Ecological Archives M075-021-A2

Nicolas Mouquet, Jeremy A. Thomas, Graham W. Elmes, Ralph T. Clarke, and Michael E. Hochberg. 2005. Population dynamics and conservation of a specialized predator: a case study of Maculinea arion. Ecological Monographs 75:525–542.

Appendix B. Condition for simple dynamics in nested model of density dependence.

In models with two levels of density dependence, the shape of the curve Nt + 1 vs. Nt is a function of the shape of the curves defining the two kinds of density dependence (Rodrigez (1988) . In M. arion, we isolated these two episodes by considering:

img
(B.1)

 

With nt for the number of caterpillars that fall from plants and Nt for the number of adult butterflies. In this model, the first episode of density dependence is contest competition (nt + 1 vs. Nt approaches an asymptote at nt + 1 = h1, Fig. B.1a and d) and the second is scramble competition (Nt + 1 vs. nt is unimodal with a max at Nt = h2, Fig. B.1b and e). In this case, Rodrigez (1988) has shown that the resulting curve of Nt + 1 vs.Nt will be unimodal and decreases approaching an asymptote higher than zero if h1>h2 (Fig. B.1c) or will be an asymptote if h1< h2 (Fig. B.1f).

 

FigB1
 
   FIG. B1. Density dependence curves obtained for two different ratios of ant nests to plants: case 1: P = 3500 right arrow A /P = 2.85 and case 2: P = 35 right arrow A /P = 282.71. The figures show the curves obtained with the equations of nt + 1 as a function of Nt, and Nt + 1 as a function of nt + 1 as defined in Eq. A2.1 (a, b, d, e) and of Nt + 1 as a function of Nt as defined in Eq. 9 (c, f). Other parameters as for scenario #1 (Table 2 in the text).

 

The condition h1< h2 means that the first episode of density dependence is strong enough for the population to remain at a sufficiently low density so as not be affected by the second episode of density dependence.

We can find h1 and h2 by:

img
(B.2)

 

and solving img , gives (considering img ):

img
(B.3)

                                                                              

Note that we have assumed img , which gives img  that is true for the high values of xA found in the field. This assumption simplifies the algebra and does not change significantly our results.

In the case where the curve of Nt + 1 vs. Nt is an asymptote, the equilibrium will be reached monotonically. This case corresponds to h1< h2 that is:

img
(B.4)

 

LITERATURE CITED

Rodrigez, D. 1988. Models of growth with density regulation in more than one life stage. Theoretical Population Biology 34:93–117.



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