262
M A R K S . B O Y C E
of most simple predator–prey models (i.e., they have no age or sex structure)
and the model is deterministic, whereas fundamentally all ecological systems
are inherently stochastic (Maynard Smith 1974).
Rather than dwelling further on the Lotka–Volterra model, I believe that
we can dismiss it as an early effort that gave useful insight. Not only do the
neutrally stable oscillations appear peculiar and inconsistent with ecological
intuition, but the model is structurally unstable, meaning that small variations
in the model destroy the neutrally stable oscillations, leading to convergence to
equilibrium, divergence to extinction, or even stable limit cycles (EdelsteinKeshet 1988).
Volterra was aware of certain limitations to his predator–prey model and
later proposed a form in which prey were limited by density dependence:
dV/dt = V [b – (b/K )V – a × P ]
(8.8)
dP/dt = P(c × V – d )
(8.9)
Now in the absence of predators the prey population converges asymptotically on a carrying capacity, K . But the model still suffers from the assumption of prey being eaten proportionally to the product of the two population
sizes; similarly, the numerical response remains linear. However, instead of
neutrally stable cycles, the populations now oscillate while converging on an
equilibrium number of predator and prey (Volterra 1931).
Kolmogorov’s equations
More useful than the Lotka–Volterra model is the more general analysis
by Kolmogorov (1936), who studied predator–prey models of the general
form
dV/dt = V × f (V, P)
(8.10)
dP/dt = P × g (V, P)
(8.11)
where we assume that the functions f and g have several properties that are generally consistent with the ecology of predator–prey interactions. These include
M A R K S . B O Y C E
of most simple predator–prey models (i.e., they have no age or sex structure)
and the model is deterministic, whereas fundamentally all ecological systems
are inherently stochastic (Maynard Smith 1974).
Rather than dwelling further on the Lotka–Volterra model, I believe that
we can dismiss it as an early effort that gave useful insight. Not only do the
neutrally stable oscillations appear peculiar and inconsistent with ecological
intuition, but the model is structurally unstable, meaning that small variations
in the model destroy the neutrally stable oscillations, leading to convergence to
equilibrium, divergence to extinction, or even stable limit cycles (EdelsteinKeshet 1988).
Volterra was aware of certain limitations to his predator–prey model and
later proposed a form in which prey were limited by density dependence:
dV/dt = V [b – (b/K )V – a × P ]
(8.8)
dP/dt = P(c × V – d )
(8.9)
Now in the absence of predators the prey population converges asymptotically on a carrying capacity, K . But the model still suffers from the assumption of prey being eaten proportionally to the product of the two population
sizes; similarly, the numerical response remains linear. However, instead of
neutrally stable cycles, the populations now oscillate while converging on an
equilibrium number of predator and prey (Volterra 1931).
Kolmogorov’s equations
More useful than the Lotka–Volterra model is the more general analysis
by Kolmogorov (1936), who studied predator–prey models of the general
form
dV/dt = V × f (V, P)
(8.10)
dP/dt = P × g (V, P)
(8.11)
where we assume that the functions f and g have several properties that are generally consistent with the ecology of predator–prey interactions. These include