untitled

Few natural systems have been studied in sufficient detail to supply all the neces-

sary parameters that we observed in the Australian kangaroo and plant system.

Fortunately, it is often possible to estimate plausible parameter values from allometric

reasoning or historical data from a variety of sources, allowing us to make educated

guesses about system dynamics in a generic sense (Yodzis and Innes 1988; Turchin

2003). We shall demonstrate this approach for moose, wolves, and woody plants in

the boreal forests of North America. This is an important system to understand, because

it occurs across much of the extensive boreal forest biome spanning North America.

We use Turchin’s (2003) parameter estimates for the interactive model.

First, we recognize from the outset that this is fundamentally a tri-trophic system,

meaning that there are three trophic levels that interact in the food chain. The frame-

work we shall use simply expands the consumer–resource model outlined at the begin-

ning of the chapter to a third trophic level (P, for wolves) that feed on the second

trophic level (N, for moose), that themselves feed on self-regulating plant resources

(V). In all cases, we shall measure density in biomass (plants) or numerical abun-

dance (for animals) per square kilometer. Mathematically, we can represent this inter-

action with the following system of equations:

V(t) =rmax 1 −−N(t)

N(t) =eN(t) −d −P(t)

P(t) =EP(t) −D

where ais the maximum rate of plant consumption by a single moose, bis the plant

biomass at which plant consumption is half of the maximum, dis the rate of plant

consumption at which moose just sustain themselves, eis the efficiency of conver-

sion of food intake into new moose, and A, B, D, and Erepresent the same set of

parameters with respect to wolves.

We should note the similarity between the tri-trophic equations and the simpler

consumer–resource model outlined at the beginning of the chapter. Resources have

a self-regulating growth term, where the density-dependent term, 1 −V(t)/K, reduces

the growth rate proportionately with plant biomass. Plant consumption by moose

is balanced against this positive contributor to resource abundance, with plant con-

sumption expressed as the Michaelis–Menten form of the Type II functional response.

Moose have a per capita growth function that depends on their intake of plants. Balanced

against this is moose consumption by wolves. Finally, wolves have a per capita growth

function that depends on their intake of moose, balanced against a constant per capita

rate of mortality (presumably due to, for example, accidents, disease, and old age).

For a large part of the year moose browse on leaves and twigs of woody plants. Many

species of plants contribute to the food supply of moose (Belovsky 1988). However,

we know little about the web of ecological interactions within this plant guild, so

we shall consider woody browse during winter (the period of the year when food is

most often limiting to moose) as a single category. Edible biomass (measured in Mg / km^2 )

J

L

AN(t)

B+N(t)

G

I

d

dt

J

L

AN(t)

B+N(t)

G

I

J

L

aV(t)

b+V(t)

G

I

d

dt

aV(t)

b+V(t)

J

L

V(t)

K

G

I

d

dt

CONSUMER–RESOURCE DYNAMICS 207

12.6 Wolf–moose–woody plant dynamics in the boreal forest

12.6.1Models of the
tri-trophic system

12.6.2Parameter
estimation for the
wolf–moose–woody
plant system