Bifurcation structure and resilience in impulsive models...
Transcript of Bifurcation structure and resilience in impulsive models...
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Bifurcation structure and resilience in impulsive models of savannas
Alanna Hoyer-Leitzel (Mount Holyoke College) Sarah Iams (Harvard University)
January 22, 2020 at Brandeis University
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Outline
• Modeling framework: disturbance and resilience
• Three savanna models • Where does our model fit in?
• Our model, bifurcations and bistability
• Back to resilience • Under what conditions does the savanna we value exist?
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Resilience is the capacity of a system to absorb disturbance and still retain its structure and function.
Modeling questions:• How do we model disturbance?
• How do we define structure and function?
B. Walker and D. Salt, Resilience Thinking: Sustaining Ecosystems and People in a
Changing World, Island Press, 2006.
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Definition. A multiplicative flow-kick map with underlying flow
for the autonomous system is the map x = f(x)<latexit sha1_base64="lVJ9E92HVY0cuUbg8f8H8f1PZzM=">AAAB83icbVDLSsNAFL2pr1pfVZdugkWom5JUQTdC0Y3LCvYBbSiT6aQdOpmEmRtpCf0NNy4UcevPuPNvnLZZaOuBgcM593DvHD8WXKPjfFu5tfWNza38dmFnd2//oHh41NRRoihr0EhEqu0TzQSXrIEcBWvHipHQF6zlj+5mfuuJKc0j+YiTmHkhGUgecErQSN1uP8J0PL0JyuPzXrHkVJw57FXiZqQEGeq94peJ0yRkEqkgWndcJ0YvJQo5FWxa6CaaxYSOyIB1DJUkZNpL5zdP7TOj9O0gUuZJtOfq70RKQq0noW8mQ4JDvezNxP+8ToLBtZdyGSfIJF0sChJhY2TPCrD7XDGKYmIIoYqbW206JIpQNDUVTAnu8pdXSbNacS8q1YfLUu02qyMPJ3AKZXDhCmpwD3VoAIUYnuEV3qzEerHerY/FaM7KMsfwB9bnD7PEkXU=</latexit>
xn+1 = �(xn) = k'(xn, ⌧)<latexit sha1_base64="ky34Ywjpa/Md/aacHuo/myxhEgU=">AAACDXicbVDLSgMxFM3UV62vUZduglVoUcpMFXRTKLpxWcE+oDMMmTRtQzOZIcmUlqE/4MZfceNCEbfu3fk3pu0stPXAhZNz7iX3Hj9iVCrL+jYyK6tr6xvZzdzW9s7unrl/0JBhLDCp45CFouUjSRjlpK6oYqQVCYICn5GmP7id+s0hEZKG/EGNI+IGqMdpl2KktOSZJyMv4Wf2pOLU+rQw8nixMnCGSETz17mjUFz0zLxVsmaAy8ROSR6kqHnml9MJcRwQrjBDUrZtK1JugoSimJFJzokliRAeoB5pa8pRQKSbzK6ZwFOtdGA3FLq4gjP190SCAinHga87A6T6ctGbiv957Vh1r92E8ihWhOP5R92YQRXCaTSwQwXBio01QVhQvSvEfSQQVjrAnA7BXjx5mTTKJfuiVL6/zFdv0jiy4AgcgwKwwRWogjtQA3WAwSN4Bq/gzXgyXox342PemjHSmUPwB8bnDyD7mvY=</latexit>
'(x, t)<latexit sha1_base64="+6RACYxTcsbsDKP451DPJfLn7YI=">AAAB83icbVBNSwMxEM3Wr1q/qh69BItQQcpuFfRY9OKxgv2A7lKyabYNzWZDMlsspX/DiwdFvPpnvPlvTNs9aOuDgcd7M8zMC5XgBlz328mtrW9sbuW3Czu7e/sHxcOjpklSTVmDJiLR7ZAYJrhkDeAgWFtpRuJQsFY4vJv5rRHThifyEcaKBTHpSx5xSsBKvj8iWg14+ekCzrvFkltx58CrxMtICWWod4tffi+hacwkUEGM6XiugmBCNHAq2LTgp4YpQoekzzqWShIzE0zmN0/xmVV6OEq0LQl4rv6emJDYmHEc2s6YwMAsezPxP6+TQnQTTLhUKTBJF4uiVGBI8CwA3OOaURBjSwjV3N6K6YBoQsHGVLAheMsvr5JmteJdVqoPV6XabRZHHp2gU1RGHrpGNXSP6qiBKFLoGb2iNyd1Xpx352PRmnOymWP0B87nD2n/kUU=</latexit>
x = 0.5x(1� x)
⌧ = 1
k = 0.3<latexit sha1_base64="V/qur6/Rx9yICyy8hEcJ4CfyLMM=">AAACDnicbZC7TsMwFIadcivlFmBkiahalYEoaUGwVKpgYSwSvUhNVTmu21p1nMg+Qa2iPgELr8LCAEKszGy8De5lgMKRLH36/3Nsn9+POFPgOF9GamV1bX0jvZnZ2t7Z3TP3D+oqjCWhNRLyUDZ9rChngtaAAafNSFIc+Jw2/OH11G/cU6lYKO5gHNF2gPuC9RjBoKWOmfO6ISSjSb7s2Oejgns6OvG8jAc4zpddTcOpUeqYWcd2ZmX9BXcBWbSoasf81PeSOKACCMdKtVwngnaCJTDC6STjxYpGmAxxn7Y0ChxQ1U5m60ysnFa6Vi+U+giwZurPiQQHSo0DX3cGGAZq2ZuK/3mtGHqX7YSJKAYqyPyhXswtCK1pNlaXSUqAjzVgIpn+q0UGWGICOsGMDsFdXvkv1Iu2W7KLt2fZytUijjQ6QseogFx0gSroBlVRDRH0gJ7QC3o1Ho1n4814n7emjMXMIfpVxsc3CYKY5g==</latexit>
