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Ecology 302: Lecture VII. Species Interactions.
(Gotelli, Chapters 6; Ricklefs, Chapter 14-15)
MacArthur’s warblers. Variation in feeding behavior allows morphologically
similar species of the genus Dendroica to coexist in eastern coniferous
forests.
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Key Points.
• Types of pair-wise interactions and examples.
• Lynx-hare cycle.
o Embedded in a complex food web.
o Supplemental hare feeding / hare predator exclusion ex-
periments suggest that both predation (lynx eats hare) and
hare food supply essential components of the cycle.
o Validated by three-level food chain model: Lynx-hare-
vegetation.
• Lotka-Volterra (L-V) predator-prey model.
o Prey grow exponentially absent predation
o Predators harvest prey in proportion to their abundance.
o Isocline analysis ⇒ oscillatory solutions.
o Global analysis ⇒ infinite number of neutrally stable pe-
riodic orbits.
• In the laboratory, one or both species of a predator-prey pair
go extinct unless special measures taken.
o Increasing microcosm size.
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o Reducing prey food supply (medium impoverishment)
o Reducing contact rate between predator and prey
o Spatially heterogeneous environment.
• L-V model biologically unrealistic.
• Consumer-resource equations cap victim density, i.e., absent
predation, victims assumed to grow logistically.
o System goes to stable equilibrium with both species pre-
sent or
o Predators die out and � → �
o Depends on whether or not predator numbers can increase
when � � �.
• Rosenzweig-MacArthur equations add predator satiation.
o Two species equilibrium can be stable or unstable
o If unstable, dynamics → a single stable limit cycle.
o Depends on degree of predator-proficiency.
o Paradox of enrichment.
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I. Pair-wise Interactions.
A. Competition (-/-).
1. Each species reduces the
growth rate of the
other.
2. Can be exploitative or
via interference.
3. Resources can be re-
newable (seeds, in-sects,
etc.) or not (space, nest
cavities).
B. Predator-Prey (+/-)
1. One species benefits;
the other suffers.
2. Includes pathogen-host,
parasite-host interac-
tions.
C. Mutualism (+/+)
1. Can be obligate – e.g.,
a. Yucca-Yucca moth.
b. Acacia-ant.
2. Or not.
Figure 1. Examples of predator-
prey, mutualistic and commensal
interactions.
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D. Commensalism (+/0)
1. One species benefits from, but does not affect the
other.
2. E.g. Cattle egrets / buffalo.
E. Amensalism (-/0)
1. One species harmed by, but does not affect, the other.
2. Understory (short) trees are shaded out by, but do not
affect, the growth of canopy (tall) species.
Figure 2. Above. Competition for space by two species of barnacles.
Next page. Ant-acacia mutualism.
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F. Nature of interaction can change with circumstamces.
G. Ricklefs gives example of saguraro-nurse plant interaction:
1. Saguaro protected by nurse plants when young (+/0)
2. Ungrateful older saguaro unaffected by, but interfereres
with, nurse plant (0/-). Or the two plants compete with
each other for water (-/-).
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II. Lynx-Hare Cycle.
A. Pairwise interactions embedded in larger networks.
1. Sometimes, behavior of the whole understandable in
terms of a limited number of key interactions.
2. Example: Boreal forest foodweb.
a. Arrows indicate flow of energy – who eats
whom.
b. Much can be understood in terms of three
variables: lynx, hare and “vegetation.”
Figure 3. The lynx-hare interaction is embedded in a larger network.
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B. Background. 1. “Ten-year” cycle
since late 1700s.
2. Lynx eat hare
(principally); hares
eat vegetation.
3. Long-standing
dispute as to nature
of the oscilation.
C. Field experiments ⇒
both important.
D. Three treatments:
1. Hares given supple-
mental food.
2. Terrestrial predators
excluded by fences
through which hares
could pass.
3. Food + Exclusion
Figure 4. Top. Lynx pursues
hare. Bottom. Time series.
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E. Results:
1. Supplemental food
increased hare den-
sities, but didn’t de-
lay population crash.
2. Excluding terrestrial
(but not avian) pred-
ators had negligible
effect.
3. Combined treatment
increased hare den-
sities and postponed
crash.
F. Three “species” model.
1. Lynx, hare, vegetation + seasonality.
2. Parameterized from known biology.
3. Qualitatively replicates experimental findings.
Figure 5. Effect of supplemental
feeding and partial predator ex-
clusion on snowshoe hare de-
mography. Top. Data of Krebs
and associates. Bottom. Output
of three species model
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III. Lotka-Volterra Predator-Prey Dynamics.
A. Equations.
���� � ��� � ��
(1) ���� � ��� � ��
1. P – predators; V – victims;
2. – baby predators per victim consumed; k – kill rate
�pd-sec���; d – predator per capita death rate
�sec���.
3. r – victim per capita rate of increase absent
predation �sec���;
B. Equilibrium:
�∗ � � ;�∗ � �
(2)
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IV. Isocline Analysis.
A. Plot zero-growth isoclines:
���� � 0:� � �
(3) ���� � 0:� � �
B. Note whether each species
increases or decreases in
different regions of � � �
plane.
���� � 0 ⇔ � � �
(4) ���� � 0 ⇔ � � �
C. Draw arrows in P and V directions; compute resultants.
D. We can conclude that the system can oscillate, but not
whether the oscillations die out, grow or tend to one or so-
called “limit cycles”.
Figure 4. Predator (�� ��⁄ � �)
and victim (�� ��⁄ � �) zero-
growth isoclines. Arrows indicate
changing densities of the two
species. There are two equilibria,
��, �� � !�"/$�, �� ⁄ %$�& and
the origin, which is a saddle.
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V. Global Dynamics.
