Ecology 8310 Population (and Community) Ecology Predator-prey theory Basics (Lotka-Volterra) Functional responses and risk Effect on phase-planes Dynamics.

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Ecology 8310 Population (and Community) Ecology Predator-prey theory Basics (Lotka-Volterra) Functional responses and risk Effect on phase-planes Dynamics Paradox of enrichment Predator interference and ratio dependence

How do predators respond to prey? Numerical response (demographic and aggregative) Functional response (feeding rate) Developmental response (predator growth)

Functional response: the feeding rate of a predator as f(prey density)

Let's look at the basic predator-prey model

Predation: P a NP N a PN  is death rate of predator, a NP (a) is attack rate c is conversion rate r is prey growth w/o predation

Solve for equilibrium: Analyze stability: …later (qualitative for now)

N P Phase planes: r/a Putting it together… dN/Ndt=0  /ca dP/Pdt=0 The equilibrium is neutrally stable

Dynamics: N P Out of phase by ¼ cycle. “Time-lags” (instant response, but numbers lag).

Other functional responses?

Functional Responses:

What is the effect on isoclines (and stability)?

N P Effect on prey isocline: dN/Ndt=0 dP/Pdt=0 I: neutral II: destabilizing III: depends Note: shape of predator isocline remains the same (but it will shift left/right)

How can we stabilize predator-prey dynamics?

P a NP N a PN

N P Predator interference: r/a dN/Ndt=0 dP/Pdt=0 The equilibrium is stable

Intraspecific competition among prey: N P r/a K dP/Pdt=0 dN/Ndt=0 The equilibrium is stable

Type II functional response and prey competition

N P Paradox of enrichment: dN/Ndt=0 dP/Pdt=0 Locally stable

N P Paradox of enrichment: dN/Ndt=0 dP/Pdt=0 Now shift the relative position of the predator isocline: Stable limit cycle

So, now enrich such a system …

N P Paradox of enrichment: Enrich system: e.g., increase production of prey (r) and its K What will happen?

Are there other types of functional responses?

Mutual interference & Ratio Dependence

Feeding rate Prey density Type II Functional Response (Holling)

Hassell-Varley model: f(N,P) = aNP -m / (1 + ahNP -m ) If m=0, then “prey dependent” If m=1, then “ratio dependent” Arditi & Akcakaya (1990) Osenberg et al. (1999)

Estimate m : Arditi & Akcakaya (1990) 15 studies: estimated m for each 15/15 led to rejection m=0 3/15 led to rejection of m=1  Prey dependence is “wrong”  Ratio dependence is “right”

Re-analyze with meta-analysis (Osenberg et al. 1999) : m = / (mean +/- 95% CI)  m ≠ 0  m ≠ 1  s 2 (m) =  4% of studies yield m>1

More recent analysis: m Frequency

Caveat: study bias These 15 (or 35) studies were not randomly drawn from all predator-prey systems.

But what about effect on isoclines?

Predator isocline: N P dP/dt=0

What about the prey isocline?

For r>aFor r<a From Arditi and Ginzburg How Species Interact

Change prey productivity: N P dP/dt=0 dN/dt=0

Ideas, once they take root, are hard to kill.…they persist not just in spite of a single inconvenient fact, but in spite of repeated theoretical refutations and whole piles of contrary facts. They are not truly alive—because they are not true—but neither are they dead. They are undead. They are zombie ideas. -Jeremy Fox (2011, Dynamic Ecology blog)

Tuesday's discussion: