Download Numerical and functional response

What does the Lotka-Volterra model indicate about basic
characteristics of the predator-prey interaction?
In predator-prey models, the joint equilibrium point for
predators and prey is neutrally stable, but it is not an
attractor.
predator isocline (dR/dt=0), at this
prey population size the predator
population size (set by r/c) doesn’t
grow
prey isocline, at
this prey
predator #s
population size (set
by d/ac) the
predator population
doesn’t grow
prey #s
The isoclines define regions where prey grow or decline in
numbers, and similarly for predators. Above the predator
isocline, predator numbers can grow; below it predator
numbers decrease.
predator isocline
predators
prey
On this copy of the diagram, you can see that when predator
numbers are lower than the prey isocline, prey can increase;
when above, prey numbers decrease.
predators
prey isocline
prey
How do cycles change when either r or c is changed?
Remember:
Peq = r/c
Req = d/ac
The prey isocline is set by r/c. If r increases or c decreases,
the prey isocline rises. The equilibrium, based on prey
growth rate and predator capture efficiency, “can stand”
more predators. The predator population grows to this
isocline. This does not pull the populations back toward the
common stable point!
If the death rate of predators, d, increased and/or either (or
both) c or a decreased, the predator isocline would move
right, and the number of prey could increase up to the
isocline. This doesn’t move the populations back toward the
stable point either!
Changing the parameters “de-stabilizes” the interaction.
Changing the determining parameters (d, a, c, r) can, like
the effect of increasing growth during lags in logistic
growth, cause the amplitude of cycles in predators and
prey to increase (it also changes the period of the cycles),
and makes the extinction of one (or both) species more
likely.
Reviewing the output you should have seen from Populus…
1. “Standard cycles” – Predators: P0 =20, r2 = 0.1, C = 0.01
Prey: N0 = 20, r1 = 0.1, C = 0.01
Increasing the value of c, the capture (or encounter)
efficiency causes the amplitude of the prey cycle to increase,
but the portion of the cycle at low abundance to increase a
lot, and can result in extinction…
Change parameters for the prey to N0 = 20, r = 0.1, c = 0.05
prey
predator
The increase in c created a predator that overexploited its
prey. In this model the prey declines to very low number for
quite a while, until the predator population had reached very
small numbers for a time. Do predators overexploit their
prey in nature (and cause their extinction)?
(Think here of the Red Queen argument. Is this likely to
occur in co-evolved natural populations?
Extremely rarely in nature, but when it happens, it usually
involves the introduction (by chance or human activity) of
an exotic predator that had not evolved in conjunction with
the overexploited prey.
One example:
Humans, thinking they could increase the biomass of fish
that could be harvested from African rift lakes (Victoria,
Tanganyika, Malawi), introduced Nile perch into Lake
Victoria. Of >300 species of endemic cichlid fish in Lake
Victoria, more than 200 were driven
extinct by the perch. This represents
the largest mass extinction of the
modern era.
Local fisherman can now harvest
less.
The basic Lotka-Volterra model can generate extinction,
rather than just low values. Now let’s try increasing the value
of c from 0.01 (initially) to 0.1 (an increase of a factor of
10…
Back to predator – prey cycles…
There are good examples of cycles without the heavy
interference of the experimenter. They support the basic
Lotka-Volterra model of cycles. One good one involves the
interaction of the azuki bean weevil and its braconid wasp
parasite.
Here are the ‘steps’ in this cycle: (The weevils got a constant
ration of seeds.)
1. The weevils grow toward a carrying capacity determined by
the size of the ration.
2. The wasp population, once introduced, increases rapidly;
wasp parasitism becomes a major source of mortality for
weevils.
3. The weevil population declines.
4. Wasps are very (but not perfectly) efficient parasites, and
continue to reduce the population of weevils.
5. When the weevil population nears extinction, there is
insufficient food for the wasps; their population decreases
rapidly.
6. Now weevil larvae that were not parasitized can increase
and begin a new cycle.
When we compare theory to predator-prey interactions in
the field (N.B. not lab environments), what are the
differences?
