5 Nutrients, Algae and Herbivoresthe Paradox of Enrichment Revisited
Rosenzweig's results might more reasonably
have been used to prompt questions such as
the following: What are the critical values of
enrichment? How does the time to extinction
of a system vary with the degree of enrichment? How do critical levels of enrichment
and time to extinction vary with other
parameters? Why does nature not collapse?
McAllister, LeBrasseur and Parsons
(1972).
Lotka (1925) and Volterra (1926) independently developed simple mathematical descriptions of the interaction between predators and their prey.
These theories later became the ancestors of a whole family of what is
collectively known as Lotka-Volterra models. Elementary ecology text
books usually concentrate on the simplest kind of Lotka-Volterra model,
where the prey have an unbounded capacity for exponential growth in the
absence of predators, and where the predators have an unbounded capacity
for killing prey. As noted by May (1975), such a system has certain pathological dynamic properties which are equivalent to the neutral stability of a
frictionless pendulum: the system oscillates forever with an amplitude that
is determined solely by the initial conditions.
In more realistic models where either prey growth is bounded by a finite
carrying capacity, or where predator capacity for capturing prey saturates
at a finite level of prey abundance, the neutral stability property is lost (that
is, the neutral stability is structurally unstable; May 1975). For two-dimensional Lotka-Volterra systems with one predator and one prey species, the
powerful Poincare-Bendixson theorem can be used to prove that essentially
all such systems will either settle down to a stable steady state, or end up in
a limit cycle with both the oscillation period and amplitude being independent of the initial conditions (May 1975).
That two-dimensional prey-predator systems exhibit only stable equilibrium points or stable cycles is related to the fact that a closed orbit in the
plane has a defined inside and outside. When stepping up to three or more
dimensions, one can no longer distinguish the inside from the outside of a
closed orbit, which again implies that no equivalent of the PoincareBendixson theorem exists in higher than two dimensions. Three-dimensional prey-predator systems can therefore display a full and rich dynamic
complexity, including strange attractors and chaos (May 1981).
Rosenzweig's results might more reasonably
have been used to prompt questions such as
the following: What are the critical values of
enrichment? How does the time to extinction
of a system vary with the degree of enrichment? How do critical levels of enrichment
and time to extinction vary with other
parameters? Why does nature not collapse?
McAllister, LeBrasseur and Parsons
(1972).
Lotka (1925) and Volterra (1926) independently developed simple mathematical descriptions of the interaction between predators and their prey.
These theories later became the ancestors of a whole family of what is
collectively known as Lotka-Volterra models. Elementary ecology text
books usually concentrate on the simplest kind of Lotka-Volterra model,
where the prey have an unbounded capacity for exponential growth in the
absence of predators, and where the predators have an unbounded capacity
for killing prey. As noted by May (1975), such a system has certain pathological dynamic properties which are equivalent to the neutral stability of a
frictionless pendulum: the system oscillates forever with an amplitude that
is determined solely by the initial conditions.
In more realistic models where either prey growth is bounded by a finite
carrying capacity, or where predator capacity for capturing prey saturates
at a finite level of prey abundance, the neutral stability property is lost (that
is, the neutral stability is structurally unstable; May 1975). For two-dimensional Lotka-Volterra systems with one predator and one prey species, the
powerful Poincare-Bendixson theorem can be used to prove that essentially
all such systems will either settle down to a stable steady state, or end up in
a limit cycle with both the oscillation period and amplitude being independent of the initial conditions (May 1975).
That two-dimensional prey-predator systems exhibit only stable equilibrium points or stable cycles is related to the fact that a closed orbit in the
plane has a defined inside and outside. When stepping up to three or more
dimensions, one can no longer distinguish the inside from the outside of a
closed orbit, which again implies that no equivalent of the PoincareBendixson theorem exists in higher than two dimensions. Three-dimensional prey-predator systems can therefore display a full and rich dynamic
complexity, including strange attractors and chaos (May 1981).