Ecology - Lecture 15

Front 1

Start with logistic growth model

Back 1

For this new model for predator-prey interactions we need two equations

- 1 for predators

- 1 for prey

Front 2

Assuming predation prevents prey from reaching K

Back 2

In the prey model, term (1 - N/K) can be dropped

If prey population reached K, predators would not have a negative effect

Front 3

Theoretical models of predator-prey interactions: Prey model

Back 3

*need to subtract prey taken by predators

*depends on three things

1. Predator density, P

2. Prey density, N

3. Attack rate of predators, a

Front 4

Predation Rate (Prey losses)

Back 4

*Attack rate x predator density x host density

Rate of predation = aNP

*Prey population growth = exponential growth minus # lost to predator

dN/dt = rN - aNP

Front 5

Lotka-Volterra models of predator-prey interactions: Prey

Back 5

Dynamics of the prey population:

dN/dt = B-D = rn - aNP

N = size of prey population

P = size of predator population

r = exponential growth rate of prey population

a = attack rate of predation

(prey captured/prey x time x predator)

aN = functional response = rate of prey capture by an individual predator as a function of prey abundance

*Losses to predation are proportional to NP, which assumes that predators find prey at random

Front 6

Lotka-Volterra models of predator-prey interactions (continued): Predator

Back 6

Dynamics of the predator population:

dP/dt = B - D = aeNP - dP

e = efficiency with which food is converted to population growth

d= death rate of predators

* e increases with the value of individual prey items

e = conversion efficiency

(predators/predator x time x prey)

aeN = numerical response of the predator population, predators produced for every prey available

Front 7

Equilibrium solutions for Lotka-Volterra predator-prey models: PREY

Back 7

dN/dt = rN - aNP | 0 = rN - aNP | Peq = r/a

Note that prey equil. is in terms of predator population

dN/dt = 0, when Pequil = r/a

* When the growth rate of the prey population (r) is high, more predators are needed to keep the prey population from growing

* When predators feed t high rates (a), fewer of them are needed to keep the prey population from growing

Front 8

Equilibrium solutions for Lotka-Volterra predator-prey models: PREY graph

Back 8

* low predators = quickly increasing prey

* high predators = quickly decreasing prey

* at Peq (pred pop) = exponential growth rate of population (r) is zero

Front 9

Equilibrium solutions for Lotka-Volterra predator-prey models: PREDATOR

Back 9

dP/dt = aeNP | 0 = aeNP - dP | Nequil = d/ae

*Predator equilibrium is in terms of prey population

dP/dt = 0, when Nequil = d/ae

* With an increasing predator death rate (d), more prey are need to keep preato population from declining.

* With greater conversation efficiency (e) or predator attack rate (a), fewer prey are needed to keep the predator population above positive population growth

Front 10

Equilibrium solutions for Lotka-Volterra predator-prey models: PREDATOR graph

Back 10

* if the numbers of prey are low = the numbers of predators decreases and v.v.

* when Neq (prey pop) = the death rate of predators (d) is zero

- shows the effects of donor control over a population

Front 11

Superimposing predator and prey isoclines

Back 11

: indicates that the population trajectories follow an elliptical counter-clockwise path. The point where the two isoclines cross is the joint equilibrium.

Front 12

Thoretical models of predator-prey interactions:

Cycles and waves

Back 12

Prey x time:

LRq: 2/4

URq: 3/4

ULq: 4/4

LLq: 1/4

Predator x time:

LRq: 1/4

URq: 2/4

ULq: 3/4

LLq: 4/4

Front 13

Lotka-Volterra Predator Models: EQUATIONS

Back 13

N = prey abundance

r = prey intrinsic growth rate

a = predation rate

(capture efficiency)

P = predator abundance

e = predator fecundity

d = predator mortality

Front 14

Amplitude of cycles depends on initial numbers of predator and prey

Back 14

: the higher the population turnover, the faster the system oscillates

Front 15

When does the model not generate cycles?

Back 15

(1) if the initial numbers of predator and prey are at the joint equilibrium

(2) if initial conditions are too extreme - either predators or prey crash

Front 16

Lotk-Volterra models are based on simplifying assumptions

ASSUMPTIONS:

Back 16

(1) No age structure
(2) No immigration
(3) Prey population limited only by predation
(4) Predators are specialists - eat only focal prey species
(5) Individual predators can consume an infinite number of prey. Type I functional response
(6) Predator and prey contact each other randomly - no prey refuges

Front 17

Functional Response

Back 17

: rate of prey capture by an individual predator as a function of prey abundance

*Lotka-Volterra assumes that individual predators eat more prey when they become abundant

Front 18

Functional response: equation and graph

Back 18

aN = functional response

a = attack rate of predation

< = slope of line

*straight line is unrealistic for two reasons:

(1) no predator satiation

(2) no time lost handling prey

Front 19

Predator satiation and time taken away from searching by handling and eating lead to functional response curves that level off.

TYPES:

Back 19

Type I: Individual predators can consume an infinite number of prey.

Type II: Initially like type I with decelerating predation rate at high density

cause: predator satiation at high prey density

Type III: Accelerating predation at low prey density, decelerating at high density

cause: predator satiation at high prey density

*Predator switching at low prey density, predators don't attack prey that are too rare

*Prey might have limited number of refuges, one they're filled, additionally prey are more vauable

Front 20

The functional response graphs

Back 20

I: Each predator consumes a constant portion of the prey population regardless of prey density

II: Predation rate decreases as predator satiation sets an upper limit on food consumption

III: Predation rate decreases at low as well as high prey densities

Front 21

The numerical response

Back 21

* Individual predators can increase consumption of prey only up to satiation

* Continued response to increase prey density can be achieved only though n increase in the size of the predator population

Front 22

Immigration

Back 22

: mobile predators can track prey over large areas

ex. Bay-breasted Warblers follow outbreaks of spruce budworm

Front 23

For less mobile species...

Back 23

...numerical response results from local population growth

Front 24

The numerical response of the predator popupulation

Back 24

aeR = numerical response

a = efficiency with which food converted to population growth

e = efficiency of predation

ae = conversion efficiency

(predator/predtor x time x prey)

Front 25

The numerical response (continued)

Back 25

The numerical response of the predator lags behind population growth

* When they are increasing, predators are scarce

* When prey are decreasing, predators are relatively plentiful

Front 26

Stabilizing factors

Back 26

(1) Reduced time delays in a predator's response to change in prey abundance

(2) Prey phenotypic plasticity

(3) Low predtor attack rate --

N* = d/ae, P* = r/a

(4) With Type II or II Functional response, other dynamics are possible

Front 27

How does temperature affect predator-prey cycles?

Back 27

Warming X Fish X Nutrients

-Nutrients destabilize algae

-Warming restablizes algae

Front 28

What do predators do for ecosystems?

Back 28

Cougars keep deer population low so they don't overgraze and cause erosion.