SUPPLEMENTARY INFORMATION (THEORY)

1."Period 6" as a benchmark for consumer-resource cycles

Consider the fairly general H-P model that includes the possibility of surviving (invulnerable) adult hosts, and an attack rate that depends on host and/or parasitoid density.

(1a)

(1b)

Equilibria are solutions of:

(2a)

(2b)

Stability matrix elements are[1]:

(3a)

(3b)

(3c)

(3d)

Define .(4)

These definitions are chosen so that “typically” we expect both  and  to be positive, as host survival should decrease as parasitoids increase, and increase as hosts increase.

Equations (5)-(8) can then be written in the less intimidating form:

(5a)

(5b)

(5c)

(5d)

The characteristic equation (with  being the eigenvalue as in Gurney and Nisbet, 1998, chapter 3) is

(6)

with , and (7)

The transition to oscillations occurs when (Gurney and Nisbet 1998, page 61), so on the stability boundary, the characteristic equation is

(8)

where .(9)

The period, P, of cycles on the stability boundary is given by

,(10)

with[2]. From equation (10), it can be shown by routine algebra:

(a)Thus the period is always greater than 6 if , which is guaranteed if and are positive.

(b) Period increases as increasesand approaches infinity as .

(c) Period approaches 6, as if and only if and . This result holds even with non-zero values of SP.

(d)Period decreases with increasing .

2. Period as a liming period for large

Result (c) from the preceding section generalizes to systems where the resource and consumer have juvenile development times with (integer) values and respectively. The model defined by equations (1) then takes the form:

(10a)

(10b)

With the same restrictions ( and ) as before, as , the period on the stability boundary approaches .

To prove this, assume as before, a non-trivial equilibrium state. Define small perturbations from equilibrium by

;.(11)

Then, the linearized dynamics near equilibrium are given by

(12a)

(12b)

The characteristic equation (with  being the eigenvalue as in Gurney and Nisbet 1998, chapter 3) is now

The characteristic equation is:

,(13)

which, in view of our assumptions that , and , reduces to:

(14)

On the local stability boundary, , so we can write

; (15)

and hence, taking the modulus of both sides of equation (14),

.(16)

The left-hand side of equation (16) does not involve ; thus as , , with the product remaining finite.

Now take the imaginary part of both sides of equation (14) in the limiting case , and . The result is

(17)

implying

(18)

We are interested in multi-generational cycles, certainly with periods greater than twice either development time, so we are interested in situations where . Thus equation (19) implies

,(19)

i.e.(20)

and period = .(21)

[1] Subscripts denote partial derivatives. For example

[2] Note that unless this inequality holds, A1.<-2, and the system is unstable