1. ASCENDENCY Scharler, U.M.; University of KawZulu-Natal, Durban, South Africa PETRA MAKORIČ March, 2015

2. 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 March, 20152

3. Conditional probabilities and ecosystem complexity The probabilities in a mechanistic world, of events following specific causes can be calculated by joint probabilities p(a i, b j ). Conditional probabilities are denoted by p(a i │b j ), and are calculated by dividing the absolute probabilities p(a i, b j ) by the marginal probability p(a i ), 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 March, 20153 Table 1 Joint probabilities and their column/row sums Table 2. conditional probabilities

4. 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. March, 20154

5. 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 March, 20155 Figure 1. Hypothetical unconstrained network Figure 2 Hypothetical constried network: higer AMI

6. Uncertainty H Uncertainty that an event occurs H = K log p(a i ) (1) And the uncertainty that an event occurs provided certain information (bj) is available H = K log p(a i │b j ) (2) The information then is the a prior uncertainty minus the uncertainty if bj is known or I = K log p(a i ) - [klog p(a i │b j ) (3) I = K log p (a i │b j ) K log p(a i ) (4) I = K log [p(a i │b j )/p(a i )] (5) March, 20156

7. 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 March, 20157 (6)

8. The formula of AMI from an ecological point of view can be reweitten as The conditional probabilitiy p(a i │b j )=p(a i,b j )/p(a i ) Can be rewritten as T ij /T i and the marginal probability as T j /T.. March, 20158 (7) (8)

9. 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 March, 20159 (9) (10) (11)

10. 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 March, 201510

11. 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 March, 201511 (12) (13)(14)

12. 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. March, 201512

13. 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 March, 201513

14. 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 pathways March, 201514

15. 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. March, 201515 (14) (15) (16)

16. 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 March, 201516 (17)

17. 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 March, 201517 (18)

18. 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 March, 201518 (19)

19. 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 March, 201519 (20)

20. 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 March, 201520

21. Thank you for your attention March, 201521

22. 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, 201522