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Predation model predictions. Initial Lotka-Volterra. Linear functional response no K - neutral stability L-V model with K - stable equilibrium No K, type II or III functional response - unstable

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predation model predictions
Predation model predictions
  • Initial Lotka-Volterra. Linear functional response no K - neutral stability
  • L-V model with K - stable equilibrium
  • No K, type II or III functional response - unstable
  • Hump shaped prey isocline, vertical predator isocline - neutral stability, stable equilibrium or prey extinction

Paradox of enrichment (Rosenweig 1971 Science 171:385-387)

…depends on the assumption of a vertical predator isocline


Modified - hump prey isocline:

Instead of a hump, the prey isocline may be expected to rise at

low prey populations because:

- prey may find spatial refuges from predation at low densities

(type III functional response)

- prey may immigrate into habitat containing low prey numbers

- kinked hump isoclines are inherently stable - predators can

consume prey only down as far as the remaining prey population

present in refuges or to the levels maintained by immigration

(where prey isocline approaches vertical)


Changing the predator numerical response:

  • Have so far assumed the predator numerical responseV
  • (per capita growth rate of predators as function of prey abundance) is a linear function of prey abundance
  • - Seems reasonable to assume that predators also have carrying
  • capacities determined by factors other than prey availability.
  • - In state space, a carrying capacity will bend the predator zero
  • growth isocline to the right (more prey does not increase
  • predator numbers)
  • - This will generate a stable equilibrium along the horizontal
  • part of the predator isocline

Predator carrying capacity: predator can no longer

drive prey to extinction. Stable coexistence


What happens if a predator has multiple prey?

- Outcome will depend on whether prey species cycle in concert

- If prey abundances are not strongly positively correlated then

as one prey species becomes scarce, the predator can continue to

feed and increase its population size.

Unstable equilibrium

predators drive prey

to extinction


Predator carrying capacity and multiple prey species:

Predator isocline with a positive slope is stable (damped


Any factor that rotates the prey or predator isocline in a clockwise direction will tend to stabilize interactions


Additional modifications to predator - prey models

So far assumed that responses of predators to prey (and vice versa)

are instantaneous.

More realistic that predator population sizes will lag somewhat

behind prey populations (the time required for consumed prey to

be transformed into new predators)

Incorporating time lags into models generally has a destabilizing

effect leading to larger oscillations in predator/prey populations.


Conclusions: predation models

• Simple predation models can generate a variety of behaviors -

Neutral stability, stable limit cycles, and equilibrium stability.

• Adding predator or prey carrying capacities tends to stabilize

models whereas incorporating non-linear functional responses and

time lags generally leads to instability.

• Multiple prey species can destabilize populations if it allows

predators to over-exploit prey, but predation may also facilitate

prey coexistence (Paine paper) - depending on predator preference

and competitive interactions among prey species

• Simple lab predator-prey experiments most often result in

extinction over a short time. This perhaps indicates the importance

of refuges from predators in natural systems


Cyclic dynamics of the arctic collared lemming

Gilg et al. (2003)

Lemming populations show superannual cycles suggestive of predator prey dynamics


Measured lemming densities, over several years, and predation rates (observations, skat samples, pellet samples, lemming fur in borrows)

Skua Fox

Functional response

Lemmings eaten/day

Lemming density


Skua Fox

Numeric response

Young produced/yr


Stoats occupy lemming

nests in winter. Only predator to show a delayed response - highest number year after lemming pop peak

Numeric response


Observed data

Model prediction



No space or food limitation for lemmings ie no prey density-dependence (resource limitation)

Only predation regulates lemming populations

Dynamics generated by destabilizing predation by the stoat (time lag effects) and stabilizing effects by other predators (density dependent effects)

Delayed numerical response of stoat driving force for numerical fluctuations

Can use the model to examine effects of removing one or more predators on prey population dynamics


What about examples when food does become limiting?

- Most famous example is cycling of Canadian lynx and snoe-shoe

hare (Elton and Nicholson 1942, Journal of Animal Ecol 11:215)

- Hare populations cycle with peak abundance ~ every 10 yr

- Major source of hare mortality is predation

- Lynx populations appear to closely track hare populations often

peaking 1-2 years after the hare peak


Classic example of Lotka-Volterra neutral stability?

