
254 Modeling and Inverse Problems in the Presence of Uncertainty
where β(t, x) is the reproduction (birth) rate of individuals with size x at time
t. We observe that for this case we have a non-local boundary co nditio n at
x.
Next we consider the equation for a p opulation density in which mortality
is present. We observe that the density of a population with size x at time
t + ∆t can be calculated by multiplying the probability that individuals with
size ζ at time t will move to x in a time increment ∆t by the density of a
population with size ζ at time t and summing over all the values of ζ. That
is,
u(t + ∆t, x) =
Z
R
u(t, ζ)(1 − ∆td(t, ζ))ρ(t + ∆t, ...