The general R-function for a population embedded in a web of interactions with other components of its environment is

R₁ = f(N₁, N₂, N₃, …, N_d; X₁, X₂, X₃, …, X_j)

where N₁ is the density of the subject population, N₂, N₃, ..... N_di are the densities of reactive elements (usually living organisms), and X₁, X₂, ..... X_j of passive elements (usually physical factors), in the environment. If only passive factors are present, then we know that the R-function will be dominated by first order feedback specified by the general equation

Time delays

When reactive factors are present in the environment, the equation above is no longer valid because the 4^th principle can be evoked. In this case we will have a system of d equations, one for each reactive element in the environment. However, it can be shown that this system can be reduced to a single equation of order d. In other words, the presence of circular causal pathways causes higher order feedbacks which can be interpreted, with some caution, as time delays in the feedback structure. These time delays can be represented in the R-function by

where d is the maximum delay, usually created by the longest feedback loop, or the feedback involving the most reactive factors. The time delay parameter d is sometimes called the dimension of the system. If this function has the same basic form as the logistic, then we have a multiple-delay nonlinear logistic R-function

According to the principle of limiting factors, however, only one feedback is likely to dominate the dynamics of a population close to equilibrium, and so we obtain the single-delay logistic

where d is now the delay created by the dominant feedback, or the dominant dimension. The general stability criteria for the time-delayed logistic equation are:

1. Smooth approach to equilibrium when AQd < 1.
2. Damped-stable approach to equilibrium when 1 < AQd < 2.
3. Periodic cycles around equilibrium when 2 < AQd < 2.692.
4. Aperiodic chaos when AQd > 2.692.

Remember that there are other hypotheses, such as the maternal effect, for time delays and resultant cycles in animal populations.

Two equilibria

When the 2^nd principle dominates the R-function over certain ranges of population density, unstable thresholds can be created (see figure). However, it may be difficult to construct models for R-functions with unstable thresholds because natural populations are almost never observed near to such equilibria, for they quickly move away from these points. Thus, data are almost never available for validating or fitting R-functions to regions where cooperation dominates. On the other hand, populations spend a great deal of time near to stabilizing equilibria and so data are frequently available for validating or fitting R-functions describing ^-feedback due to competition and trophic relationships. Hence, stabilizing R-functions are usually used to describe real populations and, when more than one stabilizing process is known to operate, separate R-function can be used to describe each process. The location of the unstable thresholds which separate stabilizing basins of attraction are determined by whatever means possible, including guesswork. The resulting logistic model for a system with two stabilizing R-functions is

where the subscripts J and K identify the parameters associated with the low-density and high-density R-functions, respectively, and E is the escape threshold (see figure). Note that each R-function can have a different time delay.

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