# Predation (Chapter 18) Predator-prey cycles Models of predation

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Predation (Chapter 18) Predator-prey cycles Models of predation
Functional vs. numeric responses Stability in predator-prey models

Two big themes: Predators can limit prey populations.
This keeps populations below K.

Predator and prey populations increase and decrease in regular cycles.

A verbal model of predator-prey cycles:
Predators eat prey and reduce their numbers Predators go hungry and decline in number With fewer predators, prey survive better and increase Increasing prey populations allow predators to increase And repeat…

Why don’t predators increase at the same time as the prey?

The Lotka-Volterra Model: Assumptions
Prey grow exponentially in the absence of predators. Predation is directly proportional to the product of prey and predator abundances (random encounters). Predator populations grow based on the number of prey. Death rates are independent of prey abundance.

R = prey population size (“resource”)
P = predator population size r = exponential growth rate of the prey c = capture efficiency of the predators

rate of change in the prey population intrinsic growth rate of the prey removal of prey by predators

a = efficiency with which prey are converted into predators
For the predators: a = efficiency with which prey are converted into predators d = death rate of predators death rate of predators rate of change in the predator population conversion of prey into new predators

Prey population reaches equilibrium when dR/dt = 0
equilibrium – state of balance between opposing forces populations at equilibrium do not change Prey population stabilizes based on the size of the predator population

Predator population reaches equilibrium when dP/dt = 0
Predator population stabilizes based on the size of the prey population

Isocline – a line along which populations will not change over time.
Predator numbers will stay constant if R = d/ac Prey numbers will stay constant if P = r/c.

Number of Predators (P) Number of prey (R) Predators are stable when:
Prey are stable when: Number of Predators (P) Number of prey (R)

Number of Predators (P) Number of prey (R) Prey are stable when: Prey
Isocline Number of Predators (P) r/c d/ac Number of prey (R)

Number of Predators (P) Number of prey (R) Predators are stable when:
isocline Number of Predators (P) d/ac Number of prey (R)

equilibrium Number of Predators (P) r/c d/ac Number of prey (R)

Predation (Chapter 18) Finish Lotka-Volterra model
Functional vs. numeric responses Stability in predator-prey cycles

Number of predators depends on the prey population.
isocline Number of Predators (P) Predators decrease Predators increase d/ac Number of prey (R)

Number of prey depends on the predator population.
Prey decrease Prey Isocline Number of Predators (P) r/c Prey increase d/ac Number of prey (R)

Changing the number of prey can cause 2 types of responses:
Functional response – relationship between an individual predator’s food consumption and the density of prey Numeric response – change in the population of predators in response to prey availability

Lotka-Volterra: prey are consumed in direct proportion to their availability (cRP term)
known as Type I functional response predators never satiate! no limit on the growth rate of predators!

Type II functional response – consumption rate increases at first, but eventually predators satiate (upper limit on consumption rate)

Type III functional response – consumption rate is low at low prey densities, increases, and then reaches an upper limit

Why type III functional response?
at low densities, prey may be able to hide, but at higher densities hiding spaces fill up predators may be more efficient at capturing more common prey predators may switch prey species as they become more/less abundant

Numeric response comes from Population growth
(though most predator populations grow slowly) Immigration predator populations may be attracted to prey outbreaks

Predator-prey cycles can be unstable
efficient predators can drive prey to extinction if the population moves away from the equilibrium, there is no force pulling the populations back to equilibrium eventually random oscillations will drive one or both species to extinction

Factors promoting stability in predator-prey relationships
Inefficient predators (prey escaping) less efficient predators (lower c) allow more prey to survive more living prey support more predators Outside factors limit populations higher d for predators lower r for prey

Alternative food sources for the predator
less pressure on prey populations Refuges from predation at low prey densities prevents prey populations from falling too low Rapid numeric response of predators to changes in prey population

Huffaker’s experiment on predator-prey coexistence
2 mite species, predator and prey

Initial experiments – predators drove prey extinct then went extinct themselves
Adding barriers to dispersal allowed predators and prey to coexist.

Refuges from predation allow predator and prey to coexist.

Prey population outbreaks
Population growth curve for logistic population growth Per capita population growth rate ro K Density of prey population

Density of prey population
Type III functional response curve for predators Per capita death rate K Density of prey population

Multiple stable states are possible.

Below A – birth rate > death rate; population increases

Point A – stable equilibrium; population increases below A and decreases above A

Between A & B – predators reduce population back to A

Unstable equilibrium – equilibrium point from which a population will move to a new, different equilibrium if disturbed

Point B – unstable equilibrium; below B, predation reduces population to A; above B, predators are less efficient, so population grows to C B

Between B & C – predators are less efficient, prey increase up to C

Point C – stable equilibrium

Predator-prey systems can have multiple stable states
Reducing the number of predators can lead to an outbreak of prey

Growth rate Death rate