Math 19. Lecture 4Introduction to Differential EquationsT. JudsonFall 20051 Modeling a Population with a Carrying Capac-itySuppose that we are to analyze the population of fish in a lake and that weknow from past experience that the maximum number of fish that the lakewill support is 200 (thousand?) fish. How can we model this population?2 The Logistic EquationThe logistic equation is commonly used to m odel population growth in aresource limited environment. It was first used by the Belgian biologistVerhulst to p redict the populations of Belgium and France. We can ad-just the model of exponential growth , dp/dt = kp, to account for limitedenvironment and limited resources. We make the f ollowing assumptions.• If th e population is small, the rate of growth of the population isproportional to its size.dpdt= kp if p is small.• If the population is too large to be supported by its environment andresources (N = carrying capacity), the popu lation will decrease. Ifp > N, thendpdt< 0.1• The correct equation might be of the formdpdt= kh(p)pThe function h(p) should be close to 1 if p is small, but negative ifp > N. The simp lest expression that has these properties ish(p) =1 −pN.• The logistic population model isdpdt= k1 −pNp ordpdt= kp −1Np2.To get the equ ation in the book, let b = 1/N .• This equation is a mathematical model. It can only be verified byexperiment!3 dfielddfield is the program th at will solve differential equations. A java versionsof dfield can be f ound at http://math.rice.edu/~dfield/4 Some Vocabulary• An equ i lib rium solution is constant for all values of the independentvariable. T he graph is a horizontal line. Equilibrium values can id en -tified by setting the derivative of the function equal to zero.• An equilibrium solution is stable if a small change in the initial con-ditions gives a solution which tends toward the equilibrium as theindependent variable tends towards positive infinity.• An equilibr ium solution is unstable if a small change in the initialconditions gives a solution which veers away from th e equilibrium asthe independent variable tends towards positive infinity.25 The Effects of HarvestingNow look at effect of different levels of fishing on a fish population. If fishingtakes place at a continuous rate of H fish per year, the fish population Psatisfies the differential equationdPdt= 2P − 0.01P2− H.6 Some Important Points• The qualitative beh avior of solutions p(t) to the logistic, and to thegeneric equationdpdt= f(p),can be obtained from information in the graph of the function f .• An equation of the formdpdt= f(p),is completely predictive. Choose any starting value for p and thereis precisely one solution th at starts at your chosen starting time withyour chosen starting value.• The points w here th e graph crosses the p-axis correspond to the con-stant solutions to the differential equation. Meanwhile, if p(t) is atime dependent solution and f(p(t)) > 0, then p(t) moves to the righton the p-axis as time increases. Conversely, if f(p(t)) < 0, then p(t)moves to the left.7 Global TemperatureLet T = T (t) be the average temperature of the earth’s surface at time t.This can be modeled bydTdt= fin− fout.• finis the contribution from incoming solar energy. finincreases withT since a greater fraction of the earth’s surface is covered with ice atlower temperatures. The ice reeflects sunlight and reduces the radia-tion reaching the surface of the earth.3• foutis the contribution from the outgoing radiation. foutincreaseswith T since a hot body radiates more than a cool one.Homework• Chapter 3. Exercises 1, 2, 3, 4; pp. 62–63.Readings and References• C. Taubes. Modeling Diffe rential Equations in Biology. Prentice Hall,Upper S addle River, NJ, 2001. Chapter
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