The final problem with the L-V model is that consumers do not compete for prey, violating attribute 3 and condition 3. One solution to this problem is to use a ratio-dependent behavioral response which gives us an L-V model modified to include intraspecific competition amongst predators

This model satisfies all conditions for a biologically plausible model. Notice that, in this version of the L-V model, the attack rate declines with the density of the consumer and satisfies condition 3.

Population Dynamics with Ratio-dependent Satiation

The zero-growth isoclines for the L-V model with ratio-dependent satiation are: For the prey,

which is a parabola through the origin intercepting the prey axis at a / b = K (see figure).

Figure. (Left) Herbivore (H) and carnivore (C) phase space showing zero-growth isoclines for the L-V model with intraspecific competition amongst prey and ratio-dependent predator satiation (prey isocline = blue parabola, predator isocline = slanting red line). (Right) Herbivore and carnivore time series plots showing damped cycle to equilibrium.

For the predator,

which is a straight line through the origin with slope (cd - m) / Fm. vNotice that ratio-dependent functional responses alter the basic form of the predator isocline, from a vertical line to a right-slanting line. Also notice that ratio-dependent L-V models can be stable even when the community equilibrium is to the left of the peak in the prey isocline.