- Predation occurs when one species consumes another.
- Volterra (1926) proposed a model of predation derived
from the exponential growth equation and the law of mass action.
Volterra wrote "The first case I have considered is that of two associated
species, of which one, finding sufficient food in its environment, would
multiply indefinitely when left to itself, while the other would perish for lack
of nourishment if left alone; but the second feeds upon the first, and so the
two species can coexist together. The proportional rate of increase of the
eaten species diminishes as the number of individuals of the eating species
increases, while the augmentation of the eating species increases with the
increase of the number of the eaten species." (Nature 118: 558-60).
- The equations Volterra proposed for this situation were:
- dP/dt =rP - aPC
- dC/dt = faPC - dC
- where P is the prey which has growth rate r, C
is the consumer (predator) which has death rate d, a is the
fraction of the prey population eaten by a single predator, and f is the
fraction of consumed prey that is converted into predator offspring.
- Volterra found that "the numbers of individuals of the two species are
periodic functions of the time, with equal periods but with different phases, so
that each species goes through a cycle relative to the other during a period, a
process which may be called the 'fluctuation of the two species.' "
In the graph, the yellow line is the predator and the blue line is the
- This model is analyzed by examining the conditions under which the
populations of the predator and prey grow or decline. The zero growth isoclines
or nullclines for the predator and prey are defined as follows:
- Predator nullcline: dC/dt = 0
- 0 = faCP - dC
- faCP = dC
- P = d/(fa)
- This is the prey population density at which predator population does not
grow. If the prey are more abundant than this, the predator grows (dC/dt >
- Prey nullcline: dP/dt = 0
- 0 = rP - aCP
- aCP = rP
- C = r/a
- This is the predator population density at which the prey population does
not grow. If the predators are less abundant than this, the prey grows (dP/dt
These relationships can be graphed in a phase plot in which the predator
population (vertical axis) is plotted against the prey population (horizontal
axis). The yellow line (horizontal)
represents the prey nullcline, above which the predator population is so
large as to cause the prey to decline, and below which the predator population
is small enough the the prey populations grow. The blue line (vertical)
represents the predator nullcline, to the right of which the prey
population is so large as to cause the predator population to grow, and to the
left of which the prey are so rare that the predator starves. This is
equivalent to Fig 10.16 in Krohne:
The red arrows on the plot represent the expected trajectories of the
- When predator populations are above the prey nullcline,
the prey population declines (arrows all point to the left), and when predator
populations are below the prey nullcline, the prey population grows (arrows all
point to the right).
- When the prey populations are above the predator
nullcline (the predator has plenty of food), the predator population grows
(arrows all point upward), and
when prey populations are below the predator nullcline (the predator has too
little food), the predator population declines (arrows all point downward).
- These patterns of growth cause the fluctuations in population sizes seen in
the plots of numbers versus time. On the phase plot, the fluctuations appear as
egg shape orbits:
Each orbit represents a different starting population size. The
outermost orbit corresponds to the numbers versus time graph at the top of this
outline, and the innermost orbit corresponds to the plot below:
- Volterra used these results to explain the change in fish populations that
resulted in the Adriatic Sea (between Italy and Yugoslavia/Serbia/Albania)
during and after the First World War, when fishing stopped because of the
hostilities. "A complete closure of the fishery was a form of 'protection'
under which the voracious fishes were much the better and prospered accordingly,
but the ordinary food-fishes, on which these are accustomed to prey, were worse
off than before."
- Since Volterra's introduction of this model, other authors have modified it,
adding for instance logistic prey populations, and predators that become
satiated with food . An example of logistic prey is Crawley's plant-herbivore
- dP/dt = rP(1-P/K) - aCP
- dC/dt = bCP - dC
- The logistic prey population tends to stabilize the model, so fluctuations
in abundance are not so extreme, and tend to dissipate over time:
- Nicholson and Bailey (1935) examined the interactions
between insect hosts and their parasitoids and found the opposite trend:
expanding oscillations in abundance .
- In these models, predator-prey interactions result in fluctuations
in the populations of both predators and prey. Such fluctuations have been
found in laboratory populations of predators and prey including Gause's
and Didinium experiments,
Huffaker's experiments with mites (Figure 10.23 in Krohne),
the Hudson Bay Company fur records,
field and laboratory populations of insects, and other cases. There
are many examples of populations of prey that are kept in check by predators.
- A current area of research involves determining the importance of
top-down (predator control) versus bottom-up (resource limitation
and competitive control) of populations in nature (p 426 in Krohne).