Population Dynamics: Modeling the Growth of Biological Species, Study notes of Electrical and Electronics Engineering

Download Population Dynamics: Modeling the Growth of Biological Species and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! 1 November 21, 2003 Start Presentation Population Dynamics I • Today, we shall look at the problem of modeling population dynamics, i.e., determining the sizes of populations of biological species as functions of time. • Such systems are modeled as pure mass flows, i.e., energy conservation laws are not being considered. • Consequently, bond graphs are not suitable for describing these types of models. November 21, 2003 Start Presentation Table of Contents • Limitations of bond graphs • Exponential growth • Limits to growth • Logistic function • Continuous-time vs. discrete-time • Chain letter • U.S. census • Curve fitting • Predator-prey models • Larch bud moth • Competition and cooperation 2 November 21, 2003 Start Presentation Limitations of Bond Graphs I • Bond graphs have been designed around the conservation principles of physics (energy conservation, mass conservation), and are therefore only suitable for the description of physical systems. • Chemistry was a border-line case. Although it is possible to model chemical reaction dynamics down to the level of physics, this is not truly necessary, since the reaction rate equations are decoupled from the energy balance equations. Hence this is rarely done. We did it, because the bond-graphic interpretation of chemical reactions offered additional insight that we could not have gained easily by other means. • Yet, as the complexity of molecules grows, especially in organic chemistry, it becomes more and more difficult to know what the elementary step reactions are, and at that level, chemistry becomes an empirical science, the knowledge of which is essentially covered by interpretations of observations alone. November 21, 2003 Start Presentation Limitations of Bond Graphs II • As we proceed to systems of ever increasing complexity, such as in biochemistry, the situation becomes truly hopeless. • Although we live in a physical universe, and although a majority of scientists would agree that the laws of nature are ultimately laws of physics, we lack the detailed understanding necessary to e.g. explain the processes of mitosis and meiosis (cell division) on the basis of the underlying physics, or worse, to explain how the genetic code directs the cells to reproduce a functioning living being from its blueprint. • With this lecture, we are taking a giant step, bypassing organic chemistry, biochemistry, molecular biology, cell biology, genetics, etc., jumping right to the level of population dynamics, i.e., taking a macroscopic look at how populations of species develop in size over time. 5 November 21, 2003 Start Presentation Continuous-time vs. Discrete-time Model • Applying the forward-Euler integration algorithm: • to the differential equation describing the population change: • we get a difference equation: • It may be justified to use this much cruder model, either because the accuracy of our model is not all that great anyway, or because a population may reproduce only in spring (h = 1.0). P = k · P· P(t+h) = P(t) + h · P(t)· P(t+h) = P(t) + h · k · P(t) = (1.0 + h·k) · P(t) November 21, 2003 Start Presentation The Chain Letter I • Population dynamics modeling techniques may also be applied to macroeconomic modeling. Let us consider the model of a chain letter. • The following rules are set to govern this (artificial) model: A chain letter is received with two addresses on it, the address of the sender, and the address of the sender’s sender. After receiving the letter, a recipient sends $1 to the sender’s sender. He or she then sends the letter on to 10 other people, again with two addresses, his (or her) own as the new sender, and the sender’s address as the new sender’s sender. The letter is only mailed within the U.S. Every recipient answers the letter exactly once. When a recipient receives the same letter for a second or subsequent time, he (or she) simply throws it away. 6 November 21, 2003 Start Presentation The Chain Letter II • Special rules are needed to provide initial conditions. • Every sender has 100 receiver’s receivers, thus is expected to make $100. • Except for the first 11, who don’t pay anything, every sender pays exactly $1. • Hence this is a wonderful (and totally illegal!) way of making money out of thin air. The originator sends the letter to 10 people without sending money to anyone. If a recipient receives the letter with only one address (the sender’s address), he or she sends the letter on to 10 other people with two addresses (his or her own as the sender, and that of the originator as the sender’s sender). No money is paid to anyone in this case. November 21, 2003 Start Presentation The Chain Letter III • We can model the chain letter easily as a discrete system. I = 10 · (1.0 - )PPmax I is the average number of new infections per recipient. R = I · pre(R) R, the number of new recipients, can be computed as the number of new infections per recipient multiplied by the number of recipients one step earlier. P = pre(P) + R P, the number of already infected people, can be computed as the number of people infected previously plus the new recruits. 