One key difference is that there are refuges in nature. The
more efficient the predator, the more important refuges are.
Another important question: Do predator-prey cycles occur
in natural ecosystems?
Answer: Only very rarely.
However there are examples (though qualification is
necessary; these are the only examples I could think of).
Here are a couple:
1. Cycles in the populations of arctic lemmings and their
predators, the snowy owl. The necessary qualification is
very similar to one that will arise in explaining example 2.
2. Cycles in the populations of arctic hares and their
predator, the lynx. This is the example usually cited, due
to the long record of fur purchases (~1710 - ~1960) kept
by the Hudson Bay Company.
Here is the part of that more than 200 year record shown in
textbooks:
And here are the participants interacting:
Note the
scales on
the two Y
axes.
The period of the cycle is ~10 years.
We expect that if we were to plot relative densities of
predator and prey on Y and X axes respectively, with
different points representing different times in a cycle, theory
says that the points should move counterclockwise. However,
when Gilpin did this, he got:
Lynx
If the cycles are running backwards, do hares eat lynx?
Not likely!
Gilpin (and others) had explanative suggestions:
1. The hares might serve as reservoirs of diseases transmitted
to lynx.
2. Trappers may have waited until hare numbers were high
before going out to trap either species. That would make
peaks appear to be nearly simultaneous (as they appear) or
even show lynx ‘leading’ hares. Recent scientific counts
suggest that this trapping bias may have occurred.
3. Krebs et al. (1995) showed that food quality for the hares
may be a key factor in explaining their cycles, and thus that
lynx are simply following along. Here is how it works:
A. Hare number increases.
B. Larger number means lower food abundance and quality
left for hares to eat.
C. Hares lose weight, juvenile mortality increases, and the
now extended foraging time necessary to find food
increases exposure to predators. Hare numbers decrease.
D. The plant community begins to recover, but the first plants
to increase in abundance are toxic to hares. Toxins have
been identified, but they do not appear until after the hare
population declines.
E. The plant community completes recovery while the hares
remain at low density.
F. Then, with non-toxic plant biomass increased, the hare
population once more increases.
So, what the the toxic plants can do is slow the recovery of
the hare population. Further, the lynx can also slow recovery
during harsh winters.
However, most of the time, the evidence suggests that the
lynx does not ‘cause’ hare cycles, but the size of the lynx
population simply ‘tracks’ the abundance of hares.
What this shows is a modern view of interactive effects in the
form of cascades. Modern ecology frequently also asks the
question of whether controlling influences work ‘bottomup’, i.e. the food is the ultimate control, or ‘top-down’,
which would assign basic control to the lynx.
It is worth noting, though they will not be considered here,
that there are alternatives to the Lotka-Volterra model, e.g.
the Nicholson-Bailey model for parasite – host interaction.
You have or will, depending on how far in advance (or after
your scheduled lab) you do lab exercises, explore the
Nicholson – Bailey model this week.
One of the problems that led to the development of
alternative models is the lack of any aspect of predator
behaviour. The model, with fixed c, assumes that the
predator does not respond behaviourally to changes in the
size and availability of the prey population.
Real predators respond in two ways to changes in prey
abundance:
1. A numerical response, and
2. A functional response
Numerical response
Numerical responses occur in many predator populations.
Here is an approximate curve describing the type of response
C.S. Holling observed using shrews as predators:
Number of
predators
Number of prey
Another kind of numerical response:
In short-eared owls, there is variation in clutch size
depending on the abundance of prey. In the field, clutch sizes
range from 4 (in ‘poor’ years) to 7 (in years of high prey
abundance. In addition to a response in reproduction, there
can be immigration to areas of high prey abundance. In
Alaska…
Functional responses are more complicated. These responses
are the result of the hunting behaviour of individual
predators.
The functional response is determined by the pattern of
consumption of prey as the abundance of those prey
increases. Specifically, it is the number of prey eaten per
predator, as prey numbers increase.
