Two further kinds of developments will be mentioned briefly in the context of differential equations, though both can be taken further using difference equations. First, some processes involve time lags. May (1973) gives an example which is yet another modification of Lotka-Volterra equations for herbivore (H) and carnivore (P) populations:
(8.30) (8.31 )
In this case, there is a time delay on the effect of resource limi tation in the growth of the prey population. A characteristic tirre can be defined for the system as
which is the geometric mean of the tirres associated with birthrates and death-rates of the prey and predator populations respectively. Then the equilibrium points of Equations (8.30) and (8.31) is stable or unstable according to whether
'"[ < T unstable) The need for age disaggregation arises when there are
periodic fluctuations in the environment of a period considerably less than the life-time of a species in the system. A variety of models to meet these situations are presented by Oster and Takahashi (1974). They can be developed in a differente equation format and are probably less important than sorre of the
a number of striking results on complex dynamic behaviour associated with very simple difference equations. This provides us with a new perspective which turns out to have direct applications to some situations in urban geography. First, however, we comment generally on the distinctions in use between differential and difference equations both in ecology and more widely. If the main state vari ables are changing continuously, then differential equations are appropriate; if the events can be considered to be discrete, then difference equations offer the correct formul at ion. In ecology, if e lemen ts of popul ati ons typically have lives longer than the period of analysis and associated events, then differential equations are appropriate. But often, a species will all die out by the end of a single time period, leaving, say, eggs to create the next generation in the next time interval. These are known as non-overlapping generations. Difference equations are then most appropriate. It should al so be added that when differential equations are integrated numerically, they are in effect being treated as difference equations and some of the results described below are applicable.