ASCENDENCYScharler, U.M.; University of KawZulu-Natal, Durban, South Africa

PETRA MAKORIČ

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What is ascendency?

• The theory of ascendeny was developd to explain ecosystem behavior and development

• Theory of ascendency tries to capture nonmechanistic behavior in a single index, indicative of ecosystem state and development, and of ecosystem health

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Conditional probabilities and ecosystem complexity

• The probabilities in a mechanistic world, of events following specific causes can be calculated by joint probabilities p(ai, bj).

• Conditional probabilities are denoted by p(ai │bj), and are calculated by dividing the absolute probabilities p(ai, bj) by the marginal probability p(ai), or the sum of all probable effects of one cause

• It is possible to calculate the conditional probability of a mechanical cause-effect pair

• In conditional probability, a couse can have more than one effect

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Table 1 Joint probabilities and their column/row sums

Table 2. conditional probabilities

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Total system throughput

• 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 and its outside.

• Total system throughput can increase either by increasing the extent of the system or by increasing the activity of the system.

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Average mutual information

• If all events are equiprobable, then the average uncertainty about what event will happen next is the highest

• 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

• One that is not equiprobable is where more material flows along some pathways, and less material along others

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Figure 1. Hypothetical unconstrained network

Figure 2 Hypothetical constried network: higer AMI

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Uncertainty H

• Uncertainty that an event occurs

H = K log p(ai) (1)

• And the uncertainty that an event occurs provided certain information (bj) is available

H = K log p(ai│bj) (2)

• The information then is the a prior uncertainty minus the uncertainty if bj is known or

I = K log p(ai) - [klog p(ai│bj) (3)

I = K log p (ai│bj) K log p(ai) (4)

I = K log [p(ai│bj)/p(ai)] (5)

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Average mutual informatiom (AMI)

• Avegare mutual information (AMI) is the amount of uncertainty reduced by knowing bj

• Results are in units of K

• As a hypothetical example above, the a priori

uncertainty about where a quantum of material flows in ecological network is given by Shannon’s formula

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• The formula of AMI from an ecological point of view can be reweitten as

• The conditional probabilitiy

p(ai│bj)=p(ai,bj)/p(ai)

• Can be rewritten as Tij/Ti and the marginal probability as Tj/T..

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Ascendency (A)

• Ascendency in the resulting index when the TST substitute the k (scalar konstant) in order to scale the AMI to the size of the system in question

So from that

We get

Or

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• There are number of influences that can change ascendancy of system

• If TST and AMI increase, then the ascendency will increase (if there are no external disturbances)

• Is limited by any constraints on the increase of TST and AMI

• Is higer when pathways are fewer in number and more articulated

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Development Capacity• MacArthur applied Shannon’s diversity index to the material

flows in an ecosystem to arrive at a measure for the diversity of flows, H

• K is the scalar constant and T.. Is the TST or sum over all combinations of Tij

• H can, like AMI, be multiplied by TST to scale the diversity of flows to the system. TST x H is called the development capacity, or limit for devepment, C

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(13) (14)

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• C is limited by two factors, namely TST and the number of compartments.

• The limits to TST are the same as in the case of ascendency.

• If a certain amount of TST is split between too many compartments, then some compartments will end up with a very small throughput

• This process is believed to reduce the number of compartments and therefore the number of flows.

• More stable systems are thus believed to show a higher C compared to systems undergoing frequent disturbances.

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Residual uncertainty

• The initial compexity H, consists of two elements:– AMI which is describing the information gained

by reducing the uncertinity in flow probabilitiy which is an index of the organized part of the system

– And of residual uncerainity Hc called also conditional diversity

• H=AMI + Hc

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Hc or Overhead

• When residual uncertainty is scaled by TST is also called the overhead

• The overhead represents the unorganized, ineffecient and indeterminate part of flow structure

• It is considered an insurance for the system• Overhead is split into four comoponents:

overhead due to imports

exports

respiration

initial pathwaysMarch, 2015

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• The combined overhead is denoted by

• Scaling Hc to the system by replacing k with TST yields

• The relationship betwen C, A and Ф so becomes

C= A. + Ф.

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Imports• Imports enter the system via fewer pathways or

compartments ascendency will increase at the expence of the overhead

• It is expected that system is a more stable environment rely on fewer import pathways compared to perturbed system

• Formula for the overhed on imports

• Imorts are assumed to originate in the fictious compartment 0

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Exports

• Similar to the overhead on imports• Depend on the amount of exporting pathways

leaving the system• Depend on amount transfered along those pathways• If it is present positive feedback via another system

then an increase in exports becomes benefical to the system

• The overhead on exports id denoted by

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Respiration

• The overhead regarding the dissipation depends on the magnitude lost to the environment, on the number of pathways, and the distribution of the magnitde transferred

• Losses through dissipation are requied by the second law of thermidynamics and are neccessary to maintain metabolisms

• The overhead on dissipation is

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Redundancy• Fourth part of the overhead• There are disadvanteages to the system in maintaining

redundant pathways• An increase in dissipations can occur• The resource transfered along different parallel pathways

might not always end up at the right time at the consumer• An obvious advantage of parallel pathways is the

insurance of having more than one route of transfer in case of disturbances

• Redundancy is denoted by

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Biomass Inclusive Ascendency

• Can be used as a theoretical basis to derive element limitations for compartments

• To identify limiting nutrient linkages• To quantify the successional trend to include

larger species with slower turnover times• AMI can be caculated between a biomass

probability and the resulting flow probability, by calculating relationship between biomass and flows

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Thank you for your attention

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Questions

• What is the principle of ascendency?• What is total system throughput?• What is average mutual information?• Which components make up the overhead?• Wat can biomass inclusive ascendency be used

for?

March, 2015