| Lecture 27: Predation
Models
Reading: Economy of Nature, pp. 485-493.
Dynamics of Predator-Prey Systems Lotka-Volterra Model
Reductions in prey number by a predator species are due to the frequency of predator-prey encounters, and the encounter frequency is a function of both C and N. Predator attack efficiency must also be a factor in the rate of prey removals by predators where a’ is the rate of successful predator attacks (assumed to be constant). If predator killing and removal of prey from the prey population is a’CN, then a prey population subject to predation will have a growth rate that is reduced by the rate of prey consumption by predators.
Increases in predator population can only occur in the presence of prey, and the rate of prey consumption by predators is a’CN. The predator efficiency of converting prey into predator offspring is f, so the predator birth rate is fa’CN.
Zero Growth Isoclines for Predator and Prey Populations Predator and prey zero isoclines are shown as constants (straight lines) on separate predator-prey density plots with population growth vectors (after Begon, Harper and Townsend, 1990, p 339, Fig. 10.2).
For prey:
For predators (consumers):
The predator and prey zero growth isoclines together define quadrants on a plot of predator and prey densities. Addition of predator and prey growth vectors in each quadrant indicates the predicted change in predator and prey densities from any starting combination of densities (after Begon, Harper and Townsend, 1990, p 339, Fig. 10.2).
Predator (consumer) population density oscillates with a time lag behind prey population density oscillations. The predicted outcome is called neutral stability (after Begon, Harper and Townsend, 1990, p 339, Fig. 10.2).
Under conditions of neutral stability, both predator and prey populations follow the same cycles indefinitely until external factors cause a shift in the population density of one population, then a new cycle begins. However, environmental conditions do change and this could cause erratic patterns of population density change over time. The overall pattern may appear to be neither stable nor cyclical. Some predator-prey interactions do exhibit neutral stable oscillation as predicted by this simple Lotka-Volterra model. Population cycles predicted by this model are observed in natural populations of lynx and snowshoe hare in Canada (based on fur pelt records of the Hudson’s Bay Company), and in laboratory populations of bean beetles and their parasitoid wasp predators (Ricklefs, 1996, pp 453 and 455, Fig. 20.5 and 20.6).
Oscillation does not characterize every predator-prey system, and oscillation may not be due to the predator-prey interaction alone. For example, the cycles in lynx and snowshoe hare densities may result from cycles in the food supply of the hares that are related or unrelated to the herbivore-plant interaction. Lynx population densities may simply be tracking the temporal variation in hare densities. The Lotka-Volterra models do not predict constant population densities, but some predator and prey populations are constant over time, showing no oscillation.
Making the Models More Realistic Self-limitation (intraspecific competition) can be introduced to the predator-prey models to make the interactions more realistic. For predators (consumers): The density (or number) of prey required to just maintain their (predator) population (zero growth condition) need not be constant at all predator densities. Predator zero growth isoclines could take four progressively more realistic forms. The vertical zero growth isocline (A) is the simple Lotka-Volterra model in which a given density of prey will support any density of predators. Large predator populations may require larger prey populations (B) in a linear relationship. However, at high predator densities, competition between predators for prey (mutual interference) could reduce consumption rates so the predator zero growth isocline would be non-linear at high predator densities (C). Finally, high predator densities could be limited by resources other than prey, so there would be a maximum sustainable predator density (carrying capacity) at which point the zero growth isocline becomes independent of prey densities (D) (after Begon, Harper and Townsend, 1990, p 345, Fig. 10.6).
Predator population zero growth isocline
with self-limitation, and well defined carrying capacity (
The location of the predator zero growth isocline and the densities at which the curve changes shape will vary with each predator species, the particulars of a given predator-prey interaction, and the environment at a given time and place.
Similar to the realistic modifications
we can make to the consumer zero growth isocline, the prey are likely
to experience little or no intraspecific competition at low prey densities
but intense intraspecific competition (and self-limitation) at high prey
densities. Consequently, prey population growth would be zero at high
prey densities (prey carrying capacity) in the absence of any predators
(when
The interaction between predator and prey
populations with self-limitation will yield a range of outcomes depending
on the relative positions of the predator and prey zero growth isoclines.
Predator populations with little self-limitation yield the least stable
interactions (
|
without predators
without prey
so: 
or, 
so: 
or, 
).
Consequently, predator population growth would be zero even in the presence
of abundant prey once the predator population grows to carrying capacity
density (when 
) (after
Begon, Harper and Townsend, 1996, p 382, Fig. 10.7).
) (after Begon, Harper
and Townsend, 1996, p 382, Fig. 10.7).
) and predator
populations with the most self-limitation yield the most stable interactions
(
) (after Begon, Harper
and Townsend, 1990, p 347, Fig. 10.7).
