1. BSC 417/517 Environmental Modeling
Predator-Prey Oscillations on the Kaibab Plateau

2. The Predator-Prey Relationship
Predator-prey relationships have always occupied a special place in ecology
Ideal topic for systems dynamics
Examine interaction between deer and predators on Kaibab Plateau
Learn about possible behavior of predator and prey populations if predators had not been removed in the early 1900s

3. Deer and Predators on Kaibab Plateau
Information on deer population irruption is not reliable
Data on predators is even more sketchy
Gain insight into predator prey relationship on the Plateau from a more well-documented system: the snowshoe hare-lynx system in Canada
Time series available on number of lynx pelts purchased by the Hudson Bay Co.

4. Snowshoe Hare-Lynx System7 6 5 4 3
Hares 2
Lynx 1

5. Snowshoe Hare-Lynx System
Records show peak in number of lynx pelts every 9-10 years
Data suggest that populations have oscillated in a cyclical manner for over 100 years
Data are viewed as a classical example of predator-prey interaction
Oscillations are not related to seasonal or other obvious annual changes
Best examples of predator-prey oscillations in mammal populations show periodicity of 3-4 or 9-10 years

6. Reference Mode for Kaibab Deer-Predator System
Use hare-lynx example to draw a reference mode for deer-predator relationship
Should the oscillations be sustained, damped, or growing?
Intuition says sustained, but many other types of behavior have been observed
For sake of simplicity, go with sustained oscillation with 9-10 year periodicity as reference mode
Peaks in predator (cougar) populations should lag behind peaks in deer population by a few years

7. Initial Model – Equilibrium Conditions
2000 50
4000
2000
0.0
0.5
800
0.0 5 40
1.0

8. Model Structure Ignore biomass impact of deer growth
Assume ample forage is present by setting fraction forage needs met equal to 1.0
Predator stock is dependent on deer density vis-à-vis deer density-dependent kill rate and kill-rate dependent net birth rate

9. Predator Kill Rate Functional Response
Number of deer killed per predator per year is 60 if there are more than 10 deer/1000 acres… ~1 kill/week = satiation limit
Shape of graphical function reflects a combination of “Type I” and “Type II” functional response
Kill rate
Kill rate
Type I
Type II
Prey density
Prey density

10. Predator Kill Rate Graphical Function

11. Predator Birth Rate Response
Net birth rate is dependent on kill rate: higher kill rate => higher net birth rate
Maximum net birth rate = 0.45/yr
Cougars start to breed young (2-3 years age)
Breed every 2 years with an average of 3 kittens
Maximum net birth rate for predators and prey are comparable and relatively high…implications for potential oscillation?

12. Predator Birth Rate Graphical Function

13. Initial Model Results Verify Equilibrium Conditions
Initial predator density = 50

14. Initial Model Results - Nonequilibrium Initial Prey Density
Set initial predator density at 45
System displays unstable behavior (as illustrated by 30 vs. 50 year simulation)
Predators virtually annihilate prey after ca. 25 year, which lead to ensuing unstable behavior
Question: why doesn’t such unstable behavior typically occur in nature?

15. Initial Model Results - Nonequilibrium Initial Prey Density

16. Natural Predator-Prey Systems
Predators don’t normally hunt prey to zero
Rather, select individuals from prey population that are easiest to catch (young, old, weak)
Minimum threshold concept: prey density limit below which predators would no longer find it profitable to hunt the prey and would switch to different prey
Threshold is determined by availability of prey hiding places (refuge) and prey social behavior

17. Revising The Model
Should we revise the model to take into account the threshold concept, effect of prey refuge, and prey social behavior?
Perhaps expand deer population to multiple stocks to simulate deer age structure, and then allow predators to concentrate on young and old deer
Sounds good, but…complexity would increase dramatically in face of limited data…
Better to consider if combined effect of these factors could be taken into account within existing, simple model structure

18. Revised Model
Try using a different functional response for density-dependent kill rate which incorporates the concept of threshold prey density
No kills if deer density falls below 2 deer per 1000 acres, e.g. because of the ability of deer to find safe refuge when overall density is low
S-shaped function response corresponds to “Type III” functional response

19. Type III Functional Response

20. Revised Model Results

21. Revised Model Results Initial predator population is set at 100
Large predator population causes an initial decline in deer population, but predator population declines quickly
Damped oscillatory behavior ensues with periodicity of ca. 10 years
Result essentially corresponds to the original reference mode

22. Further Interpretation
The initial “dynamic hypothesis” was that the cougar and deer populations could interact to produce stable cycles with a period similar to the classic 9-10 year cycle observed in other mammalian predator-prey systems
Requirement for a Type III functional response to produce stable behavior can be interpreted as an indication of the importance of prey refuge or threshold levels

23. State Space (Phase Plane) Diagram
Point attractor

24. Patterns of Oscillation
Previous simulations show possibility for both damped and growing oscillations, depending on the nature of the predator functional response
What about potential for sustained oscillation, as state in the reference mode?
Could random disturbances lead to persistent cycles?

25. Influence of Random Variation
Introduce randomness into the deer net birth rate via the following equations
net birth rate = random factor
random factor = random(-0.2,0.2,123)
The random factor allows net birth rate to vary randomly from a low of 0.3 to a high of 0.7
The value 123 is a “seed” for the random number generator

26. Influence of Random Variation
System shows sustained oscillation over long time scales, with periodicity of ca. 10 years
Reference mode has been generated

27. Policy Test: Selective Removal of Predators
Results of revised model with random variation in deer birth rate suggests that stable predator-prey interactions would have been possible if the predators had not been removed from the Kaibab Plateau
Although predator population averages 50, substantially higher numbers occur in some years, which could pose problem for ranchers livestock
Test influence of allowing hunters to kill some predators to protect live stock

28. Model With Selective Removal of Predators

29. New Equations predator_kills = IF(TIME>start_year) THEN
(predator_population-maximum_acceptable_predators) ELSE 0
start_year = 1920
maximum_acceptable_predators = 55

30. Simulation Results With Selective Removal of Predators

31. Interpretation
Results suggest that it might have been possible to reduce peak values of predator population without destroying the stability of the predator-prey system
However, managers in early 1900s had essentially no knowledge of predator-prey dynamics
Even today, other factors besides predator-prey population dynamics are know to be important in governing response of the system…

32. Current Interpretation of the Hare-Lynx Predator Prey System
Krebs et al. (Bioscience 2001) (see PDF on web-site) conclude that Lotka and Volterra were only partly correct when the concluded that the snowshoe hare cycle was the product of a predator-prey oscillation
Missed critical point that the cycle can only be understood by considering three trophic levels rather than just two
Hare cycle is produced by interaction between predation and food supplies
Dependence on food supply ripples across many species of predators and prey in boreal forest