the samecolonization 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>0dt

Name: 8.2-2a) In the Levins model, a fixed number of sites, N, are available
Uploaded: 2024-03-20T06:38:58.000Z
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Description: 8.2-2a) In the Levins model, a fixed number of sites, N, are available in thehabitat and can be occupied by a subpopulation or empty, with the fraction ofoccupied sites represented by p, the rate at which occupied sites sendpropagules to other sites

We have, dpdt=cp1-p-p2 For the equilibrium point we have, dpdt=0cp1-p-p2=0pc1-p-p=0p=0or c1-p-p=0c1-p=pc-cp=pp1+c=cp=c1+cHence, there exists two equilibrium values for p and they are, p=0 or c1+c