Lecture 7

1. Lecture 7 Bsc 417

2. Outline • Oscillation behavior model • Examples • Programming in STELLA

3. 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

4. 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

5. 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

6. 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?

7. 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

8. from Odum, Fundamentals of Ecology, Saunders, 1953 The Lotka-Volterra Model

9. 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

10. 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…

11. 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

12. 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. 

14. 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.