Introduction to the Modeling and Analysis of Complex Systems

108 CHAPTER 6. CONTINUOUS-TIME MODELS I: MODELINGExercise 6.6 There are many other more sophisticated methods for the numericalintegration of differential equations, such as the backward Euler method,Heun’s method, the Runge-Kutta methods, etc. Investigate some of those methodsto see how they work and why their results are better than that of the Euler forwardmethod.6.5 Building Your Own Model EquationPrinciples and best practices of building your own equations for a continuous-time modelare very much the same as those we discussed for discrete-time models in Sections 4.5and 4.6. The only difference is that, in differential equations, you need to describe timederivatives, i.e., instantaneous rates of change of the system’s state variables, instead oftheir actual values in the next time step.Here are some modeling exercises. The first one is exactly the same as the one inSection 4.6, except that you are now writing the model in differential equations, so thatyou can see the differences between the two kinds of models. The other two are on newtopics, which are relevant to chemistry and social sciences. Work on these exercises toget some experience writing continuous-time models!Exercise 6.7 Develop a continuous-time mathematical model of two speciescompeting for the same resource, and simulate its behavior.Exercise 6.8 Imagine two chemical species, S and E, interacting in a test tube.Assume that E catalyzes the production of itself using S as a substrate in thefollowing chemical reaction:S + E → 2E (6.31)Develop a continuous-time mathematical model that describes the temporalchanges of the concentration of S and E and simulate its behavior.