In order to understand the causes of change in populations of living organisms, we first need to know something about the general subject of dynamics. The dynamics of systems in general, from space-ships to ecosystems, are studied under the disciplines of general systems theory, dynamic systems theory, nonlinear dynamics, chaos theory and so on. From the more applied side we have what is called control theory, which engineers use to control the behavior of dynamic systems (auto-pilots, govenors, etc.). From these theories I have extracted the following important rules and tenets:
First, changes in a dynamic system can be caused by two major kinds of processes which we will call exogenous and endogenous.
Exogenous processes cause changes in a system of interest, say a population of animals, but are themselves unaffected by those changes. For example, the population of predators in the top drawing of Fig. 1 has a negative effect (as shown by the arrow with a negative sign) on the prey population because the more predators there are, the more prey will be consumed. Another way of saying this is that the number (or density) of prey at one point in time is inversely related to the number (or density) of predators at a previous point in time, or that prey density is an inverse function of predator density. Inverse functions from one variable to another are depicted as arrows with a negative sign, as in Fig. 1, and the function is usually considered to operate over a unit of time. Notice that the prey population in the top diagram has no effect on the predator population (no arrow going from the prey to the predator) and so predation is an exogenous process. Because exogenous forces are external to and independent of the system, they are sometimes called inputs into the system.
Figure 1. Diagram showing the potential interactions between a population of prey N and predators P: (Top) Predators act as a negative exogenous factor on the prey population but the prey population has a positive effect on its own numbers; (Bottom) The prey population now has a positive effect on its predators so that the interaction now forms an endogenous negative feedback loop.
Endogenous forces, on the other hand, induce changes in the system and are also affected in return by those changes and, therefore, are internal to and interdependent on the system. For this reason they are considered to be part of the system structure. For example, there is a positive arrow going from the prey population to itself in Fig. 1 which signifies that prey numbers (or density) at one point in time are directly related to prey numbers (or density) at some previous time. In other words, prey density is a positive function of past prey density. This is a reasonable proposition because the more prey there are the more there are likely to be in the future. Arrows that go from a variable back to itself are called feedback loops, and those that involve a positive function, like that in Fig. 1, are called positive feedback loops (+feedback for short). In addition, a feedback loop like this, that goes from a variable directly back to itself without involving any other variables, is said to have a dimension of one because it only involves one variable. It is also called a first order feedback loop.
Feedback loops can also involve other variables. For example, in the bottom diagram of Fig. 1 there is an arrow with a positive sign going from the prey to the predator population. This arrow is positive to show that predator numbers (or density) are a direct function of prey numbers (or density) because the more prey there are the more predators there will be in the future (more food results in more predator reproduction). This additional arrow gives rise to another feedback loop from the prey population back to itself and, of course, a loop from the predator population back to itself as well. This feedback loop passes through both predator and prey populations and links them together in a mutually causal process. When two variables are involved in a mutually causal feedback loop both variables are considered to part of a second order feedback loop and the system is two dimensional. Of course it is possible to have higher order feedback loops involving many variables. In these cases the dimension of the system is determined by the number of variables in the longest feedback loop.
When a feedback loop contains more than one function, the overall sign of the loop is obtained by multiplying the signs of all the functions. In the lower diagram of Fig. 1, for example, we multiply a positive and a negative function [i.e., (+) × (-) = (-)] to obtain a second order negative feedback loop (-feedback for short). Hence, the feedback structure of the lower figure is made up of one first order +feedback process and one second order -feedback process.
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©1997 Alan A. Berryman