Conditional probabilities and ecosystem complexity

In a mechanistic world, the probabilities of events following specific causes can be calculated by joint probabilities p(«,-,by). These describe an absolute probability of an effect occurring in response to a cause. In ecosystems, it is believed that no such direct, mechanistic cause and effect behavior exists due to the interaction with other elements which in turn influence the patterns of cause and effects between pairs. Instead of the absolute probability, Popper introduces the term propensity, which describes a bias that events might (not will) happen. Popper therefore calls for a measure of such relative or conditional probabilities. Conditional probabilities are denoted by p(aj\bj), and are calculated by dividing the absolute probabilities p(aj,bj) by the marginal probability p(a), or the sum of all probable effects of one cause (Tables 1-3). The

bi |
b2 |
b3 |
b4 | |

a1 |
4 |
5 |
7 |
9 |

a2 |
2 |
4 |
2 |
1 |

a3 |
6 |
7 |
9 |
4 |

bi |
b 2 |
b 3 |
b4 |
pa | |

a1 |
0.07 |
0.08 |
0.12 |
0.15 |
0.42 |

a2 |
0.03 |
0.07 |
0.03 |
0.02 |
0.15 |

a3 |
0.10 |
0.12 |
0.15 |
0.07 |
0.43 |

Pb) |
0.20 |
0.27 |
0.30 |
0.23 |
1.00 |

Values are obtained by dividing the number of occurrences (Table 1) by the total number of observations (60).

Values are obtained by dividing the number of occurrences (Table 1) by the total number of observations (60).

b1 |
b2 |
b3 |
b4 | |

a1 |
0.33 |
0.31 |
0.39 |
0.64 |

a2 |
0.17 |
0.25 |
0.11 |
0.07 |

a3 |
0.50 |
0.44 |
0.50 |
0.29 |

Values are obtained by dividing the values in the joint probability matrix by the column sums (p(a,-)) (see Table 2).

Values are obtained by dividing the values in the joint probability matrix by the column sums (p(a,-)) (see Table 2).

conditional probability thus describes cause and effect in the context of other absolute probabilities, considering that a cause might have more than one effect. This eliminates the pitfall of disregarding the influence of other interactions on the one in question. It is, of course, possible to calculate the conditional probability of a mechanical cause-effect pair, that is, the case of having one cause and one effect. This turns out to be 1, or in other words, there is certainty that the effect in question will follow the cause in question.

Since ecosystems are open, not all causes can be accounted for. Some of them might originate outside the system. Therefore, an open ecosystem can never evolve towards a mechanistic behavior of cause and effect. Ulanowicz states that autocatalysis, or indirect mutualism, is an important cause in ecosystem growth and development. Autocatalysis is apparent when members of a feeding loop positively enhance the following member of the loop, which eventually leads back to a positive enhancement of the starting member. Autocatalytic loops exert a selection pressure on its members in that a member of the loop might be replaced with a new constituent who has a more positive effect. Autocatalytic loops exhibit a centripetality, which enables them to attract more resources (available energy). These are reasons for the growth and development of, or increase of order in, ecosystems.

To quantify growth and development, ecosystems are portrayed as networks of material or energy exchanges. These networks of feeding transfers are believed to adequately describe an ecosystem. It is assumed that other significant aspects of ecological systems, such as behavioral aspects, are in one form or another imprinted on the amount of energy transferred, through their effect on population size and predator avoidance.

Ascendency describes both growth and development. Growth of the ecosystem is measured as any increase in total system throughput (TST), which is the sum of all exchanges within the ecosystem and between the system in question and its outside (imports, exports, respirations). Total system throughput can rise either by increasing the extent of the system (more species, or by extending ecosystem borders), or by an increased activity of the system (e.g., during phytoplankton blooms).

Ecosystem development is quantified from the same networks of material exchanges with the help of information theory. In autocatalytic loops, the trend for transferring material is as follows: those linkages which are most rewarding to the loop will transfer more material than those which are not (compartments have in general more than one outgoing link and can thus have pathways to compartments outside the loop). The latter are not necessarily discarded, but transfer only a small amount of material. If a quantum of material sits in a compartment in an autocatalytic loop, then it is therefore more likely to be transferred along a route with high material transfer than along a route with low material transfer. The probability that a quantum of material flows along the highly frequented routes is, therefore, higher compared to a network where all routes transfer the same amount of material. Conversely, the probability that a quantum of material flows along the less-frequented routes will be lower compared to a network where all routes transfer the same amount of material. Such a change in probability can be quantified with the help of information theory. Information is defined as the agent that causes a change in probability. Ulanowicz uses the term information to describe 'the effects of that which imparts order and pattern to the system'.

In the calculation of information, the starting point is to quantify ecosystem complexity. The complexity of a system is mirrored in the system configuration (amount of links and distribution of transfers along those links). According to Boltzmann, the potential of each configuration contributing to systems complexity, s, can be calculated as the negative logarithm of the probability that the event (the system configuration) will occur (s = —k log p, where k is a constant of proportionality, i.e., a scaling factor). If a system configuration (the event) will occur always (p = 1), then the contribution to complexity is diminished (log(1) = 0, uncertainty is at its lowest) and the system behavior is simple (i.e., it always behaves the same way). If a system configuration (or event) occurs only rarely, then there is a large potential for complexity (i.e., it can behave in many different ways, uncertainty is high). Behavior of a truly complex system is unique each time it functions (uncertainty is highest).