A. One can prove (but we will not) that
1. There are an infinite number of oscillations
that neither grow nor decay.
2. Amplitude and period of the oscillations
depends on the initial values of the two
species.
Figure 5. Lotka-Volterra dynamics. left. Isocline analysis indicates the
potential for oscillatory behavior. Right. Solutions to Eqs 1. Changing
the constants distorts the solution curves, but does not affect the quali-
tative picture – an infinite number of neutrally stable cycles.
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VI. In the Laboratory.
A. Early experiments by Gause and others
1. Confirmed oscillatory nature of Pd-Py dynamics, but
2. Absent immigration, one or both species invariably go
extinct.
B. Subsequently determined that oscillations stabilized by
1. Increasing size of the microcosm – no stochastic
extinction (Luckenbill)
2. Medium impoverishment (Luckenbill) – as opposed to
enrichment which destabilizes (see Gotelli, pp. 140 ff.)
3. Mobility reduction (both species) – reduces contact
rate– by making the medium more viscous (Luckinbill).
4. Spatial heterogeneity – Huffaker’s mite expts (see
Rickleffs, p. 310 ff.)
Figure 5. Left. Didinium eats Paramecium. Right. When introduced to a
population of Paramecium at carrying capacity, Didinium exterminates
its prey and then dies out. From Gause (1934)
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VII. Beyond Lotka-Volterra.
A. Unrealistic L-V assumptions.
1. No limit to numbers of
prey absent predation.
2. Per predator harvest rate
� '. Requires
a. infinite predator hunt-
ing, killing skills;
b. Infinitely distensible
predator stomachs
3. No stochastic extinction.
B. Other unrealistic assumptions.
a. Only two species.
b. Populations homogeneous – all individuals the same.
c. Space doesn’t matter – “well stirred” assumption.
d. Phenotypes fixed – no behavioral, developmental,
evolutionary responses to changing numbers of both
species.
So many mice; so little time.
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Figure 6. Potpourri of predator-prey dynamics. Representative trajectories
superimposed on zero-growth isoclines �� ��⁄ � � and �� ��⁄ � � a. Lot-
ka-Volterra model (Equations 1) reproduces the back and forth motion of a
frictionless pendulum. b. With the addition of an upper bound to victim
density, infinite numbers of periodic solutions give way to a single stable
equilibrium. c. The addition of predator satiation restores the possibility of
oscillatory behavior. d. Providing for extinction when victim densities drop
below a critical threshold converts the system into an ecological analog of
a nerve cell – stimulate and fire.
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C. Consumer-Resource Equations cap victim population.
1. Equations.
���� � ��� � ��
(5)
���� � � (� )1 � �
�+ � �,
2. If �∗ � � there is an interior equilibrium (Figure 4b) at
�^ � � )1 � �∗
� + (6)
�^ � �
3. Two scenarios:
a. �∗ � �: ��, �� → �0, ��
b. �∗ � �: ��, �� � ��∗, �∗�; ��, �� � �0, �� is a saddle.
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Figure 7. Isoclines (top) and dynamics (bottom) in the case of victim lim-
itation in the absence of predators (Eqs 5). Left. �∗ � .. Right. �∗ � ..
Note that in this and the preceding (L-V) case, the origin in as saddle.
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D. Nonlinear Functional Response (FR) – cap predator
consumption.
1. Replace ' in LV and CR
with /���, where
/��� � 0121321 (7)
and χ is called the half-
saturation constant.
2. /��� is called the pred-
ator functional response
(FR).
3. Note that
lim0→7 /��� � 1
4. Type II FR ( 8 9 1). /���
increases with V at a
diminishing rate.
5. Type III FR. (/��� � 1. /��� is sigmoidal.
Figure 8. Realizations of Eq 7
(functional response) for differ-
ent values of n (top) and χ (bot-
tom). With n = 1, FR is equivalent
to Michaelis-Menten kinetics.
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E. Rosenzweig-MacArthur (RM) model: Prey carrying capacity
+ type II FR.
1. Equations.
���: � � ) �
; < � � �� (8)
���: � � (� )1 � �
�+ � �; < �,
2. Equilibria.
��, �� � �0,0�; ��, �� � �0, ��
(9)
��∗, �∗� � ( �; � � , �
)1 � �∗
� + �; < �∗�,
3. Isoclines.
�= � 0:� � � )1 � �
�+ �; < ��
(10)
�= � 0:� � �; � �
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4. Victim isocline dome-shaped; peaks at
> � � � ;2 �11�
5. Three scenarios:
a. �∗ � �: �����, ����� → �0, ��
b. � � �∗ � >: �����, ����� → ��∗. �∗�
c. �∗ � >: �����, ����� → a limit cycle.
Figure 9. Predator-prey dynamics with victim carrying capacity and type
II FR. Left. �∗ �J. The interior equilibrium is stable. Right. Solutions tend
to a stable limit cycle.
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6. Paradox of enrichment:
Increasing victim carrying
capacity.
a. Equilibrium density of
predators increased.
b. Equilibrium density of
victims unaffected.
c. Can destabilize other-
wise stable interaction,
i.e., stable equilibrium becomes unstable; system
winds out to a stable limit cycle.
d. Consequence of shifting > � �� � χ�/2 (victim density
at which victim isocline peaks) to the right.
Figure 10. Enriching a food chain
at its base can destabilize a pred-
ator-prey interaction by shifting
the peak in the victim isocline to
the right of the predator isocline.
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Table 1. Dynamics of Predator-Prey Models.
Model ��∗, �∗� (0,0) (0,K) POs
LV Neut.
Stable Saddle –
∞ Number
Neut. Stable
CR Stable Saddle Saddle or
stable –
RM Stable
Unstable Saddle
Saddle or
stable
–
Limit cycle
