(a) Compute R0 and use Newton's method to find r. (b) What do you predict will happen to R0 and r if the survivorship for age 5 and beyond decreases by just 5%? Now compute the new values. (c) Compute R0 and r if individuals delay reproduction from year 2 to year 4 because of a food shortage. That is, assume that individuals are now 4 years old when they get the reproduction previously associated with a 2 year old, 5 years old when they get reproduction previously associated with a 3 year old, etc. Interpret your results.

Underlying all of these calculations is the schedule of survival and fecundity and it is the schedule of survival that I now want to investigate, using a theory of organismal vitality determined by Brownian motion due to Jim Anderson at the University of Washington (Anderson 1992, 2000). Survival to any age is the result of internal processes and external processes, so that we write /(a) = Pe(a)Pv(a), where Pe(a) is the probability of survival to age a associated with external causes (random or accidental mortality, we might say), and which we assume to be e—ma, and Pv(a) is the survival to age a associated with internal processes and organismal vitality.

Let us define V(t) to be that vitality, with the notion that V(t) > 0 means that the organism is alive and that V(t) = 0 corresponds to death. Anderson assumes that V(t) satisfies the following stochastic differential equation dV = —pdt + adW (8.86)

so we see that V(t) declines deterministically at a constant rate and is incremented in a stochastic fashion by Brownian motion. This is clearly the simplest assumption that one can make, but, as the work of Anderson shows, one can go a long way with it.

It may be helpful to think of vitality as the result of a variety of hidden physiological and biochemical processes which, when taken together, determine an overall state of the organism. It may also be that there is no such thing as ''external'' mortality - that all mortality is vitality driven. For example, the ability to escape a falling tree (a random event in the forest) may depend upon internal state as much as anything else.

The probability density for V(t), defined so that ^(v, t|v0, 0)dv = Pr{v < V(t) < v + dv| V(0) = v0}, satisfies the forward equation

and from the definition, we know that ^(v, t|v0, 0) = 6(v — v0); as before, one boundary condition will be ¿>(v, t|v0, 0)! 0 as v n. For the second boundary condition, since an organism starting with no vitality is dead ^(v, t|0, 0) = 0. The solution of Eq. (8.87) satisfying the specified initial and boundary conditions is not exceptionally difficult to find, but this is one of the few cases in this book in which I say ''we look it up.'' Some of the best sources for looking up solutions of the standard diffusion equation are Carslaw and Jaeger (1959), Goel and Richter-Dyn (1974), and Crank (1975) (this particular solution is computed by the ''method of images'' in which we satisfy the boundary condition at 0 by subtracting an appropriate mirror image quantity). The solution is pv, t|vo, 0) =

0 0

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