Let us return to the annual plant model from Chapter 2 and consider how density dependence may be incorporated into that density-mdependent model. Imagine that competition for space occurs between juvenile and adult plants: that there is intraspecific competition. The effect of competition on individual survival is expected to increase as plant density increases. In years of low plant density there will be relatively high survival but increasing density will result in reduced survival. We have seen that the simplest model for density dependence is a linear change represented by s = -mN + c (equation 5.1). In the following example we will use d as the density of individuals subjected to density dependence. In the density-independent model we assumed that an average of 0.2 plants survived after germination up to seed set. In the new density-dependent model, 0.2 can be taken as the value of c; that is, when the effects of density are negligible. Therefore s is close to c at the lowest population densities and, with any increase in density, s declines linearly according to the gradient m. The linear reduction assumes that there is a density, dmax, at which s = 0, so that m = 0.2/ dmax (i.e. c/dmax). Therefore the linear density dependence equation can be written as s = 0.2 - (0.2/dmax)d or s = 0.2(1 - d/dmax). A new model, incorporating density dependence, now replaces the old density-independent model:

Number of germinating seed next year (Nt+1) = number of seed germinating this year (Nt) x fraction surviving to seed set (0.2(1 - d/dmax)) x average number of seeds produced (100) x fraction surviving over winter (0.1)

If the fecundity and survival values are combined, as before, into the single value, X (=0.2 x 100 x 0.1), we produce the equation:

In this case the interpretation of density dependence is that the fraction of germinating seed that survives is reduced by increasing density, which could be caused by intraspecific competition. In year t, d will be equal to 0.2Nt. If 0.2/dmax is replaced by 1/K we obtain:

Equation 5.4, incorporating density dependence, is known as the discrete logistic equation and represents a strategic model for the population dynamics of annual species. K is the carrying capacity, defined as the maximum number of individuals a habitat can support. Equation 5.4 is sometimes written as Nt+i = XNt(1 - aNt); that is, replacing 1/K by a. Berryman (1992) and Elliott (1994) review the use of the discrete logistic and similar equations whereas May et al. (1974) and May (1981) discuss the density-dependent terms.

Modelling of intraspecific competition has lead to a variety of equations incorporating density dependence. The model of Hassell (1975), described by the equation Nt+1 = XNt(1 + aNt)-b, provides parameters a and b which can describe change from contest to scramble competition. a gives the threshold

density at which density dependence occurs and b is the strength of the density dependence. This model, derived from earlier studies of insects (Morris 1959, Varley & Gradwell 1960), was related to models of fisheries (N+i = XNt(1 + aNt)-1 (Beverton & Holt 1957). In turn, the model of Hassell was developed by Watkinson (1980) to describe the population dynamics of annual plants:

where a and b are the parameters of the Hassell model, w is the degree of self thinning and X is the finite rate of population change.

We will now consider some of the properties of equation 5.4 and compare them with the density-independent equations in Chapter 2. If we multiply out the right-hand side of equation 5.4 we see an important attribute of the density-dependent equation:

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