There are 3 patterns into which functional responses are
usually grouped…
TYPE I FUNCTINAL REPSONSE
This is the response of a filter feeding predator, e.g. a baleen
whale. As prey (krill) abundance increases, the swimming of
the whale through the higher density of krill increases the
number of prey eaten in direct proportion to prey abundance.
However, at some abundance of prey, the filter is filled, and
more prey can’t be taken. Here’s what the relationship looks
like…
TYPE II FUNCTIONAL RESPONSE
This type of functional response was observed by Holling in
studies of sawflies. It is common among invertebrate
predators. The shape of the curve is determined by the
characteristics of the predator in searching for, handling and
eating prey. As prey become more numerous, the predator
has to spend a greater
fraction of its time in
handling and eating,
leaving less time for
searching. At some
density, a balance is
reached, and more
prey can’t be taken.
TYPE III FUNCTIONAL RESPONSE
This response is seen in higher vertebrates and some higher
invertebrates. The curve shows that the predator is ‘switching’
from some other prey when this one is scarce to a high capture
rate for this prey when it is more common. Note that at high
prey density this looks very much like the type II response.
What might cause a type III response?
1. Habitat heterogeneity, with refuges available. Prey achieve
safety in refuges, but, as population size increases, there
are not enough refuges for all prey.
2. Predators form ‘search images’ that aid in prey capture. At
low prey abundance, the search images may not be
reinforced. Capture is less efficient.
3. Finding sufficient food when this prey is scarce may force
the predator to ‘switch’ to more abundant alternatives.
There is lots of experimental evidence of this switching
behaviour in predators. The text shows you an invertebrate
example…
Your text also shows you a view of these functional responses
that explains the potential of the predator population to limit
(or regulate) the size of the prey population…
Note the different
y-axes of the two
graphs: number
and proportion of
prey consumed.
To expand on the theoretical view of functional responses,
here’s evidence of a real functional response (Pentatomid
‘bug’ attacking a sawfly larval stage) plotted as prey/predator
against prey density. The curve has a clear ‘hump’.
Type I predators consume a constant proportion of the prey
population (until satiation is reached; at abundances above
that they take a declining proportion of the total prey).
Type II predators take a declining proportion of the prey as
prey abundance increases. That clearly cannot limit (or
regulate) the prey.
However, type III predators can, if they are efficient enough,
limit the prey population. It is at an intermediate prey
abundance that the proportion of prey captured is maximum.
Only type III has this maximum.
A type III predator functional response can increase the
likelihood of predator and prey persistence. What tends to
reduce the amplitude of the oscillations predicted by theory
(and increases the ‘stability’of the system)?
1. Predator inefficiency – a lower value for c increases the
equilibrium population size for both prey and (therefore) the
joint equilibrium for predators.
2. Density-dependent limitation of either species by factors
other than their interaction.
3. Alternative food sources for the predator, so that switching
can occur.
4. Refuges for the prey, at least when they are at low density.
5. Reduced time lags in predator response to prey density.
Your text demonstrates that, while the simple model has a
single equilibrium point, in theory, at least, multiple
equilibrium points in a predator-prey system are possible.
How?
Look at two curves shown on the next slide (and text figure
18.17): a) a curve showing per capita recruitment rate for prey
as a function of prey density, and b) a curve for predation rate
(alternatively you can think of this as probability of predation)
per individual prey, also as a function of prey density.
If this is a ‘switching’ predator, then the predation rate curve
has some sort of ‘hump’ in the middle. As a result, there can
be multiple equilibria…
A and C are stable
equilibrium points, but
B is an unstable
equilibrium (i.e. not
an attractor).
This general pattern
would occur if the
recruitment rate curve
were the straight line
we saw for simple
logistic growth.
Your text, in Figure 18.18, shows you how different
persistence of predator and prey can occur as the efficiency
of the predator changes, and as the density of the predator
population relative to the abundance of its prey changes.
Make sure you understand what’s different to produce each
of the results.
How are the theory and models of predator – prey interaction
applied?
Insect predator – prey systems are among the most
intensively studied.
The reason for this intensive study is that many insect
species (especially exotic species) cause significant (both
ecological and economic) damage to agricultural crops.
