BIOS 3010: Ecology Lecture 9: Dynamics of predationhomepages.wmich.edu/~malcolm/BIOS3010-ecology/Lectures/L09-Bios... · BIOS 3010: Ecology Lecture 9: slide 1 ... • The prey population

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Dr. S. Malcolm BIOS 3010: Ecology Lecture 9: slide 1

BIOS 3010: Ecology Lecture 9: Dynamics of predation

•  1. Lecture summary: –  Modelling population

dynamics. –  Lotka Volterra model:

•  Isoclines. •  Cycles. •  Effects of competition. •  Effects of functional

responses. •  Spatial heterogeneity. •  Pseudo-interference. •  Aggregative reponses.

J. Kobalenko. 1997. Forest Cats Of North America. Firefly Books

http://www.americazoo.com/goto/index/mammals/134.htm

Snowshoe hare and lynx.


2. Modelling the dynamics of predation:

•  Two approaches to describe the dynamics of interactions such as the cycles of Fig. 10.1: – 1) Differential equations:

•  Lotka-Volterra model for continuous breeding. •  A mass-action model with no logistic limitation.

– 2) Difference equations: •  Nicholson-Bailey model for host-parasitoid dynamics

in discrete generation times. •  We will ignore this model here and focus on the Lotka-

Volterra model.


3. The Lotka-Volterra model:

•  Where P = number of predators, N = numbers or biomass of prey. •  Prey increase exponentially in the absence of predators at:

–  dN/dt = rN (r is the intrinsic rate of increase of prey) •  Prey removed by predators at an attack rate a (encounter frequency = searching

efficiency) so prey mortality due to predators will be aPN and so, –  Prey dynamics are: dN/dt = rN - aPN

“births - deaths” •  Without prey, predators starve to death at a rate q, and so,

–  dP/dt = -qP •  Predators increase as a function of food consumption rate (aPN) & predator’s

efficiency (f) at turning this into offspring to give predator increase as faPN , so balance of predator birth and mortality gives: –  Predator dynamics of: dP/dt = faPN - qP

“births - deaths”

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4. Lotka-Volterra prey-predator isoclines and the model dynamics:

•  When prey population increase (dN/dt = rN - aPN) is zero,

–  dN/dt = 0 and rN = aPN, or P = r/a (Fig. 10.2a). •  When predator population increase (dP/dt = faPN -

qP) is zero, –  dP/dt = 0, faPN = qP, or N = q/fa (Fig. 10.2b).

•  Combined, these equations generate: –  1. Anticlockwise circles as in Fig. 10.2c and coupled,

indefinite oscillations in abundance (cycles) with, –  2. A time delay between prey and predator abundances

(Fig. 10.2d) and, –  3. Neutral stability (Fig. 10.2e).


5. Prey-predator cycles:

•  Coupled oscillations as in Fig. 10.1c or Fig. 10.4 may or may not be a product of prey-predator interactions alone, but may already exist in the absence of predators.

•  “Bottom-up” (from food) and “Top-down” (from natural enemies) influences on population cycling in the northern American community of plants, hares, grouse and predators (Fig. 10.5).

•  Cycles can also show delayed density dependent mortality (Fig. 10.3) in which mortality appears to be density independent, but when plotted as a time series it spirals inwards for damped oscillations.


6. Competition and the Lotka-Volterra predation model:

•  Crowding modifies the prey and predator zero isoclines because mutual interference among predators increases with their density and increased prey density (Fig. 10.7a) and predators reach an upper density level.

•  The prey population can also be limited by crowding through intraspecific competition (logistic limitation) to generate the zero isocline of Fig. 10.7b. –  Together these isoclines are no longer neutrally

stable, but are damped to converge to a stable equilibrium.

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7. Functional responses and the Lotka-Volterra predation model:

•  Predator functional responses to prey density also modify the model in a similar way:

–  For type 2 functional responses (rising at a decelerating rate to an asymptote): •  Prey isocline is humped (Fig. 10.12). •  At low prey density this can lead to instability and extinction

–  Unlikely because predator handling times have to be very long. •  At high prey density this leads to damped oscillations & stability

–  The “Allee effect” »  Disproportionately low rate of recruitment at low population

density. »  Important in conservation and resource management.


8. Functional responses and the Lotka-Volterra predation model:

•  For type 3 functional response (sigmoidal: in which predators are less efficient at low prey density) the prey isocline does not intersect with the predator axis and different predators generate dynamics like: –  Fig. 10.11a(i) for efficient predators, and –  Fig. 10.11a(ii) for less efficient predators.

•  But where the predator switches effectively from prey to prey then its abundance may be independent of prey density (Fig. 10.11b).


9. Spatial heterogeneity:

•  Aggregated prey or prey occurring in crevices or other refuges (like the prey mites in Huffaker’s orange experiment) show spatial heterogeneity (clumped distributions) and the prey isocline can look like those in Figs 10.11a & b which tends to generate stable equilibria quickly (hence the coexistence of the prey and predator mites).

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10. Pseudo-interference:

•  “Pseudo-interference” also generates an aggregation of risk among hosts of parasites.

•  At high parasitoid density, attacked hosts are more likely to have been parasitized already (like Fig 10.7c iii).


11. Aggregative responses:

•  Aggregative responses of parasitoids to hosts can either be the same at all host densities (Fig. 10.14a) or density dependent (Fig 10.14 b & c) or density independent (Fig. 10.14d).

•  Such aggregations of risk led Pacala, Hassell et al. to generate the CV2>1 rule in which: –  If the coefficient of variation (standard deviation/mean) of the risk of

being parasitized is greater than 1 then the interaction is more likely to be stable.

–  Especially when aggregated risk was host density independent.


Figure 10.1:

Abundance cycles of prey and their predators: (a) rodents and tawny owls, (b) ragwort and cinnabar moth

larvae, (c) snowshoe hare and Canada

lynx

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Figure 10.2:

The Lotka-Volterra prey-predator model: (a) Prey zero isocline. (b) Predator zero

isocline. (c, d) Anticlockwise

cycles. (e) Neutral stability.


Figure 10.4 (3rd ed.) : Lab dynamics of moth larvae with or without parasitoids in deep medium (a,b) or shallow medium (c,d) (see Fig 8.16, 4th ed.).

deep

shallow


Figure 10.5 (3rd ed.):

Influences on the abundances of populations in different trophic levels in an Alberta community

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Figure 10.3: Delayed density dependence in a host-parasitoid model (a-d) and for the winter moth (e).


Figure 10.7:

Predator (a) and prey (b) isoclines subject to crowding, and the effects of increasing predator crowding (c) on stability and the equilibrium abundances of prey and predators.


Figure 10.11a: Effects of a type 3 functional response on the prey isocline and the dynamics of (i) efficient and (ii) less efficient predators.

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Figure 10.11b: Effects of a combination of a type 3 functional response and predator switching behavior.


Figure 10.12: Effects of a humped prey isocline caused by a predator type 2 functional response or an Allee effect


Figure 10.14: Aggregative responses of parasitoids and the aggregation of risk

(a) Constant ratio, (b) Increased aggregation

of risk, (c) inverse density

dependence, (d) density independence

with some aggregation of risk

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BIOS 3010: Ecology Lecture 9: Dynamics of predationhomepages.wmich.edu/~malcolm/BIOS3010-ecology/Lectures/L09-Bios... · BIOS 3010: Ecology Lecture 9: slide 1 ... • The prey population