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Viability and resilience Guillaume Deffuant, Sophie Martin

This article examines the concepts of resilience and viability in the context of attractors and constraint sets in dynamic systems. It explores the relationship between good and bad attractors, resilience based on returning to attractors, and resilience based on viability with different constraint sets. The article also introduces the possibility of incorporating action in the system, and discusses the advantages and limitations of the approach.

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Viability and resilience Guillaume Deffuant, Sophie Martin

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  1. Viability and resilience Guillaume Deffuant, Sophie Martin

  2. Outline • From the reslience based on attractors to resilience based on viability • Example on savanna dynamics (Anderies et al. 2002) • Resilience based on returning to attractors • Resilience based on viability, first constraint set • Resilience based on viability, second constraint set • Resilience based on viability, second constraint set with action • Resilience based on viability, third constraint set with action • Conclusion

  3. Resilience of based on attractors • Hypothesis: • some attractors provide desired properties of the system (good attractors) • some attractors don’t (bad attractors). • The system is resilient to a perturbation if the perturbation keeps the system in the attraction basin of a « good » attractor, it is not resilient if the perturbation drives the system to the attractor basin of a « bad » attractor. • Resilience value connected with the size of the « good » attractor basins.

  4. Simplified example of savanna dynamics • Taken from Anderies et al. 2003 • Two variables: • shoot biomass (grass) s • grazing g • We suppose that once the value of grazing g is decided, it remains fixed.

  5. Good and bad attractors good attractors bad attractors (no grass)

  6. Definition of resilience as a function of grazing Resilience as distance from unstable to stable equilibrium Perfect resilience

  7. Viability (without action) • Viability has been developped by J.P. Aubin in the 90ies. • Define a desirable property of the system as a subset of the state space (constraint set) • The viability kernel is the set of states from which the trajectory remains in the constraint set.

  8. Constraint set: s > 0;05 0.6 shoot biomass s 0.05 0.0 0.45 grazing g N°8

  9. Viability kernel 0.6 shoot biomass s 0.05 0.0 0.45 grazing g

  10. Resilience based on viability (without action) • A resilient state is a state from which the trajectory goes to the viability kernel. • The measure of resilience is the inverse of the integral of a cost per unit of time along the trajectrory to the viability kernel.

  11. Resilience 0.6 shoot biomass s 0.05 0.0 0.45 grazing g

  12. Comparing the definitions • Most states which are resilient in the attractor definition are viable in the viability definition • The measure of resilience is different (depends on the dynamics) • Difference due to the choice of the constraint set ?

  13. New constraint set 0.6 shoot biomass s 0.05 0.0 0.45 grazing g

  14. Viability kernel 2 0.6 shoot biomass s 0.0 0.45 grazing g

  15. Resilience 2 0.6 shoot biomass s 0.0 0.0 grazing g 0.45

  16. Comparing the definitions • The resilient states almost coincide. • The resilience values are close but in some places depend more on the dynamics in the resilience viability definition.

  17. Introducing a possibility to act on the system • At each time step, we suppose that it is possible to act on the system. • For instance, we suppose that the grazing pressure can be modified of a value dg, with -0.02 < dg < 0.02 • The viability kernel definition becomes: the set of states from which a policy of action keeps the system within the constraint set. • Resilient state definition: state from which there exists a policy of action which brings the system into the viability kernel. • Resilience value: inverse of the minimum cost to go back to the viability kernel.

  18. Viability kernel 3 0.6 shoot biomass s 0.0 0.0 grazing g 0.45

  19. Resilience 3 0.6 shoot biomass s 0.0 0.0 grazing g 0.45

  20. No attractor in the constraint set • The dynamics do not necessary lead to an attractor. • Suppose now that we want to keep the level of grass between 0.05 and 0.18 • We still can change the grazing of at most 0.02 (positive or negative)

  21. Constraint set : 0.1 < s < 0.18 0.6 shoot biomass s 0.18 0.1 0.0 0.0 grazing g 0.45

  22. Viability kernel, with no stable equilibrium 0.6 shoot biomass s 0.18 0.1 0.0 0.0 grazing g 0.45

  23. Resilience 0.6 shoot biomass s 0.18 0.1 0.0 0.0 grazing g 0.45

  24. Conclusion • The resilience based on viability can be seen as an extension of resilience based on attractors • The « good » attractor is replaced by the viability kernel defined on the constraint set of the desired property • The attraction basin is replaced by the « capture basin » of the viability kernel (i.e. the points for which the exists a policy of action leading to the viability kernel) • Advantages • Can include naturally an action in the approach • Does not necessitate equilibirum in the dynamics.

  25. Problem • To compute viability kernels and resilience values, one must discretise the space • When the dimension of the space grows, the number of points of the grid grows exponentially. • The method cannot be applied on dynamical system with a state of many dimensions.

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