Ecological Archives E086-055-A1

Colleen K. Kelly and Michael G. Bowler. 2005. A new application of storage dynamics: differential sensitivity, diffuse competition, and temporal niches. Ecology 86:1012–1022.

Appendix A. The model: biological assumptions and mathematical development. A pdf file of this appendix is also available.

 

In a model in which two species compete directly and the recruitment parameters  satisfy the condition

 = constant =   
(A.1)

the species with the smaller mean recruitment will have an age profile which fluctuates more about a smooth curve than that of the commoner.  This is because the recruitment fluctuations are equal in magnitude and opposite in sign

and so if species 1 has the smaller mean recruitment

  
(A.2)

This is illustrated in Fig. A1.

If the two species system has long term stability,

        
(A.3)

(where the bar denotes long-term averages) and if lifetimes are not dissimilar, the species with smaller mean recruitment is the rarer.

Condition (A.1) is realized in the two component lottery model (Chesson and Warner 1981), exactly for the case of equal d parameters.  From this, we devised a model to explain the results of Kelly et al. (2001), as follows.

Suppose that when a site that can support an adult becomes available through the death of an adult there is some probability that it is most suitable for species in class A (which might be a genus) and that such sites are at once colonised by species  which are members of class A.  We have in mind spatiotemporal partitioning into such site classes, as in Pacala and Tilman (1994).  Species  and  compete for such class A sites.  In the absence of differential recruitment caused by environmental fluctuations, the system would relax to a configuration in which one species is wholly dominant.  However, if  goes to zero on a quasiperiodic basis (as a result of environmental fluctuations) then  and  can coexist, provided  has sufficient competitive advantage to win over  during those periods in which young  are safe.  The necessary assumptions are not extreme – merely that class A sites are colonised by class A faster than by species in another class and that  can colonise vacant A sites in a time short in comparison with the typical length of an  recruitment crisis.

In the two component lottery model a location suitable for a new adult becomes available at a rate equal to the rate at which adults die and the probability of species  taking it is

where  is the product of a reproduction rate and a competitive factor (Chesson and Warner 1981).  The quantity  is the fraction of sites occupied by  and the site is at once taken by either  or .  Then

(A.4)

with analogous equations for  (a difference equation is given below).  Addition of the equations for  and  demonstrates at once that  +  = 1 at all times in a mature forest. 

The second term on the right hand side of Eq. A.4 is the recruitment .

The sum  +  is given by

For fluctuations on a time scale much less than , the fraction of sites occupied does not deviate much from  (although the recruitment fluctuations can be large) so

(A.5)

and this becomes exact for .  For the moment we treat the case , which is not at odds with the data of Kelly et al. ( 2001) and Kelly and Bowler (2002).  Then (A.4) becomes

(A.6)

Suppose that there are periods in which juveniles of  are taken out (by freezing or baking or being eaten).  The variable  is zero (OFF) in such periods,  and  leaps to .  At other times  is ON and suppose it has a value  at such times. Suppose further that   constant with value ; that is,  is taken as being relatively invulnerable to the climatic variations which directly or indirectly take out the young of .  The long term average of  is zero if the system is long term stable and the values of  in the denominator of (7) can be replaced by  for fluctuations on timescales much less than .  Then

and  for long term stability.

Then    

and

(A.7)

where  is the fraction of time during which  ON.  * and  are both restricted to the range 0 1.  Equation A.7 is the same as Eq. 2 in the main body of the text.  This equation only has an admissible solution provided that

and as  is decreased so .  If the competitive factor  does not sufficiently exceed  then species  drives species  to extinction.  Conversely  can never exceed .  For admissible solutions to Eq. A.7 the two species coexist.

 

DISCRETE FORMULATIONS

Equation 1 may be presented in discrete form as

 (where in  t  is an integer).

Equation A.4 may also be presented in discrete form, as

The equation was originally presented in this form, suitable for numerical simulations, by Chesson and Warner (1981).

The discrete equation equivalent to (A.4) is

with age profile

where  t  and  g  are integers.  For long term storage the difference and differential formulations are equivalent.

 

LITERATURE CITED

Chesson, P. L., and R. R. Warner. 1981. Environmental variability promotes coexistence in lottery competitive systems. American Naturalist 117:923–943.

Kelly, C. K., H. Banyard Smith, Y. M. Buckley, R. Carter, M. Franco, W. Johnson, T. Jones, B. May, R. Perez Ishiwara, A. Perez-Jimenez, A. Solis Magallanes, H. Steers, and C. Waterman. 2001. Investigations in commonness and rarity: a comparative analysis of co-occuring, congeneric Mexican trees. Ecology Letters 4:618–627.

Kelly, C. K., and M. G. Bowler. 2002. Coexistence and relative abundance in forest tree species. Nature 417:437–440.

Pacala, S. W., and D. Tilman. 1994. Limiting similarity in mechanistic and spatial models of plant competition in heterogeneous environments. American Naturalist 143:222–257.

 

 

   FIG. A1. A simple illustration of the content of the inequality (A.2).  The sum of recruitments R1 and R2 is constant.  When R1 is OFF, R2 extends from the top to the bottom of the diagram.  When R1 is ON, both species recruit at the same rate but in the figure, R1 is OFF half the time.  The result is that the mean  is three times  and that

On the y-axis, R2 is measured from the top of the figure and R1 from the bottom (left side of figure).  On the right side of the figure, the bars show relative mean values for R2 and R1 over the entire time period.




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