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Re: Triceratops and Psittacosaurus etc.

In response to Steve  Brusatte's post:
>I find it hard to believe, in a strict biological sense, that four
>different species of the same genera would inhabit the same area.
>Dodson, in his book, repeats [that] ... if there were 8 or 10, then
>all of the species would be competing for the same general foods and
>something would go extinct.

Brian Choo wrote:
>Well, such a high diversity of one species is unusual but far from
>unprecedented. Using extant macropod marsupials as an analogy, the
>following species of Macropus (large grazing wallaby/kangaroo species
> - similar in size or larger than Psittacosaurus) can be found in
>warm-temperate east-coastal Australia, not always co-habiting but
>present >n the same general region...

Here finally a place for me, a mathematician to butt in!  According to
Gause model for competing species, it is possible for two species of
organisms with similar niche to coexist in the same region.  According
to the modern theory, the population N[1] of species 1 and the popu-
lation N[2] of species 2 inhabiting the same region are governed by
the system of differential equations:

dN[1]/dt = a[1]N[1](K[1] - N[1] - bN[2])/K[1],
dN[2]/dt = a[2]N[2](K[2] - N[2] - cN[1])/K[2].


a[1], a[2] are the proportinality factors obtained from the birth and
           death rate of species 1 and 2 resp.,
K[1], K[2] are the saturation population of the two species under
           the absence of the other,
b, c; 0 < b, c < 1 are the so-called competition factors.

In the unrealistic situation that b = c = 1, it is shown that one
of the two species must necessarily become extinct at some future
time.  This, supposedly is what Dodson is assuming in his book for
the case species 1 and 2 are in the same genus.  However, even in
the same genus, unless the two species are identical in every way
(in which case, I would suppose they would be considered to be the
same species, for they would be, to human eyes, be indistiguishable!)
b and  c would always be < 1.  In this case, if the

K[1]/b > K[2] and K[2]/c > K[1],

it can be proven that the two species would approach some happy
coexisting equilibrium.  Roughly, this condition means that the
habitat is moderate enough to allow a large population of the two
species to populate relative to how similar they are in their

Even in the case, one species does become extinct, the constants
K[1]/b and K[2]/c determine how fast or slowl this process takes
place.  If K[1]/b is close enough to K[2] and K[2]/c is close enough
to K[1], then if fact it might take such a long time for one species
to die out that for an extended amount of time, they continue to
coexist.  Since we know that Triceratops did become extinct, who
knows how the Triceratops population might have changed if they
happened to have existed beyond the K-T boundary?

Now, to venture on a subject I am far less knowledgeable than any
of you, someone wrote that all Ceratopian bonebeds discovered thus
far are Centrosaurine and not Chasmosaurine.  According to a book
I've read (by Prof. Phil Currie), it says that there have been
discovered (not one but) two bonebeds of Anchiceratops in Alberta.
Could someone clarify?

Yoe Itokawa
Dept. of Information and Communication Engineering
Fukuoka Institute of Technology -- Japan