Regular oscillations of populations of snowshoe hare and its predator, the lynx, were recorded by pelt counts in Canada by the Hudson Bay Company from 1845 to 1935. As the hare population starts to grow exponentially, the population of lynx grows so fast that the numbers of hares are reduced again. With less to eat, the lynx population declines again. The traditional model and graph of this oscillating system are given in Figure III-6.

In the diagram E is a constant-pressure source of food for the prey. H is the quantity of prey and C quantity of predators. K1*E*H is the growth rate of the prey; K4*H*C is the rate of consumption of prey by predators; K5*C*H is the growth rate of predators; and K8*C is the death or emigration rate of the predators. DT is put in this program so the time intervals can be changed. When DT is 0.1 the change in quantities is calculated every tenth of a time unit. For further explanation, refer to Part V, "How to Scale the Graphs for Quantity and Time".

This is the standard textbook model, but with the prey given an outside energy source (E) rather than an intrinsic rate of increase. It is not very realistic in that there is no limit to the energy source for the prey, there is no recycle or feedback control from the predators back to the prey, and there is no death of the prey except for those eaten by the predators.

Another problem with this model is that the time between oscillations depends on the quantity of Q and H at the start. In real prey-predator systems, the time between oscillations depends more on relations between prey and predators and less on starting quantities.

A more realistic model for this system is given in Figure III-7. Here we replace E, a relatively unlimited energy source, with J, the steady renewable energy source of the sun and rain. Also added are a storage, P, of grasses and depreciation of the prey, K6*H.

Examples of Oscillating Prey-Predator Models

Oscillations are observed in arctic populations. When the plants are plentiful, the small herbivore mammals (lemmings) become abundant and eat the vegetation until it becomes scarce. After this the lemming population has to decrease until the plants regrow. Then the lemmings become abundant again. The predators of the lemmings, foxes and owls, oscillate with the lemming population. Thus, the populations of producers and two levels of consumers go up and down, following each other.

Similar oscillations are observed in host-parasite relationships and in other carnivore-herbivore relationships. Examples are spruce trees and spruce budworms in Canada, and week-to-week pulses of phytoplankton and zooplankton in the sea.

Supply and demand curves in economics sometimes show these oscillations. When a product (Q) is made, it increases on the market until consumers (H) buy it; then its supply goes down until more is produced.

"What if" Experimental Problems

1. In the model in Figure III-7 change to a system in a lower latitude where there is more sun: increase J to 40,000 kilocalories per square meter per year. What happens to the oscillations? Explain why. Remember the oscillations in this model are about every ten years.
   


2. Now move to higher latitude where the sun's energy is less: decrease J to 2,000. What happens to the oscillations? Why?
   


3. In Figure III-7 change the herbivores (H) to species that have shorter lives by increasing their depreciation: increase K6 to 0.3. What happens to the herbivores (H) and carnivores (C)? What happens to the oscillations? Run both graphs together (K6 = 0.05 and K6 = 0.3). Why is the first peak of herbivores higher when the depreciation is higher (K6 = 0.3) than when it is lower (K6 = 0.05)?

COMPUTER MINIMODELS AND SIMULATION EXERCISES FOR SCIENCE AND SOCIAL STUDIES

Howard T. Odum* and Elisabeth C. Odum+
* Dept. of Environmental Engineering Sciences, UF
+ Santa Fe Community College, Gainesville

Center for Environmental Policy, 424 Black Hall
University of Florida, Gainesville, FL, 32611
Copyright 1994

Autorização concedida gentilmente pelos autores para publicação na Internet
Laboratório de Engenharia Ecológica e Informática Aplicada - LEIA - Unicamp
Enrique Ortega
Mileine Furlanetti de Lima Zanghetin
Campinas, SP, 20 de julho de 2007