A unified framework model of H/P ratio
Based on the Lotka-Volterra equations, biomass dynamics of primary
producers (P ) and herbivores (H ) are described as follows:
dP/dt = g(P)P − xP −f(P)PH,dH/dt = kf(P)PH − mH, (1)
where g (P ) is the per capita rate of biomass-specific
primary production and may be a function of P due to the
density-dependent growth, f (P ) is the per capita grazing
rate of herbivores and also may be function of P depending on the
functional response, x is the biomass-specific loss rate of
primary producers other than grazing loss, k is the conversion
efficiency of herbivores that is a fraction of ingested food converted
into the herbivore biomass, and m is the per capita mortality
rate of herbivores due to predation and other factors. If we assume bothg (P ) and f (P ) are constant values, eq. (1)
is basically the Lotka-Volterra model, while it is an expansion of the
Rosenzweig-MacArthur model if we assume the logistic growth forg (P ) and the Michaelis-Menten equation (Holling type II
functional responses) for f (P ). At the equilibrium state,
i.e. dP/dt =0 and dH/dt = 0, abundance of producers
(P ) and consumers (H ) can be represented as:
H = [g (P ) − x ](k/m )P. (2)
Thus, the relationship between H and P is not affected by
types of the functional response in herbivores (f (P )). If
we set g(P ) as the biomass-specific primary production rate at
the equilibrium state as in simple Lotka-Volterra equations (i.e.g (P ) = g ), then H /P ratio can be
expressed with log transformation as:
log(H/P) = log(k) + log(g − x) −
log(m). (3)
At the equilibrium state, (g − x )P is the amount of
primary production that herbivores consumed
(f (P )PH ). Thus, if we define β =f (P )PH /(Pg ) (= 1 − x/g ) as the
grazed fraction of primary production that herbivores consume, i.e.
edible fraction of the producers or inefficiency of the producers’
defensive traits, the H /P ratio can be expressed as:
log(H/P ) = log(k ) + log(β ) + log(g ) −
log(m ). (4)
This equation implies that the H/P biomass ratio on a log scale
is affected additively by the specific primary production rate
(log(g )), the grazable fraction of primary production
(log(β )), the conversion efficiency (log(k )), and
mortality rate of herbivores (log(m )). According to this
equation, communities having few amounts of carnivores (with smallm ) will exhibit high herbivore biomass relative to producer
biomass (H/P ) while those with low primary production (with smallg ) due to, for example, low light supply will show low H/Pratio. Increase in defended producers such as armored plants or decrease
in edible producers will decrease β by increasing the loss ratex due to the cost of defense), and will result in decreasingH /P ratio. Finally, when nutritional values of producers
decrease, the conversion efficiency of herbivores (k ) should be
low, which in turn decrease the H/P biomass ratio.