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A further simple modification of Equation 5.12 allows us to incorporate a range of types of competition, as follows (Maynard Smith & Slatkin, 1973 ; Bellows, 1981):

(5.18)

We can see how this works from Figure 5.22, which plots k against log N_t, as in Figure 5.17, but now k is log₁₀[1 + (aN_t ) ^b ]. The slope of the curve, instead of approaching 1 as it did previously, now approaches the value taken by b in Equation 5.18. Thus, by the choice of appropriate values, the model can portray undercompensation (b <1), perfect compensation (b = 1), scramble‐like overcompensation (b > 1) or even density independence (b = 0). This model has the generality that Equation 5.12 lacks, with the value of b determining the type of density dependence that is being incorporated.

Figure 5.22 The intraspecific competition inherent in Equation 5.19. The final slope is equal to the value of b in the equation.

dynamic patterns: R and b

Equation 5.18 also shares with other good models an ability to throw fresh light on the real world. Analysing the population dynamics generated by the equation, we can draw guarded conclusions about the dynamics of natural populations. The mathematical method by which this and similar equations can be examined is described by May (1975a), but the results of the analysis (Figure 5.23) can be appreciated without dwelling on the analysis itself. Figure 5.23a sets out the conditions under which we get the various patterns of population growth and dynamics that Equation 5.18 can generate. Figure 5.23b shows what these patterns are. Note first that the pattern of dynamics depends on two things: (i) b, the precise type of competition or density dependence; and (ii) R, the effective net reproductive rate (taking density‐independent mortality into account). By contrast, a determines not the pattern of fluctuation, but only the level about which any fluctuations occur.

Figure 5.23 The range of population fluctuations generated by Equation5.19. (a) Reflecting the various possible combinations of b and R. (b) The patterns of those fluctuations.

Source: After May (1975a) and Bellows (1981).

As Figure 5.23a shows, low values of b and/or R lead to populations that approach their equilibrium size without fluctuating at all (‘monotonic damping’). This has already been hinted at in Figure 5.20a. There, a population behaving in conformity with Equation 5.12 approached equilibrium directly, irrespective of the value of R. Equation 5.12 is a special case of Equation 5.18 in which b = 1 (perfect compensation). Figure 5.23a confirms that for b = 1, monotonic damping is the rule whatever the effective net reproductive rate.

As the values of b and/or R increase, the behaviour of the population changes first to damped oscillations gradually approaching equilibrium, and then to ‘stable limit cycles’ in which the population fluctuates around an equilibrium level, revisiting the same two, four or even more points time and time again. Finally, with large values of b and R, the population fluctuates in an apparently irregular and chaotic fashion.