## Equations for modelling density dependence

The essence of density-dependent mechanisms for models of population dynamics is that, as the density increases, there is an alteration in the

Fig. 5.2 Detection of density dependence by linear regression. In this example the population change in common sardine (Strangomera bentincki) from one year to the next (Rt = log Xt - log Xt-1) is regressed against its density in year t - 1 (Pedraza-Garcia & Cubillos 2008). Note that the use of t and t - 1 could be replaced by t + 1 and t.

Fig. 5.2 Detection of density dependence by linear regression. In this example the population change in common sardine (Strangomera bentincki) from one year to the next (Rt = log Xt - log Xt-1) is regressed against its density in year t - 1 (Pedraza-Garcia & Cubillos 2008). Note that the use of t and t - 1 could be replaced by t + 1 and t.

Fig. 5.3 Exponential and linear decline in fraction surviving (s) with density (N). Notice the change in shape with increasing values of a (see text for details). The intercept of the exponentially declining function can be easily altered (dashed line).

fecundity or fraction of individuals surviving and therefore a change in X, the finite rate of population change. In open systems we can invoke density-dependent changes in migration, but these are not considered in this chapter. In Fig. 5.3 a set of lines have been drawn, illustrating some possible density-dependent functions (the vertical axis is labelled s to indicate the fraction surviving). In order to include these in population models we need to describe them in mathematical terms. For example, the linear decline in s with increasing density is expressed as the equation of a straight line:

where c is the intercept on the vertical (s) axis and m is the magnitude of the gradient of the line. c must lie between 0 and +1 as we are considering the fraction surviving. Equation 5.1 tells us that at the lowest (non-zero) density the value of s is close to maximum (c) and that with increasing density the fraction of individuals surviving decreases linearly according to the gradient m. The linear reduction in s with N assumes that there is a density, Nmax, at which s = 0. In other words there is an upper (Nmax) and lower boundary (0) of possible extinction. If a population goes above Nmax it will become extinct. This upper boundary clearly creates some problems for modelling purposes. To overcome this we will consider a second mathematical function of exponential decline:

The change in s with increasing values of N for three different values of a (0.5, 0.1 and 0.01) is shown in Fig. 5.3 (recall that equation 5.2 can be natural-log transformed to give ln(s) = -aN, showing that the natural log of s declines linearly with N). The parameter a can be thought of as denoting the strength of density dependence. At any given value of N, the fraction surviving will decrease as a increases. When N = 0, s will be equal to 1, regardless of the value of a; that is, the model is designed so that at very low densities s tends towards a maximum value. Conversely, at very high densities, the value of s tends towards 0 but never reaches it (an example of an asymptote) unlike the linear density dependence. The function can be altered to s = be~aN to give a maximum value different from 1 at N = 0.

Some of the different curves in Fig. 5.2 can be considered to represent different types of intraspecific competition. An important distinction is between scramble and contest competition (Hassell 1975). In pure scramble competition, resources are divided equally among competing individuals. The consequence of this is that above a certain density the mean resource per individual is too low for survival, and therefore s plummets to zero. The upper boundary in the discrete logistic model (Section 5.4) could be interpreted as representing perfect scramble competition. The other extreme is contest competition, in which the superior competitors monopolize the resource. Consequently, a certain number of individuals always survive, even at high densities. In this case, s would approach zero at high densities, but never reach it, in agreement with the exponentially declining curve.

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