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Density dependence may operate at one or more points in the life cycle, which is especially relevant when dealing with species with more complex life cycles For example, plant species may experience density dependence at the seed, seedling and reproductive stages Density dependence can operate among seeds if there is a limited number of microsites for germination and only one seed can germinate per microsite If this is the case, as the density of seeds increases there will be a decreasing fraction that can germinate Similarly, at the other end of the life cycle, if plants are insect pollinated and there is a xed number of pollinators, higher densities of owers may result in a decreased fraction of insect visits In this case, there may be the opposite effect of enhanced visits to higher-density clumps, although the overall visitation decline will occur at some point of overall ower density.

The two-point limit cycle is also stable; for a given value of (within the range of values giving twopoint cycles) the population will always settle out to uctuate between the same two densities so that there are now two stable equilibriums In contrast, the chaotic dynamics do not have this property Here, the particular sequence of values is dependent upon initial conditions, although the size of the uctuations will be determined by the values of and K The possibility of chaotic dynamics means that if random or unpredictable dynamics are recorded this does not necessarily imply that the underlying mechanisms are random (stochastic) Some or all of the randomness could be produced by predictable deterministic processes expressed as chaos Thus, if population change is described by the discrete logistic equation each population size at t + 1 (Nt+1) is given by a particular value of Nt.

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We can see this clearly by plotting Nt+1 against Nt (Fig 55a) using the parameter values for and K of 4 and 100 (Fig 54e) The chaotic system shows the mathematical relationship of the discrete logistic: a quadratic equation A t through the points gives (as expected) a perfect t indicated by the r2 of 1 The coef cients of 004 and +4 agree with equation 55 ( /K for Nt2 and for Nt) This can be compared with a truly random sequence of values where Nt+1 plotted against Nt is a cloud of points (Fig 55b) The challenge of detecting chaos in real population dynamics is therefore to distinguish it from random events The rst study to try to detect chaos in laboratory and eld populations was by Hassell et al (1976).

where y is a dependent variable that is in uenced by x, the independent variable, and the random error is e, then regression analysis can be employed to estimate the parameters a and b. The most common method of estimation for regression analysis is ordinary least squares (OLS), which can be used given that the following assumptions hold: 1. The regression model is speci ed correctly; that is, in the above case y is a linear function of x. 2. e has a zero mean, a constant variance for all observations, and is uncorrelated between observations. 3. x is given, not correlated with the error term, and in the case where there is more than one independent variable, the independent variables are not highly correlated with each other. 4. There are more observations than the number of parameters being estimated. These assumptions basically state that the parameters can be estimated from data, and that all the available information is used. These assumptions

They used the technique of assuming an underlying mathematical model (described by the equation Nt+1 = Nt(1 + aNt) b discussed above) and determining the values of , a and b for different populations of insects They were then able to compare these values with those known to produce limit cycles and chaos (Fig 56) So Hassell and colleagues were testing whether the model that is tted to the data has parameter values which would give chaos The parameter values of b and for each species were superimposed on the regions of different dynamic behaviour predicted by the model; for example, stable equilibrium, limit cycles and chaos Only one species had values consistent with chaotic dynamics and one consistent with limit cycles All the other populations were in the stableequilibrium region.

Some studies have actively sought to investigate density dependence at different stages (Gillman et al 1993) An example of the inclusion of density dependence in a stage-structured model is that of de Kroon et al (1987) They investigated the effects of mowing as a management regime on the perennial rosette-forming Hypochaeris radicata Density dependence was incorporated at both germination and seedling establishment, which were in turn functions of gaps in the vegetation Their stage-structured model had four stages (seeds and three stages of rosette, the latter two of which were further divided into owering and non- owering; Fig 510a) The time steps were between seasons rather than years.

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