Viability and resilience Guillaume Deffuant, Sophie Martin, Justin Calabrese

1. Viability and resilience Guillaume Deffuant, Sophie Martin, Justin Calabrese

2. Definitionfromecology: The magnitude of disturbancethatcanbeabsorbedbeforethe system changes its structure by changing the variables and processesthat control itsbehavior(Gunderson and Holling 2002) Operational concept, descriptive, minimisingambiguity: • Linearisation close to equilibrium • Regime shifts • resilience of what to what(Carpenter et al. ) Resilience as a descriptive concept

3. Resilience as boundaryobject Object thatcanbeseendifferently by different disciplines and favours exchanges • Maintainambiguity • tends to be normative • Examples: • A perspective or approach to analyze social-ecologicalsystems (Folke 2006) • Flexibility over the long term

4. Outline Resiliencebased on attractors (regime shifts) Resiliencebased on viability • Without management actions and attractors in desired set of states • With actions and attractors in desired set of states • With actions and attractors not in desired set of states Conclusion

5. Resilience of based on attractors Hypothesis • Dynamics characterised by a set of attractors • someattractorsprovidedesiredproperties of the system (good attractors) • someattractorsdon’t (badattractors). The system isresilient to a perturbation if the perturbation keeps the system in the attraction basin of a « good » attractor, itis not resilient if the perturbation drives the system to the attractor basin of a « bad » attractor. Resilience value connectedwith the size of the « good » attractor basins, or to the speed of return in the vicinity of a good attractor.

6. Simplifiedexample of savannadynamics TakenfromAnderies et al. 2003 Two variables: • shoot biomass (grass) s • grazingg We suppose that once the value of grazinggisdecided, itremainsfixed.

7. Good and badattractors good attractors bad attractors (no grass)

8. Resiliencebased on attractors Dynamical system defined for instance in discrete time: x(t+dt) = x(t) + F(x(t))dt A point xais an attractor if: • F(xa) = 0 • There existsa subsetAsuchthat, for all x(0) in A: • x(t)→xawhent→∞ • The largest set Ais the attraction basin Resiliencedefined as • the size of the attractor basin • the velocity for going to the attractor (based on a linearisation of the dynamics close to the equilibrium)

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

10. Limits of the attractorbasedresilience Need to define the desiredfunctioning of the system as a set of attractors. Difficult to introducethe problem of choosing a management policy (itissupposedincluded in the system dynamics)

11. Differentview: Desiredproperty as state set Resilience supposes to define a desiredproperty (functioning) of the system Main idea: define the desiredproperty as a subset of the state space (independentlyfrom the presence of attractors)

12. Constraint set: s > 0.05 N°12

13. Link withviability Weneed to determine all points of spacefromwhich the desiredpropertyismaintained – i.e. fromwhichtrajectorystays in the desired set Viabilitytheory, developed in the 90ies by J.P. Aubin: • considers a system that collapses or badlydeteriorates if itgoesbeyond a state subsetK. • It needsalso to determine the points fromwhich the trajectoryremains in K, thatisx(0)suchthat, for all value t, we have x(t) in K, iscalled the viabilitykernel.

14. Viabilitykernel

15. Resiliencebased on viability (without action) A resilient state is a state fromwhich the trajectorygoes to the viabilitykernel (becausethenitwillstaythere and keep the property) In viabilitytheory, the set of points fromwhichtrajectories go to a target set is the capture basin of thistarget set. The resilience basinis the capture basin of the viabilitykernel The measure of resilienceis the inverse of the cost (time) to reach the viabilitykernel.

16. Resilience

17. Comparing the definitions Most states which are resilient in the attractordefinition are viable in the viabilitydefinition Is the difference due to the choice of the property to bekept (constraint set) ?

18. New property to bekept (constraint set)

19. Viabilitykernel 2

20. Resilience 2

21. . Comparing the definitions The resilient states almostcoincide. The resilience values dependmore on the dynamics in the viabilitybasedresilience

22. Introducing a possibility to act on the system Ateach time step, we suppose thatitis possible to act on the system. It canbeconsidered as an adaptation capacity. For instance, we suppose that the grazing pressure canbemodified of a value dg, with-0.02 < dg < 0.02

23. Viabilitytheorywith action Viabilitykernel : the set of states fromwhich a policy of action keeps the system within the constraint set. Capture basin of a target set: set of states fromwhichthereexists a policy of action leading to the target. Resilient basin: states belonging to the capture basin of the viabilitykernel. Resilience value: inverse of the time (or cost) to go back to the viabilitykernel. There existgeneralalgorithms to computeviabilitykernels and capture basins (P. Saint-Pierre)

24. Viabilitykernel 3

25. Resilience 3

26. Defining actions The action is the one that drives the system as close as possible in the normal direction of the nextlevel line.

27. Resilienttrajectory

28. No attractor in the constraint set The dynamics do not necessary lead to an attractor. Suppose nowthatwewant to keep the level of grassbetween 0.05 and 0.18 Westillcan change the grazing of atmost 0.02 (positive or negative)

29. Constraint set : 0.1 < s < 0.18

30. Viabilitykernel, with no stable equilibrium

31. Resilience

32. Trajectorycoming back to kernel

33. Conclusion The resiliencebased on viabilitycanbeseen as an extension of resiliencebased on attractors • The « good » attractorisreplaced by the viabilitykerneldefined on the constraint set of the desiredproperty • The attraction basin isreplaced by the « capture basin » of the viabilitykernel (i.e. the points for which the exists a policy of action leading to the viabilitykernel) Advantages • Can includenaturally an action in the approach and provides actions to make • Does not necessitateequilibirum in the dynamics.

34. Limits 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. Need to extend the framework to stochastic dynamics (Rougé, J.D. Mathias, G. Deffuant. Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication. Ecological Indicators 29, 2013)

35. Discussion Conceptual clarification about the differencebetweenviability and resilience Viabilityapproachfundamentallyadapted to sustainability, because: • focus on whatshouldbeavoidedratherthan on whatshouldbemaximised • possibility to gathercirtieriawhich are in differentunits Defining the property to bekept (or restored) is a politicaldecision Could the concepts beuseful to stakeholders ?