To calculate how much a rare configuration contributes to system complexity, it is weighted by the (low) frequency of its occurrence. The potential contributions (or events) are averaged by the configurations of the system by weighting each si by its corresponding pi (Shannon's formula: H = - K^pj logpj). In other words, each potential contributioin of occurrence is weighted by its corresponding probability that it will occur, which is summed over all system configurations. A high H corresponds to high uncertainty, complexity, and diversity.

The above discussion serves to illustrate how events can contribute to the complexity of a system. Next to consider is whether these events contribute to an ordered pattern in the system, or whether they contribute to random behavior. If all events are equiprobable, then the average uncertainty about what event will happen next is the highest. This hypothetical situation can serve as a starting point to calculate how much less uncertainty there is under circumstances where not all events are equiprob-able. The decrease in uncertainty from a situation of equiprobability to any other is called information. From an ecosystem perspective, a situation of equiprobability is one where material flows in equal amounts along all pathways (Figure 1a). One that is not equiprobable is where more material flows along some pathways, and less material along others (Figure 1b). Thus, the most indeterminate network is one where all compartments are connected with each other and where, in proportion to the compartmental throughput, equal amounts of material flow along the ingoing and outgoing pathways. Quantifying the information which is gained by transferring material along more- and less-frequented routes thus gives a clue about the unevenness of material flowing along pathways.

The change in probability from a situation where a quantum of material flows along an equiprobable pathway and along a pathway which is not equiprobable is

12 | ||||

C |
B | |||

12 |

C |
23 |
B |

1 |

Figure 1 (a) Hypothetical unconstrained network: low AMI. (b) Hypothetical constrained network: higher AMI.

calculated using conditional probabilities. To start with, the uncertainty that an event occurs is

and the uncertainty that an event occurs provided certain information (by) is available is

The information then is the a priori uncertainty minus the uncertainty if by is known or

I is not positive for all pairs of occurrences. The sum of all I's which have been weighted by the corresponding joint probability turns out always to be positive, however. The joint probability of each occurrence serves, as in Shannon's formula, as a weighting for the frequency of occurrence of each event (i.e., each co-occurrence of ai and by). The result is called the average mutual information (AMI) or

AMI is the amount of uncertainty reduced by knowing by. Results are in units of K.

As in the hypothetical example above, the a priori uncertainty about where a quantum of material flows in ecological networks is given by Shannon's formula. The additional information (by) to calculate the conditional probability is the knowledge of the outputs from each compartment in the flow network a time step earlier.

Since, from an ecological network point of view, joint and conditional probabilities refer to transfers of material from compartment i to compartment y, the above formula can be rewritten as

where the joint probability of a quantum of material (p(aj,by)) flowing from species i to species j can be denoted as Tj/T, remembering that the events in an events table are material flows in a system. T is the total system throughput, or the sum over all combinations of Tj. The summation among all rows of the matrix is denoted by the first dot, while the second dot stands for summation among columns.

The conditional probability p{a,\by) -p{a,, b,)/p(a,)

can be rewritten as Ty/T; and the marginal probability (sum of all probable outcomes, p(a;)) as Ty/T..

To summarize, the AMI describes the information gained by knowing the outputs from each compartment in the flow network a time step earlier (by) in addition to the a priori situation describing the flow of a quantum of energy or material between two compartments (a). The uncertainty of where a quantum flows is calculated through Shannon's index of flow diversity. The uncertainty of where a quantum of material will flow by knowing by is calculated by the conditional probability.

The scalar constant, k, has been retained throughout all calculations. To be able to combine growth and development into one single index, k is substituted by the 'total system throughput' or TST in order to scale the AMI to the size of the system in question. The resulting index is called ascendency and is denoted by

Besides indirect mutualism there are a number of influences that can change the ascendency of a system. These influences are thought to not have any favored direction of change, whereas indirect mutualism is believed to drive development toward increased ascendency. Mutualism is furthermore not a result of events elsewhere in the system's hierarchy but can arise at any level. Therefore it is theorized that in the absence of overwhelming external disturbances, the ascendency of a system has a propensity to increase, that is, both activity (TST) and structure (AMI) increase. The theoretical behavior of mutual information conforms to most of the 24 ecosystem properties originally put forward by Odum to characterize mature ecosystems.

Ascendency is limited by any constraints on the increase in either TST or AMI. Limits to TST are set by the finite imports from outside system boundaries and by the second law ofthermodynamics, which requires that a portion of the compartmental throughput be lost as dissipation. Therefore the TST cannot increase indefinitely via recycling. The limits to the AMI, or system development, are set by the flow structure. It limits the extent to which the flows can be organized without a change to the structure itself. Further limits to the AMI in real networks are discussed in the section titled 'Overhead'.

In theory, ascendency is higher when pathways are fewer in numbers (more specialization) and more articulated (few pathways transport most of the material). The highest theoretical value of ascendency is achieved when all players in the system have one input and one output only, and are thus joined in one big single loop. This configuration mirrors highest specialization, and in this case AMI = H (diversity of flows, see below). This situation cannot be achieved in real systems, due to reasons discussed in the section titled 'Overhead'.

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