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Appendix S2
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We proceed below to demonstrate that all species in an arbitrary n species system will
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have a singular strategy at (and thus, can potentially evolve towards) the minimum
4
threshold required for persistence, β0, when competitive exclusion is strictly deterministic
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(i.e. k   ). In order to determine the singular strategy of an n species community,
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β  1* ,,  n* , when k is arbitrarily large we take the limit as k   for the system of
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equations represented by (9), here represented by the rth equation:
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

lim  
 Cr ,1 
 C 
1
*
*
  pˆ 1   (1  k r* )  pˆ r   Cr ,n  (  r*   n* )   r ,n   pˆ n   0 .
 x  Cr ,1  (  r  1 )  
k    
2

 1 
 1 

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For any arbitrary species r the limit reduces to:

*

 lim
pˆ r   0
k  

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 r*  
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The expression for the equilibrium abundance of the rth species, p̂r , can be found by
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solving equation (2) when dpi/dt = 0, determining the expression for the first species p̂1 ,
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and then substituting it in to the expression for the abundance of the second species p̂2 . A
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process of sequential substitutions can be continued until an expression is determined for
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the rth species. Alternatively, we can solve for the rth species abundance using Cramer’s
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rule for the linear system represented by equation (4): pˆ r  det Ar , where Ar is the matrix A
(B.1)
det A
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with its rth column vector replaced with vector b from (4). We thus get
lim
det Ar
k 
pˆ r 
lim
k 
det A
k 
lim
(provided lim det A  0)
k 
(B.2)
lim
det A  1   2   n .
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The expression in the denominator can be shown to reduce to
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For the numerator, by sequentially taking the Laplace expansion starting on the last
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column of Ar one can show the solution to be
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where
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
r

r
lim
k 
k 
det A  1   r 1  r   r 1   n ,
represents the expression:
r 1 
r 1 

  

 i1

 (  r x  d )   (  m x  d )  (1  r )    (1  r )   (1  j 1 )   .
 m im1 
 i j m
 j  
m1 



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(B.3)
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Substituting the numerator and denominator into (B.2) gives us
lim
k 
pˆ r 
1   r 1   r   r 1   n  r .

1   r   n
r
(B.4)
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When we substitute (B.4) into (B.1) we can see the linear system represented by (9)
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reduces to
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
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Starting with the first species then sequentially solving for β and substituting the values
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of each β into the expression for the next species, it can be readily seen (by inspection of
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(B.3)) that for all species equation (B.5) reduces to (r x  d ) = 0, or alternatively βr = d/x,
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which is the expression for β0, the minimum or zero abundance threshold. That is, the
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d
d d
*
evolutionarily singular strategy β   , ,,  as k   ; therefore the singular
x
x x
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strategy of all species is at the zero abundance threshold when competitive exclusion is
r
 0,
for 1  r  n.
(B.5)
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the outcome of a strictly deterministic competitive process, which under condition of
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convergence stability implies that all species will evolve to extinction.
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