05 predator prey dynamics and activity3

[Population dynamics] Population dynamics

Pattern of change in population size over time

[Population dynamics] Hare-lynx cycle

Classic ~10-year population cycle

[Population dynamics] Best explanation for hare-lynx cycles

A combination of food limitation and predator limitation

[Population dynamics] Discrete-time growth

N(t+1) as a function of N(t)

[Population dynamics] Continuous-time growth

dN/dt = limit of change in N over change in t as delta t approaches 0; N grows if dN/dt > 0 and shrinks if < 0

[Lotka-Volterra] Prey equation

dN/dt = rN - aNP

[Lotka-Volterra] Predator equation

dP/dt = baNP - mP

[Lotka-Volterra] N

Prey population density

[Lotka-Volterra] P

Predator population density

[Lotka-Volterra] r

Intrinsic (exponential) growth rate of prey without predators

[Lotka-Volterra] a

Attack rate

[Lotka-Volterra] aNP

Rate prey are eaten; mass action (encounters proportional to N times P)

[Lotka-Volterra] b

Conversion rate of eaten prey into predator reproduction

[Lotka-Volterra] baNP

Predator population growth from eating prey

[Lotka-Volterra] m

Predator mortality rate

[Lotka-Volterra] rN term

Prey natural population growth (+)

[Lotka-Volterra] mP term

Predator natural mortality (-)

[Lotka-Volterra] What happens with no predators (P = 0)?

Prey grow exponentially

[Lotka-Volterra] What happens with a = 0?

Prey grow exponentially and predators go extinct

[Lotka-Volterra] Key assumption of simple LV model

Prey experience no food limitation (no carrying capacity)

[Lotka-Volterra] Typical LV output

Coupled oscillations with predator peaks lagging prey peaks

[Lotka-Volterra] Why is LV still useful?

Simple way to think about predator-prey interactions and their dynamics even though it doesn't fully explain hare-lynx cycles

[Lotka-Volterra] Individual-based models

Simulate consumption, reproduction and death of individuals using rules instead of equations; give similar results (e.g., NetLogo)

[Lotka-Volterra] Rosenzweig-MacArthur [Extra]

LV plus prey carrying capacity and predator saturation (type II functional response); can produce stable equilibria or limit cycles

[Lotka-Volterra] Paradox of enrichment

Making a system more productive increases oscillation amplitude and can lead to predator extinction

[Lotka-Volterra] Paradox of enrichment consequence

Highly productive environments can be less diverse than less productive ones

[Activity 3] Default NetLogo dynamics

Cycles: sheep rise, wolves rise, wolves overeat sheep and starve, cycle repeats; grass opposes sheep

[Activity 3] Slider matching r

sheep-reproduce (sheep reproduction without predators)

[Activity 3] Slider matching b

wolf-gain-from-food (energy wolves gain per sheep); alternate candidate wolf-reproduce

[Activity 3] 50 sheep and 250 wolves

Wolves crash early, sheep drop drastically (often both fail) because too many wolves overconsume prey

[Activity 3] Why did 50 sheep/250 wolves crash?

Wolves overconsume sheep, don't gain enough energy to reproduce, and starve when sheep become scarce

[Activity 3] Wolf reproduction to 20%

Cycle amplitude increases; higher extremes in wolf numbers

[Activity 3] Why can wolves go extinct at high reproduction?

Wolves grow fast, wipe out sheep, then starve with few survivors

[Activity 3] wolf-gain-from-food = 0

Sheep increase, grass decreases, then both stabilize with many sheep and little grass

[Activity 3] Trophic cascade in wolf-sheep-grass

Wolves don't eat grass but indirectly protect it by reducing sheep

[Activity 3] Lowering grass-regrowth-time (first effect)

Amplitude of oscillations of all three populations increases

[Activity 3] Lowering grass-regrowth-time (eventual effect)

Wolves go extinct

[Activity 3] After wolves vanish

Sheep rise then stabilize; grass limits sheep; grass and sheep slowly oscillate around stable values

[Activity 3] Is the system more or less stable after wolf loss?

Less stable (bigger oscillations) and lower biodiversity