Ecologists believe that it should be possible to find and use
predators of these crop pests to limit and control the damage
to crops.
This belief is the basis of biological control. You’ve already
seen examples of the success of biological control.
An example of a multi-level predator-prey system:
Oaks – leaf-chewing insects – bird predators of the insects –
the birds provide a ‘biological control’
The experiment:
A control group – oak trees unprotected from
herbivores and freely exposed to the birds
An experimental group – branches of trees or saplings
protected from birds by netting, but still exposed to
the leaf-chewing insects.
Experimental group 2 – branches or saplings treated
with insecticide
The results:
After 2 years leaf damage amounted to 34% inside bird
exclosures, where insect densities had increased to almost 2x
controls; 24% on controls, but only 9% where branches had
been treated with insecticide.
Effects of intense herbivory were evident in the third year in
reduced production, as well as subsequent growth of oak
saplings.
Another example:
Think back to the example of the mutualism between ants
and acacias. The stinging ants protected the tree from
herbivorous insects that might damage the tree or reduce its
growth.
Aggressive ants protect other plants, as well. In Brazil,
Tachigalia trees harbor colonies of the stinging ant
Pseudomyrmex concolor in the hollow petioles of their
leaves.
When ants were experimentally removed from seedlings
and saplings, the number of leaf-chewing herbivores
increased fourfold, removal of leaf biomass increased
tenfold, individual leaf longevity was cut by half, and
apical growth was reduced by one-third.
Biological control has not always been successful. Sometimes
it takes many years (and a lot of damage) before a suitable
control agent is found and is able to exert control.
Successful control of pest insects by predators (other insects)
has been estimated to occur in 20 – 40% of cases. Successful
control of plant pests occurs at a similar frequency.
It is important to remember that almost all these control
programs leave non-native species in the environment.
Damage may be caused by these non-native species. Here is
a case study that gives ecologists pause about importing
biological control agents…
The Eurasian thistle (a non-native plant) was accidentally
introduced to North America; it is now widespread over
Canada and the U.S. A flowerhead weevil (non-native) was
introduced in 1968 to control the Eurasian thistle.
Problem: the weevil has expanded its geographic range and
is severely damaging native thistles in addition to the target
species. It has a much broader diet than was believed at the
time of introduction. Scientists didn’t know its diet breadth
when many species of thistle were available. Detailed
knowledge of food preferences are important before
introducing any control agent.
This insect is now a real threat to native biodiversity.
The weevil has been successful because it has strong
numerical and functional responses to thistles. Those
responses are what is needed to be a good biological control
agent. Such species are more likely to control populations of
their prey.
But the message of the flowerhead weevil is that without
adequate knowledge of the proposed agent, there may be
important detrimental effects on native species.
The evolutionary race between predators and their prey has
implications for the success of both groups. It provides
another example of the potential importance of group
selection.
Imagine an interaction in which predator-prey interactions
occur in small, isolated groups. Voracious predators drive their
prey to extinction in some groups. Those are the most effective
predators; Darwinian (individual) selection has occurred
producing those effective behaviours. What then happens to
the predator?
However, groups with somewhat less effective predators may
be the ones in which both predators and prey persist.
The most voracious predator wins locally, but loses globally
to a more ‘prudent’ predator.
It is evolution that also produces those remarkable defensive
adaptations in prey species. A full-blown consideration of the
interaction of predation and the genetic systems/structure of
the interacting populations is prohibitively complex.
Diagrammatically:
Krebs, C. J., S. Boutin, R. Boonstra, A. R. E. Sinclair,
J. N. M. Smith, M. R. T. Dale, K. Martin, and R. Turkington.
1995. Impact of food and predation on the snowshoe hare
cycle. Science 269:1112-1115.
Marquis, R. J., and C. J. Whelan. 1994. Insectivorous birds
increase growth of white oak through consumption of
leaf-chewing insects. Ecology 75:2007-2014.
Tayler, R.J. 1884. Predation. Chapman and Hall, N.Y., N.Y.