The role of a ceiling on population size

One of the difficulties of the MacArthur-Wilson theory is that the density dependence of demographic interactions and the population ceiling are confounded in the same parameter K. We now separate them. In particular, we will assume that there is a population ceiling Nmax, in the sense that absolutely no more individuals can be present in the habitat of interest. (My former UC Davis, and now UC Santa Cruz, colleague David Deamer used to make this point when teaching introductory biology by having the students compute how many people could fit into Yolo County, California. You might want to do this for your own county by taking its area and dividing by a nominal value of area per person, perhaps 1 square meter. The number will be enormous; that's closer to the population ceiling, the carrying capacity is much lower.)

We now introduce a steady state population size Ns defined by the condition

With this condition, Ns does indeed have the interpretation of the deterministic equilibrial population size, or our usual sense of carrying capacity in that birth and death rates balance at Ns. This steady state will be stable if l(n) > p(n) if n < Ns and that l(n) < p(n) if n > Ns. This is the simplest dynamics that we could imagine. There might be many steady states, some stable and some unstable, but all below the population ceiling.

Why bother to contain with a population ceiling? The answer can be seen in Eq. (8.12). In its current form, this is a system of equations that is ''open,'' since each equation involves T(n — 1), T(n), and T(n + 1). It is closed from the bottom - as we have already discussed - since p(0) = 0, but introducing the population ceiling is equivalent to 1(Nmax) = 0, in which case Eq. (8.12) becomes, for n = Nmax

-1 = — (1(Nm max ))T (N m p(Nmax)T (N max — 1) (8.16)

and now the system is closed from both the top and the bottom.

Because the system is now closed, and because the population is being measured in number of individuals, the mean extinction time can be viewed as a vector

and we can write Eq. (8.12) as a product of this vector and a matrix (Mangel and Tier 1993, 1994).

Before doing that, let us expand the framework in Eq. (8.12) to include catastrophic changes in population size. That is, let us suppose that catastrophic changes occur at rate c(n) in the sense that

Prfpopulation size changes in the next dt|N(t) = n} =

Prfchange is caused by a catastrophe| change occurs} = ■

and that, given that a catastrophe occurs, there is a distribution q(y|n) of the number of individuals who die in the catastrophe

Prfy individuals die|catastrophe occurs, n individuals present} = q(y|n)

We now proceed in two steps. First, you will generalize Eq. (8.12); then we will use the population ceiling and matrix formulation to solve the generalization.

Show that the generalization of Eq. (8.12) is

-1 = 2(n)T(n + 1) - ((l(n) + ^(n) + c(n))T(n)) + ^(n)T(n - 1)

in which we allow that no individual or all individuals might die in a catastrophe. (This is an unlikely event, chosen mainly for mathematical pleasure of starting the sum from 0, rather than a larger value. In practice, q(y|n) will be zero for small values of y. Although, it is conceivable, I suppose, that a hurricane occurs and there are no deaths caused by it.)

Now we define s(n) by s(n) = l(n) + ^(n) + c(n)(1 - q(0|n)) and a matrix M whose first four rows and five columns are

c(«e + 3)q(2|ne + 3) M(«e + 3)+ c(«e + 3)q(1 |n + 3)

and we define the vector -1 by

m(«e + 4)+c(ne + 4)q(1|ne + 4) -s(ne + 4) 1(«e + 4)

Once we have done this, Eq. (8.20) takes the compact form

and if we define the inverse matrix M -1 then Eq. (8.23) has the formal solution

Now we take advantage of living in the twentyfirst century. Virtually all good software programs have automatic inversion of matrices, so that computation of Eq. (8.24) becomes a matter of filling in the matrix and then letting the computer go at it.

In Figure 8.4, I show the results of this calculation for the flour beetle model (Peters et a/. 1989) in which l(n) = b0(n + 6)exp( -b1n) and yu(n) = d1n for the case in which there are no catastrophes and three different cases of catastrophic declines (Mangel and Tier 1993, 1994). For the parameters b0 = 0.13, b1 = 0.0165, S = 1, d1 = 0.088 the steady state is at about n = 26, so a population ceiling of 50 would be much larger than the steady state. As seen in the figures, whether the

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