Evaluating resilience in models of socio-ecological systems AES Conference Warwick
Sophie Martin Laboratoire d’Ingénierie des Systèmes Complexes
April 15th 2015
Oligotrophic lake
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Characterized by clear-water, suck a lake can be used for irrigation, municipal water supplies, pollution dilution, or recreation. Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Eutrophic lake
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Characterized by turbid-water, rich in organic and mineral nutrients, supporting an abundant plant life and oxygen depletion for animal life. Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Aim of the talk
Considering socio-ecological systems with different modes of operation, showing that :
Evaluating resilience in models of socioecological systems
How to preserve desired properties of a socio-ecological system is a viability problem
Evaluating the impact of perturbations on the ability of preserving these properties is a measure of resilience
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The concept
Used in different fields :
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physics of materials
psychology
computer science
economics
ecology
social sciences
Common idea : the ability of a system to maintain or restore properties despite disruptions caused by perturbations
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The measures...
... in models made of differential equations : I
I
based on the eigenvalues of the linearization of a nonlinear system near an equilibrium point (Pimm and Lawton 1977) as inversely proportional to the size of attraction domains (Ludwig et al. 1997)
... in individual-based models : the time needed after some kind of disturbance to return to its original state or to a given neighborhood (Ortiz and Wolff 2002)
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The measures...
... in models made of differential equations : I
I
based on the eigenvalues of the linearization of a nonlinear system near an equilibrium point (Pimm and Lawton 1977) as inversely proportional to the size of attraction domains (Ludwig et al. 1997)
... in individual-based models : the time needed after some kind of disturbance to return to its original state or to a given neighborhood (Ortiz and Wolff 2002)
HOWEVER, Evaluating resilience in models of socioecological systems P. 7 / 39
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The measures...
... in models made of differential equations : I
I
based on the eigenvalues of the linearization of a nonlinear system near an equilibrium point (Pimm and Lawton 1977) as inversely proportional to the size of attraction domains (Ludwig et al. 1997)
... in individual-based models : the time needed after some kind of disturbance to return to its original state or to a given neighborhood (Ortiz and Wolff 2002)
HOWEVER, Evaluating resilience in models of socioecological systems
the set where the desired property holds, which does not necessary correspond to an attraction basin of the dynamics
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The measures...
... in models made of differential equations : I
I
based on the eigenvalues of the linearization of a nonlinear system near an equilibrium point (Pimm and Lawton 1977) as inversely proportional to the size of attraction domains (Ludwig et al. 1997)
... in individual-based models : the time needed after some kind of disturbance to return to its original state or to a given neighborhood (Ortiz and Wolff 2002)
HOWEVER, Evaluating resilience in models of socioecological systems P. 9 / 39
the set where the desired property holds, which does not necessary correspond to an attraction basin of the dynamics
the system state is not necessary in the vicinity of an equilibrium
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience : The measures...
... in models made of differential equations : I
I
based on the eigenvalues of the linearization of a nonlinear system near an equilibrium point (Pimm and Lawton 1977) as inversely proportional to the size of attraction domains (Ludwig et al. 1997)
... in individual-based models : the time needed after some kind of disturbance to return to its original state or to a given neighborhood (Ortiz and Wolff 2002)
HOWEVER, Evaluating resilience in models of socioecological systems P. 10 / 39
the set where the desired property holds, which does not necessary correspond to an attraction basin of the dynamics
the system state is not necessary in the vicinity of an equilibrium
to obtain such deterministic systems, the feedback law has to be defined first, whereas the issue is precisely to derive action policies that enhance resilience.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience A value of resilience should provide information on the impact of a possible disturbance on the system’s ability to maintain certain properties.
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if as a result of any anticipated disturbances, the property can be preserved, the value of resilience should be infinite
if any of the anticipated disturbances causes irremediable loss of the property, the value of resilience should be null
if following an anticipated disturbance the property will be lost, but can always be restored, the value of resilience should be finite but not zero and should be a decreasing function of the maximal cost of restoration.
The aim of the study of resilience is then to avoid situations in which perturbations can lead to irreversible situations and facilitate the restoration of the essential properties, where possible.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Viability kernel (Aubin 1991, Aubin et al. 2011) Given a constraint set in the state space, the viability kernel gathers all states from which there exists at least one control function that governs an evolution which remains in this constraint set.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Resilience is infinite
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Resilience is null
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Measuring resilience... ... as the impact of a possible disturbance on the system ability to maintain certain properties.
