From: eb3@world.std.com (Edwin Barkdoll)
Subject: Re: New Subscriber and a look on hunting
Organization: The World Public Access UNIX, Brookline, MA
Date: Wed, 15 Apr 1992 17:07:18 GMT

In article <1992Apr10.122751.4935@ac.dal.ca> robinson@ac.dal.ca writes:
>In article <BMCCHw.B87@world.std.com>, eb3@world.std.com (Edwin Barkdoll) writes:

>	[...]
>> 	40% plus a fudge factor.  Lets consider 3 real situations and
>> two imaginary ones:
>> Mid '50's michigan deer herd - 35% male
>> 1989/1990 PA deer herd - ~45% male
>> 1989/1990 CA deer herd - 20-25% male
>> Imaginary case #1 - 10% male
>> Imaginary case #2 - 95% male
>> 
>> Pop quiz for John: Imagine that each herd is on the same habitat with
>> the same carrying capacity, K, and the total herd populations are the
>> same.  Why will a fixed mortality rate across the herds, i.e.
>> (#animals killed)/(total herd population) NOT result in no population
>> change for some of the herds?

>	[John's calculations deleted]
>Only population 5's numbers drop in all cases, which is not surprising given 
>the buck/doe ratio.

	We differ on how we would do the calc's but our conclusion up
to this point is pretty much the same, i.e. a fixed mortality in the
above pop'ns will not yield equivalently stable populations.
	However, I don't agree with the following handwaves to justify
a fixed mortality producing stability, via changes in productivity due
to different sex ratios in the herds.

>some proportion of #fawns/doe throughout the population and it is also 
>reasonable to assume that populations with more bucks will have 
>correspondingly more multiple births.

	Never heard of such a thing.

>In population one, there would have to be a predominance of two fawns/doe, 
>with a few at three fawns/doe to produce zero growth given a 40% mortality 
>factor.  This is about average for fawn production and so it is quite 
>reasonable to assume that a 40% mortality factor would produce 
>approximately zero growth.

	If you mean that average fawns/doe is about 2.0 you are wrong
-- yes twins will predominate but the average will probably be 1.4-1.6-ish

>In population two, there are more bucks.

	Pop'n 2 has more (barely) _does_

>While there are fewer does, a 
>higher ratio of fawns/doe can be expected as a result of the increased 
>number of rutting bucks.

	Never heard of such a thing either.  There is however some
evidence that higher numbers (densities) of bucks actually _decrease_
average doe birth rates.

>In population three, the high number of does and corresponding low number 
>of bucks makes the possibility of three fawns/doe fairly remote. Two fawns/
>doe would be more reasonable, with a few around one fawn / doe, making for 
>a growth rate of approximately zero.
>
>In population four, the extreemly high number of does / buck makes it 
>unlikely that there will be many, if any, cases of triplets.  It is 
>reasonable to assume that there will be a large majority of single fawn 
>births, with some double fawn births and that is all that would be required 
>to maintain approximately zero growth for the population.

	You keep mentioning that none:triplet:twin:singleton ratios
will change significantly as buck:doe ratio varies.  Indeed you base
your entire argument upon that assumption.  I can guess the answer
but, got any references?

	Now lets see just how much the birth rates *must* change
according to John's argument to maintain all the pop'ns stable given
an across the board 40% mortality.  (Actually any fixed across the
board mortality)

	Just to recap our experimental pop'ns:
>> Imagine that each herd is on the same habitat with
>> the same carrying capacity, K, and the total herd populations are the
>> same.  Why will a fixed mortality rate across the herds, i.e.
>> (#animals killed)/(total herd population) NOT result in no population
>> change for some of the herds?

	In order to maintain teh various populations stable despite
the sex ratio differences, John argues that the proportion of
singletons, twins and triplets would vary with the sex ratio
suffieciently tjo maintain the different pop'ns at the same level.

	John's calculations above state in essence that the
proportional change in population size, given no mortality will equal
the (birth rate) times the (fraction of the herd which is
reproductively capable female).  This is a reasonable first pass, and
I'll accept it.  [Actually a more realistic model including the
carrying capacity explicitly doesn't change the following conclusions]
If John's argument is true then as the doe fraction fraction of the
herd varies, the birth rate must vary to compensate since we have
already stated above that the herds are identical in all other
respects.
	Just for simplicity lets call (birth rate) = b; (doe fraction)
= q.  Then for different herds, say herd 1 and herd 2, b1xq1 = b2xq2,
since the argument is that the fractional increment is the same (i.e.
0 in this special case but the actual value is unimportant).

	Assume for the sake of argument that for a "normal" herd with
q1 = 0.5 (50% female) the birth rate b1 is 2.0.  Consider another herd
with q2 = 0.1.  This requires that the birth rate b2 of the 2nd herd =
(q1 x b1)/q2 = 10!!!  This is an impossibly large number so lets try
something else.

	In order to have a theoretically *possible* birth rate b2 <=
3.0 we have 3.0 = (b1 x 0.5)/0.1, requiring b1 the birth rate of our
_normal_ population to be an anemic 0.6!  [An average birth rate of
3.0 is probably not even theoretically possible given a finite
proportion of young does.  Furhtermore the average birth rate of even
food supplemented older deer is not much higher than 2.0] To get an
idea of the actual normal range of fawns/doe see Verme (1983) "Sex
ratio in Odocoileus: a critical review_ F. Wildl.  Manage.  47(3) 1983
-- of 29 cited studies *none* had greater than 2 fawns/doe and only 2
studies had less than 1 fawn/doe (0.54 and 0.95).

	If birth rate b2 (of the q2 = 0.1 herd) is set to be a more
realistic 2.0 the required expected birth rate for our normal
population is required to be 0.4 which is pure fantasy.  Lowering b2
even more simply makes the birht rate of our "normal" herd even more
unrealistic.

	At the other end of the scale for a 3rd pop'n with q3 = 0.9 we
have 3.0 = (b3 x 0.9)/0.1 requiring a birht rate b3 = 0.33!  This is
totally unrealistic nor is it consistent with Gruver et al.'s J.
Wildl. Manage. 48(2) 1984) predictions of about 1.5 fawns/doe in a
Mississippi herd with a buck:doe ratio of 1:10 (i.e. q even greater
than 0.9).

	In other words, the handwaves although reasonably *sounding*
cannot be reconciled with reality.

-- 
Edwin Barkdoll
barkdoll@cattell.psych.upenn.edu
eb3@world.std.com
-- 
Edwin Barkdoll
eb3@world.std.com