Various hypotheses: Elton - variation in solar radiation resulting

from periodic sunspot activity

Keith (1963) Wildlife’s ten year cycle. Univ Wisconsin Press:

Starvation/disease at high population sizes or predation?


Lemming-style predator-prey model does not work:

- On Anticosti Island, Quebec there are no lynx but hare populations

cycle anyway…

Keith (1983,1984) Oikos 40:385-395; Wildlife Monog. 90:1-43

looked at the cycle in detail and concluded:

Hare populations are co-limited by food availability and by

predation from many sources (goshawks, owls, weasels, foxes,


Hare population growth rates are sufficient to produce

rapid depletion of food resources (principally buds and stems of

shrubs and saplings). Furthermore, induced defenses of hare forage

results in additional declines in food availability


Krebs et al. 1995

  • Definitive study of snoeshow hare dynamics in Yukon
  • Only way to understand this system is through multifactorial manipulation experiment
  • 1 km2 plots. Monitoring of populations for 8 years
  • Fertilizer to increase plant growth (N,P,K)
  • Electric fences to keep large predators out (permeable to hares)
  • Food addition treatments
  • Predator exclusion - 2x hare density
  • Food addition - 3x hare density
  • Predator x Food addition - 11 x hare density
  • Some unresolved questions: mechanism? Why hares low so long?

Refuges from predation:

Gause (1934) set out to look for evidence of population cycles

between Paramecium and Didinium in a microcosm experiment.

- Didinium quickly consumed all the Paramecium and then went


-When sediment was placed in the bottom of the microcosm

Paramecium was able to hide, Didinium went extinct and

Paramecium population recovered.

- Ability of prey to escape predators suggests that dispersal could

within a metapopulation could provide opportunities for prey and

predators to coexist


Huffaker (1958) Experimental studies on predation: dispersion

factors and predator-prey oscillations Hilgardia 27:343-83

-Looked at populations of the six-spotted mite (Eotetranychus) that

feeds on oranges and the predatory mite (Typhlodromus)

-prey mite disperses by foot or by ballooning on silk strands

-predator mites disperse on foot

- Set up arrays of oranges with mini launch-pads on them for prey

mite dispersal.

- In large arrays predators and prey coexisted and showed population




examples of plants

constrained to

refuges by insect

herbivores - coming

soon to a discussion



Predation escape through predator satiation:

If predation rates decline at high prey densities (type II and III

functional responses) then predator satiation may allow prey

species to maintain their populations.

Many examples:

Janzen suggested that seed predation is a major selective force favouring ‘masting’ (massive super-annual seed production)

e.g. Janzen (1976) Why bamboos wait so long to flower. Ann. Rev. Ecol. Syst. 7:347-391

Bamboos are the most dramatic mast fruiters, with many species

Fruiting at 30-50 yr intervals and some much longer e.g. Phyllostachysbambusoides fruits at 120 year intervals!

Other masters: oaks, beech, conifers, Tachigali


Tachigali spp.

“Suicide tree”

Monocarpic - only dicot canopy

tree in the world with this trait

Large trees reproduce synchronously ~ every 5 years

Seeds often lost to predispersal predation

Trees infected by decay fungus before death (murder not suicide??)


Williams et al (1993) Predator satiation by periodical cicadas

Ecology 74:1143-1152

Magicicada spp emerge once every 13 or 17 yrs up to 4 million/ha

= 4 tonnes of cicadas/ha - the highest biomass of a natural population

of terrestrial animals ever recorded!

- Monitored Cicada populations in N. Arkansas

- Estimated emergence rates by capturing insects in traps on the ground

as they came out of the soil

- Captured dead adults in traps and determined predation rate by birds

by counting discarded wings


- Estimated 1,063,000 cicadas emerged in 16 ha

- 50% of population emerged on just four nights

- Birds killed a large proportion of the first emergents when

Populations were low, but only killed 15 % of the population overall