Presentation on theme: "BSC 417/517 Environmental Modeling"— Presentation transcript:
1BSC 417/517 Environmental Modeling Predator-Prey Oscillations on the Kaibab Plateau
2The Predator-Prey Relationship Predator-prey relationships have always occupied a special place in ecologyIdeal topic for systems dynamicsExamine interaction between deer and predators on Kaibab PlateauLearn about possible behavior of predator and prey populations if predators had not been removed in the early 1900s
3Deer and Predators on Kaibab Plateau Information on deer population irruption is not reliableData on predators is even more sketchyGain insight into predator prey relationship on the Plateau from a more well-documented system: the snowshoe hare-lynx system in CanadaTime series available on number of lynx pelts purchased by the Hudson Bay Co.
5Snowshoe Hare-Lynx System Records show peak in number of lynx pelts every 9-10 yearsData suggest that populations have oscillated in a cyclical manner for over 100 yearsData are viewed as a classical example of predator-prey interactionOscillations are not related to seasonal or other obvious annual changesBest examples of predator-prey oscillations in mammal populations show periodicity of 3-4 or 9-10 years
6Reference Mode for Kaibab Deer-Predator System Use hare-lynx example to draw a reference mode for deer-predator relationshipShould the oscillations be sustained, damped, or growing?Intuition says sustained, but many other types of behavior have been observedFor sake of simplicity, go with sustained oscillation with 9-10 year periodicity as reference modePeaks in predator (cougar) populations should lag behind peaks in deer population by a few years
7Initial Model – Equilibrium Conditions 200050400020000.00.58000.05401.0
8Model Structure Ignore biomass impact of deer growth Assume ample forage is present by setting fraction forage needs met equal to 1.0Predator stock is dependent on deer density vis-à-vis deer density-dependent kill rate and kill-rate dependent net birth rate
9Predator 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 limitShape of graphical function reflects a combination of “Type I” and “Type II” functional responseKill rateKill rateType IType IIPrey densityPrey density
11Predator Birth Rate Response Net birth rate is dependent on kill rate: higher kill rate => higher net birth rateMaximum net birth rate = 0.45/yrCougars start to breed young (2-3 years age)Breed every 2 years with an average of 3 kittensMaximum net birth rate for predators and prey are comparable and relatively high…implications for potential oscillation?
13Initial Model Results Verify Equilibrium Conditions Initial predator density = 50
14Initial Model Results - Nonequilibrium Initial Prey Density Set initial predator density at 45System displays unstable behavior (as illustrated by 30 vs. 50 year simulation)Predators virtually annihilate prey after ca. 25 year, which lead to ensuing unstable behaviorQuestion: why doesn’t such unstable behavior typically occur in nature?
15Initial Model Results - Nonequilibrium Initial Prey Density
16Natural Predator-Prey Systems Predators don’t normally hunt prey to zeroRather, 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 preyThreshold is determined by availability of prey hiding places (refuge) and prey social behavior
17Revising The ModelShould 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 deerSounds 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
18Revised ModelTry using a different functional response for density-dependent kill rate which incorporates the concept of threshold prey densityNo 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 lowS-shaped function response corresponds to “Type III” functional response
21Revised Model Results Initial predator population is set at 100 Large predator population causes an initial decline in deer population, but predator population declines quicklyDamped oscillatory behavior ensues with periodicity of ca. 10 yearsResult essentially corresponds to the original reference mode
22Further 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 systemsRequirement 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
23State Space (Phase Plane) Diagram Point attractor
24Patterns of Oscillation Previous simulations show possibility for both damped and growing oscillations, depending on the nature of the predator functional responseWhat about potential for sustained oscillation, as state in the reference mode?Could random disturbances lead to persistent cycles?
25Influence of Random Variation Introduce randomness into the deer net birth rate via the following equationsnet birth rate = random factorrandom 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 0.7The value 123 is a “seed” for the random number generator
26Influence of Random Variation System shows sustained oscillation over long time scales, with periodicity of ca. 10 yearsReference mode has been generated
27Policy 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 PlateauAlthough predator population averages 50, substantially higher numbers occur in some years, which could pose problem for ranchers livestockTest influence of allowing hunters to kill some predators to protect live stock
30Simulation Results With Selective Removal of Predators
31InterpretationResults suggest that it might have been possible to reduce peak values of predator population without destroying the stability of the predator-prey systemHowever, managers in early 1900s had essentially no knowledge of predator-prey dynamicsEven today, other factors besides predator-prey population dynamics are know to be important in governing response of the system…
32Current 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 oscillationMissed critical point that the cycle can only be understood by considering three trophic levels rather than just twoHare cycle is produced by interaction between predation and food suppliesDependence on food supply ripples across many species of predators and prey in boreal forest