Outline Oscillation behavior model Examples Programming in STELLA
Oscillation: the basics Two interdependent reservoirs – Predator and prey, “Consumer” and “Resource” The resource enables the consumer to grow Oscillation around equilibrium values Negative feedback: the system counteracts positions away from the equilibrium value
The math: Renewable resource and consumers whose growth is dependent upon the resources RATE EQUATIONS dC(t)/dt = G x R(t) – D – G = growth rate of consumer, D = death rate of consumer, R(t) = resources at time = t dR(t)/dt = W – Q x C(t) – W = resource growth per unit time, Q = resource consumption rate, C(t) = consumers at time = t
Steady state Occurs at some equilibrium value around which R(t) and C(t) oscillate For consumers, this is W/Q For resources, this is given by D/G Period and amplitude of wave given by D, G, W, Q, R(t=0) and C(t=0) The system will run at steady state only if both reservoirs begin at their respective equilibrium values
Examples Predator-prey systems Herbivore-plant systems Parasite-host systems Any two populations that are linked by consumers and renewable resources Can you think of social or economic systems that may exhibit this behavior?
Predator-prey systems intro Predation, a "+/-" interaction, includes predator-prey, herbivore-plant, and parasite-host interactions – These linkages are the prime movers of energy through food chains and are an important factor in the ecology of populations, determining mortality of prey and birth of new predators Mathematical models and logic suggests that a coupled system of predator and prey should cycle: predators increase when prey are abundant, prey are driven to low numbers by predation, the predators decline, and the prey recover, ad infinitum Some simple systems do cycle, particularly those of the boreal forest and tundra, although this no longer seems the rule In complex systems, alternative prey and multi-way interactions probably dampen simple predator-prey cycles
from Odum, Fundamentals of Ecology, Saunders, 1953 The Lotka-Volterra Model
Questions On average, what was the period of oscillation of the lynx population? On average, what was the period of oscillation of the hare population? On average, do the peaks of the predator population match or slightly precede or slightly lag those of the prey population? – If they don't match, by how much do they differ? – Measure the difference, if any, as a fraction of the average period
Interpretation Strong counteracting (negative) feedback loop that forces the system to oscillate around an equilibrium value The further one reservoir is from the equilibrium value, the more the system works to counteract the perturbation Think about growth rates and how these may affect the shape of the oscillations – Predators may ‘regenerate’ slower, and therefore…
Changes… Predator-prey systems are potentially unstable, as is seen in the lab where predators often extinguish their prey, and then starve In nature, at least three factors are likely to promote stability and coexistence – Due to spatial heterogeneity in the environment, some prey are likely to persist in local "pockets" where they escape detection. Once predators decline, they prey can fuel a new round of population increase – Prey evolve behaviors, armor, and other defenses that reduce their vulnerability to predators – Alternative prey may provide a kind of refuge, because once a prey population becomes rare, predators may learn to search for a different prey species
Stabilization of predator-prey systems in nature Observing that frequent additions of paramecium produced predator-prey cycles in a test-tube led to the idea that in a physically heterogeneous world, there would always be some pockets of prey that predators happened not to find and eliminate Perhaps when the predator population declined, having largely run out of prey, these remaining few could set off a prey rebound. Spatial heterogeneity in the environment might have a stabilizing effect A laboratory experiment using a complex laboratory system supports this explanation. A predaceous mite feeds on an herbivorous mite, which feeds on oranges. A complex laboratory system completed four classic cycles, before collapsing.
Prickly pear cactus Observations of prickly pear cactus and the cactus moth in Australia support this lab experiment. This South American cactus became a widespread nuisance in Australia, making large areas of farmland unusable. When the moth, which feeds on this cactus, was introduced, it rapidly brought the cactus under control. Some years later both moth and cactus were rare, and it is unlikely that the casual observer would ever think that the moth had accomplished this. Once the cactus became sufficiently rare, the moths were also rare, and unable to find and eliminate every last plant. Inadequate dispersal is perhaps the only factor that keeps the cactus moth from completely exterminating its principal food source, the prickly pear cactus.