7 November 21, 2003 Start Presentation The Chain Letter IV • We can easily code this model in Modelica. November 21, 2003 Start Presentation Simulation Results Initially, every participant makes exactly $99 as expected. However, already after seven generations, the entire U.S. population has been infected. Thereafter, everyone who still participates, loses $1. The energy conservation laws are not violated! No money is being made out of thin air! Those who participate early on, make money at the expense of the many who jump on the band wagon too late. 10 November 21, 2003 Start Presentation U.S. Census III Pmax = 402.59 · 106 Inflection point = 1971 The inflection point is fairly sensitive. Yet, however we compute it, we have already passed it. ⇒ We can no longer rely on an increasing number of children to pay for our retirement benefits. November 21, 2003 Start Presentation Curve Fitting I • Let us look how the curve fitting was done. Since we only have measurement data for the population itself, not for its derivative, we first need to approximate the population gradient. • To this end, we lay a quadratic polynomial through three neighboring population data points: Pi-1 = c1 + c2 · t i-1 + c3 · t i-12 Pi = c1 + c2 · t i + c3 · t i 2 Pi+1 = c1 + c2 · t i+1 + c3 · t i+12 11 November 21, 2003 Start Presentation Curve Fitting II • In a matrix-vector form: Pi-1 t i-10 t i-11 t i-12 c1 Pi = t i 0 t i 1 t i 2 · c2 Pi+1 t i+10 t i+11 t i+12 c3} V = Vandermonde matrix p = V · c ⇒ c = V -1 · p = V \ p Matlab notation November 21, 2003 Start Presentation Curve Fitting III • Now, that we have the coefficient vector, we can approximate the population gradient: Pi = c1 + c2 · t i + c3 · t i 2 ⇒ Pi ≅ c2 + 2c3 · t i· • We could equally well have used other interpolation polynomials, such as cubic splines, or inverse Hermite interpolation. 12 November 21, 2003 Start Presentation Curve Fitting IV • We are now ready to curve-fit the logistic model: • We have n equations in the two unknowns a and b. P1 ≅ a · P1 + b · P12 · P2 ≅ a · P2 + b · P22 · Pn ≅ a · Pn + b · Pn2 · ··· We can solve for a and b only in a least-square sense.⇒ November 21, 2003 Start Presentation Curve Fitting V • In a matrix-vector form: P1 P1 P12 a · ··· P2 P2 P22 b · Pn Pn Pn2 · ··· ≅ · } V = Vandermonde matrix 15 November 21, 2003 Start Presentation The Larch Bud Moth I • The larch bud moth is an insect that lives in the upper Engiadina Valley of Southeastern Switzerland, at altitudes between 1600 – 2000 m. • Its larvae feed on the needles of the larch trees. The population has a cycle time of exactly nine years, i.e., once every nine years, the insect population is larger by several orders of magnitude, and all the larch trees turn brown because of them. • Hence the larch bud moth population was curve-fitted to the predator population of a Lotka-Volterra model. November 21, 2003 Start Presentation The Larch Bud Moth II • The curve fit is excellent indeed. Does this mean that we now understand the population dynamics of the larch bud moth? Unfortunately, the answer to this question is a decided no. 16 November 21, 2003 Start Presentation The Larch Bud Moth III • The larch bud moth is also plagued by parasites. Thus, if the insect population is large, the chances of spreading the parasites among them grows drastically. • Thus, it may make equally much sense to curve-fit the larch bud moth population to the prey population of a Lotka- Volterra model. • This was attempted as well. November 21, 2003 Start Presentation The Larch Bud Moth IV • The curve fit is equally excellent. Thus, we cannot conclude from the quality of the curve fit alone that the underlying model represents correctly the cause-effect relationship of the biological system. 17 November 21, 2003 Start Presentation The Dangers of Curve Fitting • Curve fitting can only be used for the purpose of interpolation in space and extrapolation in time (as long as the predicted variables stay within their observed ranges). • Models obtained inductively by curve-fitting a mathematical model to a set of observed data should never be used to explain the internal variables of the model. • Such a model has no internal validity. • A better (internally valid) larch bud moth model shall be presented later. November 21, 2003 Start Presentation Competition and Cooperation I • Two species can also interact with each other in other ways. • They can e.g. compete for the same food source: • or they can cooperate, e.g. in a symbiosis: x1 = a · x1 − b · x1 · x2··x2 = c · x2 − d · x1 · x2 x1 = -a · x1 + b · x1 · x2··x2 = -c · x2 + d · x1 · x2