Douglas C. Speirs and William S. C. Gurney. 2001. Population persistence in rivers and estuaries. Ecology 82:1219-1237.
Appendix B. Discrete space-time models.
To avoid the computationally onerous
task of solving Eqs. (1) and (12) numerically, we use an equivalent discrete
space-time representation described by Gurney et al. (1998). We divide space
into a contiguous series of equally spaced quadrats of width and
denote the average population density in quadrat
at time by . This density distribution is updated at intervals according to
is the redistribution
matrix and represents
the number of survivors and descendents of the population of quadrat at time
who are present at time . Following Gurney
et al. (1998) we equate with the solution of the logistic equation, so that
Well away from the domain boundaries
we again follow Gurney et al. (1998) and write the redistribution matrix
as a displaced
The coefficients , , and are chosen so that the distribution is properly normalized with a mean displacement of and a displacement variance of . That is
At the boundaries we use the method of images to define an appropriately modified redistribution matrix. In the case of the closed boundary we generate a mirror image of the distribution defined by Eqs. (B.3) to (B.5) reflected in a plane midway between the last node in the domain and its (notional) left-hand neighbor. The part of this reflection within the domain is added to the original distribution. The area thus added is exactly equal to the part of the original distribution falling outside the domain, so the distribution remains properly normalized. At the open boundary we employ a related strategy in which that part of the reflection falling within the domain is subtracted from the original distribution, and added to that falling beyond the open boundary.
This model explicitly represents
both distance from the river source ()
and depth below the river surface (), so we denote
position by a vector
. We define a contiguous series of rectangular
quadrats of width and
depth and denote the average population
density in quadrat at
time by . This density distribution is updated at intervals according to a generalisation
of Eq. (B.1)
The local growth function is given
by Eq. (B.2), and the redistribution matrix
is a generalisation
of the tent distribution given in Eq. (B.3)
For each originating depth () the coefficients , , and are chosen
so as to ensure that the distribution is properly normalised and has the appropriate
mean and variance for the
and displacements, that is
where the advection velocity () is related to depth and the overall average river velocity by Eq. (14).
In a tidally driven situation, the deterministic flow fields vary with time and it seems obvious that the redistribution matrix must be time-dependent likewise. However, if we set the update increment to exactly one tidal period, represents the residual motion averaged over the tidal cycle. If we further (strategically) ignore the spring-neap cycle, becomes time-independent.
Population processes typically have timescales of days rather than hours, so we expect day to be a viable choice of update increment. To check this we repeated a selection of our earlier simulations (which used day) and observed that the results were weakly affected by the increased . This result is in accordance with the findings of Gurney et al. (1998) in the context of a discrete spatial predator-prey model.
With set equal to the tidal period, our model is defined by Eqs. (B.6)
and (B.7) with the coefficients of the redistribution matrix chosen so that
In these equations, and represent the x- and z-components of the net displacement of a neutrally buoyant particle starting at over exactly one tidal cycle. We compute this quantity by tracking such a particle in the flow field defined by Eqs. (C.16) and (C.19) using a fourth order Runge-Kutta algorithm (Press et al. 1986) with a time increment of 10 minutes.