Ecological Archives E086-099-A1

Jason Matthiopoulos, John M. Halley, and Robert Moss. 2005. Socially induced Red Grouse population cycles need abrupt transitions between tolerance and aggression. Ecology 86:1883–1893.

Appendix A. Local stability analysis of the general model equation (Eq. 11).

We introduce the functions

(A.1)

with partial derivatives,

(A.2a)
   
(A.2b)
   
(A.2c)
   
.
(A.2d)

 

Specifying the partial derivatives to the extinction equilibrium  gives the characteristic equation . This gives the condition for population extinction in this system

.
(A.3)

 

We now shift our attention to the non-trivial equilibrium. Using Eq. 12 and the condition in Eq. 11 that either , we can write the Jacobian for this system at the non-trivial equilibrium

,
(A.4)

 

where . This gives the characteristic equation

.
(A.5)

 

In general, the non-trivial equilibrium will be stable when

.
(A.6)

 

Note that the discriminant is a quadratic in c with a positive coefficient for the quadratic term. Therefore setting the discriminant to zero and solving for c gives us the values between which the discriminant is negative,

(A.7)

 

when it is, condition (A.5) becomes

.
(A.8)

 

When the discriminant is positive, we distinguish two cases. In the first case, the quantity  is positive and condition (A.5) simplifies to

,
(A.9)

 

which is always true. In the second case, the quantity  is negative and condition (A.6) becomes

.
(A.10)

 

If the quantity on the r.h.s. of (A.10) is negative the condition is never true. Therefore, if

,
(A.11)

 

the equilibrium is unstable. If the r.h.s. of (A.10) is positive then the condition for stability becomes

.
(A.12)

 

If   then this is always true. If  then the equilibrium is stable when

.
(A.13)

 

This concludes the local stability analysis of the two equilibria.



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