Introduction to the Modeling and Analysis of Complex Systems

54 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGf(x)af(x) = –a – 1Kx + a10KxFigure 4.5: The simplest example of how the growth ratio a = f(x) should behave asa function of population size x.this case, the equation becomes x t = 0, so there is no growth. This makes perfect sense;if there are no organisms left, there should be no growth. Another extreme case: Whathappens when x t−1 = K? In this case, the equation becomes x t = x t−1 , i.e., the systemmaintains the same population size. This is the new convergent behavior you wanted toimplement, so this is also good news. Now you can check the behaviors of the new modelby computer simulations.Exercise 4.9 Simulate the behavior of the new population growth model for severaldifferent values of parameter a and initial condition x 0 to see what kind ofbehaviors are possible.(x t = − a − 1 )Kx t−1 + a x t−1 (4.23)For your information, the new model equation we have just derived above actuallyhas a particular name; it is called the logistic growth model in mathematical biology andseveral other disciplines. You can apply a parameter substitution r = a − 1 to make the