Next Change the various values of r, a, f, and q and see what
happens. Make a note each time of the changes that you have made
and of the effects observed When you do this, change each variable
one at a time double it and half it.
Slide 3
An increase in r leads to an increase in the P in order to
maintain an equilibrium N Rate of change in prey population growth
reflected by similar rate of change in predator population growth.
Note Lags the bigger the r the shorter the lag is this reasonable?
The bigger the r the greater the amplitude of fluctuations, the
larger the predator population and the greater the instabilities in
the model
Slide 4
r = 0.05 r = 0.25 r = 0.30
Slide 5
An increase in q leads to an increase in the N required to
maintain an equilibrium P An increase in q leads to an increase in
the amplitude of predator and prey fluctuations. Time lags become
less: prey population larger
Slide 6
q = 0.05 q = 0.10 q = 0.15
Slide 7
An increase in a leads to a reduction in the N required to
maintain an equilibrium P AND a reduction in the P required to
maintain an equilibrium N An increase in a leads to a reduction in
numbers of predators and prey time lags between peaks stay more or
less same: both large and small changes lead to instabilities
Slide 8
a = 0.005 a = 0.010 a = 0.020
Slide 9
An increase in f leads to a reduction in the N required to
maintain an equilibrium P An increase in f leads to a reduction in
prey numbers: both large and small changes lead to increased
magnitude of fluctuations in both predators and prey
Slide 10
f = 0.004 f = 0.008 f = 0.016
Slide 11
Changes to neither N 0 nor P 0 have any effect on the values of
P or N that are required to maintain equilibrium populations of
prey or predator alike.
Slide 12
N 0 = 500 N 0 = 2000 N 0 = 1000
Slide 13
P 0 = 10 P 0 = 40 P 0 = 20
Slide 14
Adding Intra-Specific Competition - aggregation dN dt = r.N t K
- N t K ( ) Logistic Equation dN = N t+1 - N t N t+1 = N t + r.N t
K - N t K ( ) For Prey - Incorporate losses due to predation: a.P
t.N t ( ) N t+1 = N t + r.N t K N - N t KNKN - a.P t.N t P t+1 = P
t + a.f.P t.N t ( ) K p - P t KpKp - q.P t For Predator -
Incorporate losses due to starvation: q.P t
Slide 15
Set up a spreadsheet with the following reference points
Project prey and predator populations into the future for 100 time
units both displaying intra-specific competition ( ) N t+1 = N t +
r.N t K N - N t KNKN - a.P t.N t P t+1 = P t + a.f.P t.N t ( ) K p
- P t KpKp - q.P t Plot line graphs of predator and prey numbers
over time Plot X-Y graphs of predator numbers against prey numbers
BUT.. MUST constrain BOTH populations so that if numbers drop to
zero, they remain at zero. Use an IF argument
Slide 16
Plot line graphs of predator and prey numbers over time Plot
X-Y graphs of predator numbers against prey numbers Change the
values of r, a, f, N 0, P 0, K N and K P as you did previously -
and note what happens to the relationship between predators and
prey. Use the same changes as previously and draw comparisons
between the models
Slide 17
Equilibrium Solution - Prey What is the population size of the
predators that induces no change in the prey population size? dN dt
= 0 ( ) N t+1 = N t + r.N t K N - N t KNKN - a.P t.N t dN = N t+1 -
N t dN dt ( ) = r.N t K N - N t KNKN - a.P t.N t ( ) r.N t K N - N
t KNKN = a.P t.N t IF dN dt = 0 ( ) 0 = r.N t K N - N t KNKN - a.P
t.N t THEN ( ) r K N - N t KNKN = a.P t. ( ) r/a NtNt KNKN = P t 1
- r/a r.N t aK N = P t
Slide 18
r/a r.N t aK N = P t r/a = P t NOTE The latter is a constant.
The new equilibrium is not in other words, it will vary with the
numbers of prey and the carrying capacity of the prey population
How does this compare with the equilibrium solution for prey
growing without intra-specific competition? Equilibrium Solution -
Predator P t+1 = P t + a.f.P t.N t ( ) K p - P t KpKp - q.P t What
is the population size of the prey that induces no change in the
predator population size? dN = N t+1 - N t a.f.(K p P t ) q.K p
NtNt = q/a.f = N t NOTE The latter is a constant. The new
equilibrium is not in other words, it will vary with the numbers of
predators and the carrying capacity of the prey population
Slide 19
Next Change the various values of r, a, f, q, and the values of
k and see what happens. Make a note each time of the changes that
you have made and of the effects observed When you do this, change
each variable one at a time double it and half it.
Slide 20
Varying the parameter values has an effect on the persistence
of the fluctuations, and models can mimic the range of patterns
observed in nature Regardless of what you do to the various
parameters, the relationships between predator and prey are more
stable now than they were when you looked at populations growing
exponentially in most cases. Intra-specific competition stabilizes
the predator-prey relationship
Slide 21
Refuges Predator - Prey relationships can also be stabilised if
the prey have a refuge in time or space either permanently (e.g. in
cracks and crevices on a shore may provide refuges for small
gastropods from large predatory whelks) or temporarily (e.g. a
population that spans a large area but is subject to predation only
in part of the range has a temporary refuge). Prey Predator Overlap
zone Refuge
Slide 22
GET STUDENTS TO WORK OUT HOW THEY WOULD BUILD REFUGES INTO A
PREDATOR-PREY MODEL OF POPULATIONS GROWING WITHOUT INTRA-SPECIFIC
COMPETITION
Slide 23
P t+1 = P t + f.a.P t.N t q.P t N t+1 = N t + r.N t a.P t.N t
Open a spreadsheet in MSExcel How do different values of r, a, q, N
0, P 0, f and refuge size influence the outcomes of species
interactions,? Next Project a prey and a predator population into
the future for 100 time units using these two equations BUT.. MUST
constrain BOTH populations so that if numbers drop to the refuge
size, they remain at that number. Use an IF argument
=IF(H2+(B$4*H2)-(E$4*I2*H2)>C$4,
(H2+(B$4*H2)-(E$4*I2*H2),C$4)
Slide 24
Meta-populations Up to now considered populations as
distributed evenly across an area of habitable space At a larger
scale of observation (landscape), suitable habitats for a
population are rarely evenly distributed Suitable Habitat,
Unsuitable Habitat
Slide 25
Not ALL suitable habitats may be occupied by a population at
the same time These discrete populations are known as meta-
populations, and each may well be behaving independently of the
others. There is no reason to suppose that all will be the same
size! The number of suitable habitats occupied depends on the
probabilities of local population extinction and local population
creation the latter being dependent on immigration
(dispersal).
Slide 26
Prey populations can be seen as suitable habitats, in so far as
predators are concerned some being more suitable (bigger) than
others. The differences between patches (asynchrony) lends
stability to the predator-prey interactions. At peaks in the
population cycles, losses through emigration from a patch will be
greater than gains through immigration. Conversely at troughs,
gains through immigration will exceed losses through emigration. *
**
Slide 27
Eotetranychus sexmaculatus In isolation on single orange Time
(days) Prey Numbers Typhlodromus occidentalis Time (days) Numbers
Predator and prey grown together on single orange Extinction!
Delayed introduction of predator Ellner et al (2001) Nature 412:
538-543
Slide 28
When populations of predator and prey grown together in
meta-population structure Sustained oscillations OVERALL patchiness
stabilizes predator-prey relationships, though extinction occurs in
individual patches Time (days) Numbers Eotetranychus
sexmaculatusTyphlodromus occidentalis
Slide 29
THE END Image acknowledgements http://www.google.com
