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Unformatted text preview: 4.5. LINEARIZATION AND DIFFERENCE EQUATIONS 545 4.5 Linearization and Difference Equations As we have seen in earlier chapters, difference equations x n +1 = f ( x n ) are useful for describing biological dynamics of varying complexity. The simplest dynamics occur at an equilibrium because by definition these are the solutions of the difference equation that remain constant for all time. Specifically if a given first value x satisfies x = f ( x ) then our difference equation implies x 1 = x . Repeated application results in x n +1 = x n = ··· = x for all n > 0. While equilibria may be easily to identified by solving the equation x = f ( x ), their biological relevance depends on their stability . Many biological systems, when perturbed, naturally return to their the equilibrium state around which they operate. The temperature of our bodies is a case in point. If our temperature is perturbed because of an infection, it returns to its equilibrium value of 98.6 ◦ F once we are well again. Not all equilibria, however, are stable. If we stand up a 6 month old child, it may stay upright for a second or two, but until the child is around a year old it will soon fall over. The reason for this is that standing vertically, without feedback control from our muscles constantly moving us to correct our tendency to fall over, is an unstable situation. Thus, when a biological system is perturbed away from equilibrium, it may do one of two things. First, it may return to the equilibrium state, in which case the equilibrium is considered stable . Alternatively, even if the perturbation is small, the system may continue to drift away from the equilibrium. In this case, the equilibrium is unstable . In this section, we make the notion of stability precise and provide a simple algebraic method for checking stability. This method relies on linearizing the difference equation near the equilibrium. These ideas and methods are applied to models of population growth and population genetics. We conclude the section by considering another application of linearization and difference equations. Namely, numerically solving a non-linear equation. This numerical method is a commonly employed alternative to the bisection method presented in Example 9 of Section 2.3. Equilibrium stability We begin with the following example which motivates the notion of a stable equilibrium. Example 1. Logistic equation In Example 7 of Section 2.5, we introduced the discrete logistic equation which is a simple model of population growth. If x n denotes the population density ( e.g. average number of individuals per acre) in the n-th generation, then the model is given by x n +1 = x n + rx n (1 − x n /K ) x specified, © 2010 Schreiber, Smith & Getz 546 4.5. LINEARIZATION AND DIFFERENCE EQUATIONS where r is the per-capita growth rate at low densities and K is the environmental carrying capacity of the population....
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- Fall '10
- Population Ecology, difference equations, Stability theory, Smith & Getz, ©2010 Schreiber