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Resilience is finite but non null
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 1
The dynamics : I
model of (Anderies, 2002) : the grass plant consists of two parts, the crown and the shoots. Growth occurs through the interaction of these two parts : c 0 (t) s 0 (t)
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= =
rs s(t) − c(t) (ac c(t) + rc c(t)s(t))(1 − s(t)) − γg (t)s(t)
where c represents crown biomass, s shoot biomass, rs , ac and rc are parameters that describe the rate at which crown or shoot biomass grow when crown and shoot are present. γg represents the grazing pressure.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 1
The dynamics : I
farmers can not adjust stocking rates instantaneously. Thus, we consider that the variations of the stocking rate are bounded : ¯] γg0 (t) = u(t) ∈ [u; u
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 1
The dynamics : I
3-dimensional state space (c(t), s(t) and γg (t)) with one control variable (u(t)) : c 0 (t) s 0 (t) γg0 (t)
= = =
rs s(t) − c(t) (ac c(t) + rc c(t)s(t))(1 − s(t)) − γg (t)s(t) ¯] u(t) ∈ [u; u
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 2
The property under study I
a minimal grazing pressure,
I
and a minimal quantity of shoot biomass. γg (t) s(t)
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
≥ ≥
γg s
Interest of assessing resilience : illustration with an example 2
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The property under study
Constraint set as a subset of the three-dimensional state space (c, s, γg ) with s = 0.1 and γg = 0.65.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example
The first step consists in studying the compatibility between :
dynamics c 0 (t) = rs s(t) − c(t) s 0 (t) = (ac c(t) + rc c(t)s(t))(1 − s(t)) − γg (t)s(t) γg0 (t) = u(t) ∈ [u; u¯]
Evaluating resilience in models of socioecological systems
and the desired property
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
γg s
≥ ≥
γg s
Interest of assessing resilience : illustration with an example
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Viability kernel for rs = 3, ac = 0.1, rc = 1, γg = 0.65, s = 0.1 and u ¯ = −u = 0.05.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 3
The anticipated disturbances A period of drought causes sudden reduction of shoot biomass. We represent a drought event as a jump in the state space from (c, s, γg ) to (˜ c , ˜s , γ˜g ) where :
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I
c˜ = c, we assume that the drought event does not affect crown biomass,
I
γ˜g = γg , the drought event has no direct impact on the grazing pressure,
I
˜s = s − αs where α ∈ [0, α] ¯ represents the severity of drought, the maximal anticipated severity is α ¯ ≤ 1.
Thus, anticipated disturbances are jumps in the state space from (c, s, γg ) to (c, s − αs, γg ) where α ∈ [0, α]. ¯
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 3
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The anticipated disturbances
Drought event : The system state jumps from the state (c = 2, s = 0.7, γg = 0.8) to the state (c = 2, s = 0.35, γg = 0.8). The shoot biomass has been divided by 2, the severity, α, of such a drought equals 0.5.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Interest of assessing resilience : illustration with an example 4
The cost function The damage is evaluated by the time required to find a safe situation : I
biomass and grazing pressure are greater than the minimum values
I
these minimum values can be met indefinitely by appropriate grazing practices.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Evaluation of resilience
Evaluating resilience in models of socioecological systems
The second step consists in evaluating the resilience of the rangeland from the impact of drought events on its ability to preserve minimal levels of shoot biomass and grazing pressure. Resilience measure of the rangeland is then evaluated at any point of the state space as the inverse of the worst damage which corresponds to the drought event of maximal anticipated severity α ¯.
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience values
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Two sections (γg = 0.65 (left) and γg = 0.9 (right)) of resilience values of the rangeland toward drought events. rs = 3, ac = 0.1, rc = 1, γg = 0.65, s = 0.1, u ¯ = −u = 0.05 and α ¯ = 0.5.
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Resilience values
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Left : resilience to drought events smaller than 0.1 (colored black area for γg = 0.65 + hatched area for γg = 0.9). Right : infinite resilience (colored black area for γg = 0.9 + hatched area for γg = 0.65).
Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
Conclusion and perspectives
To develop this way of measuring resilience in the viability framework, we have made hypotheses on the mathematical form of the model :
Evaluating resilience in models of socioecological systems
the dynamics : controlled dynamical system
the desired property : a subset of the state space
the perturbations : a single shock
the cost function : an intertemporal cost
All these assumptions are restrictive, and the way of measuring resilience has to be extended at least to :
dynamics with uncertainties
repeated perturbations
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Sophie Martin | Laboratoire d’Ingénierie des Systèmes Complexes
