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)PxPf(P)PH,dH/dt = kf(P)PHmH, (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(gx) − log(m). (3)
At the equilibrium state, (gx )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.