How can I replicate the model of a paper that describes an animal's foraging radius?

Question

I'm trying to follow the algebra of a paper that describes a model used to estimate a bird's foraging radius. It takes an energetics approach.

$T$ is the foraging cycle, $E_{ma}$ is adult basal metabolism, $E_{mc}$ is chick basal metabolism (divided by two because two parents feed it) and the energy required to fly a distance $2r$ is given by $E_t(2r)$. So you get a model showing how much energy the bird needs.

$T(E_{ma}+\frac{E{mc}}{2}) + E_t(2r)$

So the bird must have a set amount of energy $e$ to match these costs:

$e= T(E_{ma}+\frac{E{mc}}{2}) + E_t(2r)$

It's given that there's a linear relationship between distance and cost of flight with $k$ some constant:

$E_{t}(2r)=2kr $

Substituting and rearranging for $r$ to get the maximum radius the bird could fly:

$r = \frac{e-T(E_{ma}+\frac{E_{mc}}{2})}{2k}$

If I plug in the provided values:

$e = 7.8 x 10^6 Joules$

$T = 48 hours $

$E_{ma} = 24 Watts$

$E_{mc} = 42 Watts$

$k = 2.0 Jm^{-1}$

I get the answer in the paper of 6000m (see my R code below).

The model is then modified to allow the birds to digest some food and eat again.

The time spent in a feeding area is set as the time away from the nest minus time travelling where $V$ is some constant speed.

$\tau_f = \tau - \frac{2r}{V}$

The estimate for the amount of energy the bird can get is the:

$e(r) = Q(C+\delta\tau_f)$

where $Q$ is energy density of food, $C$ is crop capacity and $\delta$ is rate of digestion.

The energetic costs are unchanged:

$T(E_{ma}+\frac{E{mc}}{2}) + 2kr$

They then equate energy costs with gain and solve for $r$

$r = \frac {QC + Q\delta\tau-T(E_{ma} + \frac{E_{mc}}{2} )} {2k + \frac{2Q\delta}{V} }$

They supply these values and get a value of 205km:

$\tau = 12 hours$

$C =1.5kg$

$Q=5.2*10^6J$

$\delta =0.055kg^{-1}hour$

$V=45km^{-1}hour$

I cannot get the value of 205 that they provide and I can't see where I'm going wrong. I suspect I'm messing up my units but any insight would be much appreciated.

R code for models

first model

e = 7.8 * 10 ^ 6
Time = 48 * 60 * 60 ## to get it in seconds
Ema = 24
Emc = 42
k = 2

radius <- function(e, Time, Ema, Emc, k) {
  R <- (e - Time * (Ema + Emc / 2)) / (2 * k)
  return(R)
}
radius (e, Time, Ema, Emc, k)

second model

Q <- 5.2 * 10 ^ 6
C <- 1.5
d <- 0.055
t <- 12 * 60 * 60
Time <- 48 * 60 * 60
Ema <- 24
Emc <- 42
k <- 2
V <- 45


radius2 <- function(Q, C, d, t, Time, Ema, Emc, k, V) {
  R <- (Q * C + Q * d * t - Time * (Ema + Emc / 2)) / (2 * k + ((2 * Q * d) /
                                                                  V))
  return(R)
}

radius2(Q, C, d, t, Time, Ema, Emc, k, V)