1. 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

2. How do predators respond to prey?
Numerical response (demographic and aggregative)
Functional response (feeding rate)
Developmental response (predator ind. growth)
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

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

4. Let's look at the basic predator-prey model
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

5. Predation: aNP aPN μ is death rate of predator, aNP (a) is attack rate
c is conversion rate
r is prey growth w/o predation

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

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

8. Dynamics: 2 N P 3 1 4
Out of phase by ¼ cycle. “Time-lags” (instant response, but numbers lag). 1 1 2 2 2 3 3 3 4 4

9. Congruence has been very appealing…
Congruence has been very appealing….we'll return to this later when we discuss population cycles specifically.

10. Other functional responses?

11. Functional Responses:

12. What is the effect on isoclines
(and stability)?
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

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

14. How can we stabilize predator-prey dynamics?
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

15. P
aNP
aPN N

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

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

19. Type II functional response and prey competition
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

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

21. Now shift the relative position of the predator isocline:
Paradox of enrichment:
Now shift the relative position of the predator isocline:
dP/Pdt=0 N P
Stable limit cycle
dN/Ndt=0
Equilibrium is unstable, but at high prey densities, dens-dep is strong enough to constrain the system and provides a source of stability (neg feedback). This limit cycle is an "attractor".

22. So, now enrich such a system …
Numerical: the classic demographic response in predator-prey models (more prey=lower mortality or higher birth), but has also been extended to include migration (aggregation of preds to locally high densities of prey).
Developmental: Murdoch 1971
Functional: a series of paper by Buzz Holling in 1950's and 1960's

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

24. Are there other types of functional responses?

25. Mutual interference & Ratio Dependence

26. Type II Functional Response (Holling)
Feeding rate
Prey density

27. 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)

28. 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”

29. Re-analyze with meta-analysis (Osenberg et al. 1999) :
m = / (mean +/- 95% CI)
m ≠ 0
m ≠ 1
s2(m) =
4% of studies yield m>1

30. More recent analysis:
Frequency m

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

32. Caveat: depletion Feeding rate Initial prey density
Estimates of "mutual interference" have not accounted for depletion (e.g., Abrams .
What do you expect if you allow prey to deplete in the trials in which you estimate feeding rate?
Feeding rate
(per predator)
Initial prey density

33. 8 preds x 4 prey/pred = 20% depletion
Thus, these patterns may have been misinterpreted. Feeding rate is intended to measure an "instantaneous" rate, not a long-term feeding rate.

34. But what about effect on isoclines?

35. P N Predator isocline: dP/dt=0
It goes through origin: essentially, the feeding rate is constant for a fixed value of N/P; so if you keep N/P the same, the feeding rate is the same. Hence there is a single N/P at which the birth rate balances the death rate. Increasing the predator mortality would reduce the slope of this line.

36. What about the prey isocline?

37. From Arditi and Ginzburg 2012. How Species Interact
For r>a
Why does it have this shape? A bit complicated… In ratio dep, as P—infinite (or ast N0) the RISK (per capita, imposed by the entire predator popualation: i.e., FR*P/N) increases to a. I.e., the highest the risk can be is 'a'. IN prey dep, the maximum risk (for a given P) was found as N infinite, so we could always impose more mortality by simply increasing P. But in Ratio Dep, that's not possible (because predators interfere); instead the RISK from the ENTIRE population has a limit, thus increasing P more does not generate more mortality! So, NL is the prey density at which prey per capita growth is =a (this will be some value>0, but less than K, if r>a).
From Arditi and Ginzburg How Species Interact

38. Effect of prey productivity?

39. P N Change prey productivity:
Thus P* and N* both increase with increasing productivity (keep this in mind for next lecture) N P
dP/dt=0
Prey growth is logistic with Ratio-Dep functional response
dN/dt=0

40. Ideas, once they take root, are hard to kill
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)

41. But can ratio dependence explain empirical patterns…