Model rationale

Kynard et al., 2007, performed experiments on dispersal of pallid sturgeon embryos. They recommended a passive dispersal model in which embryos simply float at the bottom of the river until they develop into larvae, at which time they begin feeding at the bottom and actively determine their position. The main determinants of how far embryos drift are the velocity of water at the bottom of the main channel, combined with the temperature of water, which determines how long embryos develop.

Input variables

River velocity

Kynard et al. used 32 cm per second, which is 1.152 km per hour.

rivVelMO <- 1.152

Time of dispersal

The time of year of dispersal is important because this determines the river temperature, which controls the amount of time it takes for embryos to develop. Kynard et al. states that dispersal occurs in June and July.

River temperature

Using the same method as in Kynard et al. 2007, river temperature is linked to daily development of fish using the daily cumulative temperature units (CTU) that the embryo experiences while developing. Day 0 fish accumulate 0 temperature units, day 1 fish accumulate the mean temperature experienced by a day 0 embryo in degrees Celsius, day 2 fish accumulate the mean of day 0 plus the mean of day 1, and so on.

Development into larva occured on day 11 at 198 CTU and day 12 at 204-205 CTU. Days begin with age 0 embryo. Recommend using 200 CTU for criterion for reaching larval stage.

Spawning locations

Furthest downstream spawning location is river kilometer (rkm) 503, which is between the Old River Control Structure and Natchez, MS. Furthest upstream spawning, and upstream limit of population occur at rkm 1533, at the mouth of the Ohio River.

lastSpawnLoc <- 503
firstSpawnLoc <- 1533

The location of the proposed Mid-Barataria sediment diversion is at rkm 97.69

divLoc <- 97.69

The distance from the last spawning location to the sediment diversion is 405.31 km.

lastSpawnLoc - divLoc

## [1] 405.31

Spawning locations are assumed to be uniformly distributed upriver from the furthest downstream spawning location.

Models

Model 1: Simplest point estimate using data from Kynard et al. 2007

Simply multiply the hours of development by the river velocity

maxDays <- 12
maxHours <- 12 * 24
maxDistPoint <- maxHours * rivVelMO
maxDistPoint

## [1] 331.776

rkmPoint <- lastSpawnLoc - maxDistPoint
rkmPoint

## [1] 171.224

Model 2a: Gamma distribution for development days. Only for last spawning location.

Gamma distribution for number of development days. Parameterized arbitrarily to give a mean of 12 days.

devDaysDist <- rgamma(1000, 36, 3)

Distribution of distances traveled for embryos fertilized at a given km of river.

distDist <- devDaysDist * 24 * rivVelMO
hist(distDist, xlab = 'Distance traveled (km)', main = 'Distribution of larval settlement', col = 'gray')

Distribution of river km traveld to, that is river km of settlement.

rkmDist2a <- lastSpawnLoc - distDist
hist(rkmDist2a, xlab = 'River km', main = 'Distribution of larval settlement', col = 'gray')

The proportion of larvae settling at or below the proposed diversion is

sum(rkmDist2a <= divLoc) / length(rkmDist2a)

## [1] 0.093

Model 2b: Gamma distribution for development days. All spawning locations.

Use the same distribution of days of development and the same distribution of distances traveled for embryos fertilized at a given km of river as for 2a.

spawningLocations <- lastSpawnLoc:firstSpawnLoc
rkmDist2b <- NA
for(i in 1:length(spawningLocations)){
  distDist <- devDaysDist * 24 * rivVelMO
  rkmDist <- spawningLocations[i] - distDist
  rkmDist2b <- c(rkmDist2b, rkmDist)
}
rkmDist2b <- rkmDist2b[-1]
hist(rkmDist2b, xlab = 'River km', main = 'Distribution of larval settlement', col = 'gray', freq = FALSE)
abline(v = divLoc, col = 'red', lwd = 2)