Logistic Equation: Growth Population Dynamics

Logistic Equation Lab Script: Growth Population Dynamics. An Interdisciplinary Lab Script for Math, Physics and Non Linear Dynamic Courses.

Instructions

To the Instructor

• Discuss with students all theory relevant to the experiment.
• Feel free to modify the experiment title, description and any required observation.
• Use conditions and results to perform any graphical analysis or additional computations.

Theory

The growth population dynamics of a colony of a single species in a controlled environment (a lab) can be modeled by ecologists in terms of the Logistic equation (1 - 4),

P_{n + 1} = cP_n(1 - P_n)

where

P_n = percentage of a limiting population L that is alive at generation n. By definition, 0 < P < 1.
P_{n + 1} percentage of the limiting population L that is alive at another generation n + 1
c = growth population constant; depends upon several parameters such as amount of food, temperature, and the like. It is usually defined by the ecologist between 0 < 4.

This is an idealized model for growth population without external influences. The species under investigation could be a colony of formosan subterranean termites, potatoe weevils, or fire ants -the exact nature of the colony is irrelevant.

The limiting population that is alive (L) is governed, for example, by the physical size of the laboratory or colony in which the species is confined; After all, there can be no more of the species alive than can physically fit into the laboratory (1 - 4). Thus if P_n = 1/2 = 0.5, the exact population is P_nL = L/2 at generation n. If there is no species present (initial population P₀ = 0) or if the lab is completely full (P₀ = 1), then there is no species present during the ensuing generations. Furthermore, if P₀ is small, the population tends to increase, whereas if P₀ is large, the population tends to decrease, as expected. From the numerical dynamics standpoint the exact value of L is inmaterial; what is important is the fraction of survival species, P_n, and how this fraction changes between generations.

However, assuming that the ecologist can count accurately the population of the species during each generation, can he or she use this knowledge to predict in advance the population of future generations? How small changes in the initial conditions (small changes in growth rates and initial population values) affect the outcome of the experiment (prediction of future populations)?

Enters Chaos Theory. The study where a dynamical behavior changes as a condition is varied is called bifurcation theory. Chaos Theory pretends to describe why small changes on initial conditions (e.g., P = 0.50, P = 0.51; c = 3.70, c = 3.71) produce unpredictable behaviors. Understanding when and why these unpredictable behaviors occur should enable an ecologist to develop better growth population models or experimental conditions suitable for replication and modeling. Under chaotic conditions growth population (or the outcome of any experiment, for that matter) cannot be predicted (1 - 4).

Notes

• An attractor is an unique outcome for disimilar experimental conditions.
• A fixed point is a point that doesn't change under iteration.
• An orbit is the numerical sequence of an iterated point.
• An attracting fixed point is a point that attracts orbits.
• A repelling fixed point is a point that repels orbits.
• Attracting and repelling periodic points are cyclical points present in an orbit.
• A basin of attraction consists of all points whose orbits tend to a given attracting fixed point.
• Chaos is a condition in which an iterated orbit never settles down.
• Unpredictable behaviors upon small changes on initial conditions is called sensitive dependency. It is the signature of chaotic systems.

Possible Experiments

1. Math, Physics and Non Linear Dynamics: Numerical Analysis of Attractors and Basins of Attractions for the Logistic Equation.
2. Biology, Pest Integrated Management and Environmental: Growth Population Models for Formosan Subterranean Termites, Cockroaches and Fire Ants.

Suggested Exercises

Find similarities and differences for three different growth population models in which c = 3, L = 30,000, n = 50 and

1. P₀ is 0, 0.5, and 1, respectively.
2. P₀ is any three values between 0 < P₀ < 1.
3. P₀ < 0, P₀ > 1 or P₀ = 2c? In each case, what is the enviromental significance of the result?

An ecologist is interested in studying a single species present in a residential neighborhood whose growth population is known to oscillate every 4 generations. The species affect other species and the neighborhood real estate property value. He examines four possible growth population models using P₀ = 0.5, L = 20,000, and n = 100 at three different growth rates; i.e., c = 1.5, c = 3.2 and c = 3.5

1. Compare the growth population dynamics in each case.
2. Which of the three is a suitable working model and why?
3. Suggest a working model that experience chaos, thus is not suitable for modeling the ecologist's case study.

References

1. An Introduction to Chaotic Dynamical Systems; Robert L. Devaney, Addison, 1988.
2. Chaos and Fractals; Robert L. Devaney and Linda Keen, Editors, American Mathematical Society, 1989.
3. Chaos, Fractals and Dynamics; Robert L. Devaney, Addison, 1990.
4. A First Course in Chaotic Dynamical Systems; Robert L. Devaney, Addison, 1992.

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