Modeling Disturbance
fixed point of map
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Definition. An impulsive differential equation has the formx = f(t, x) for t 6= tn
x(t
+n ) = x(t
�n )� I(t
�n , x
�n )
<latexit sha1_base64="LvCjDKMwgUav+Ja90f+qyo9zHFY=">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</latexit>
x = 0.5x(1� x)
tn = n
I(t�n , x�n ) = 0.3x(t�n )
<latexit sha1_base64="egZvegD3Rfo7wlWrIb/VKqGg9yg=">AAACInicbZBLS8NAEMc39VXrq+rRS7BYWrAl8YF6KIhe9KZgH9DWstlu26WbTdidSEroZ/HiV/HiQVFPgh/GTduDtg7s8uM/M8zM3/E5U2BZX0Zibn5hcSm5nFpZXVvfSG9uVZQXSELLxOOerDlYUc4ELQMDTmu+pNh1OK06/cs4X32gUjFP3MHAp00XdwXrMIJBS630WaPtQRQOsyWreBzm7EKYbzRS0BLZktBwndN4X9gP4z8fFx2GYynfSmesojUKcxbsCWTQJG5a6Q89iwQuFUA4VqpuWz40IyyBEU6HqUagqI9JH3dpXaPALlXNaHTi0NzTStvseFI/AeZI/d0RYVepgevoShdDT03nYvG/XD2AzmkzYsIPgAoyHtQJuAmeGftltpmkBPhAAyaS6V1N0sMSE9CuprQJ9vTJs1A5KNqHxYPbo8z5xcSOJNpBuyiHbHSCztEVukFlRNAjekav6M14Ml6Md+NzXJowJj3b6E8Y3z/DOaAS</latexit>
}
periodic equilibrium
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Maintaining Structure and Function
A stable flow-kick FP is in the same basin of attraction.
From Figure 3 in Meyer, Katherine, et al. "Quantifying resilience to recurrent ecosystem disturbances using flow–kick dynamics." Nature Sustainability 1.11 (2018): 671.
fixed point of map
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Adapted from Figure 2 in Meyer, Katherine, et al. "Quantifying resilience to recurrent ecosystem disturbances using flow–kick dynamics." Nature Sustainability 1.11 (2018): 671.
(flow time)
(kic
k)
Resilience (same basin)
Not Resilient
k <latexit sha1_base64="S3l8nnBLDerjgmOsR9KRP3N3+lk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5rhfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f09OM8w==</latexit>
⌧<latexit sha1_base64="Wa8bCywmcQCIXorlDsXU45DPVf0=">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120y7d3YTdiVBC/4IXD4p49Q9589+YtDlo64OBx3szzMwLYiksuu63s7a+sbm1Xdop7+7tHxxWjo7bNkoM4y0Wych0A2q5FJq3UKDk3dhwqgLJO8HkLvc7T9xYEelHnMbcV3SkRSgYxVzqI00Glapbc+cgq8QrSBUKNAeVr/4wYoniGpmk1vY8N0Y/pQYFk3xW7ieWx5RN6Ij3Mqqp4tZP57fOyHmmDEkYmaw0krn6eyKlytqpCrJORXFsl71c/M/rJRje+KnQcYJcs8WiMJEEI5I/TobCcIZymhHKjMhuJWxMDWWYxVPOQvCWX14l7XrNu6zVH66qjdsijhKcwhlcgAfX0IB7aEILGIzhGV7hzVHOi/PufCxa15xi5gT+wPn8ASJ7jkw=</latexit>
Disturbance Spacek⌧�<latexit sha1_base64="r7HIMNjHDgJjtqySPXZUJKLjnO0=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBiyVRQY9FLx4r2A9oQ9lsN+3azSbsToQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4WV1bX1jeJmaWt7Z3evvH/QNHGqGW+wWMa6HVDDpVC8gQIlbyea0yiQvBWMbqd+64lrI2L1gOOE+xEdKBEKRtFKzVEXaXrWK1fcqjsDWSZeTiqQo94rf3X7MUsjrpBJakzHcxP0M6pRMMknpW5qeELZiA54x1JFI278bHbthJxYpU/CWNtSSGbq74mMRsaMo8B2RhSHZtGbiv95nRTDaz8TKkmRKzZfFKaSYEymr5O+0JyhHFtCmRb2VsKGVFOGNqCSDcFbfHmZNM+r3kX1/P6yUrvJ4yjCERzDKXhwBTW4gzo0gMEjPMMrvDmx8+K8Ox/z1oKTzxzCHzifP1Z9jvg=</latexit>
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A stable flow-kick FP is in a socially-valued region of the phase space.
Adapted from Figure 3 in Zeeman, Mary Lou, et al. "Resilience of socially valued properties of natural systems to repeated disturbance: A framework to support value-laden management
decisions." Natural Resource Modeling 31.3 (2018): e12170.
Socially-valued region⌧<latexit sha1_base64="Wa8bCywmcQCIXorlDsXU45DPVf0=">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120y7d3YTdiVBC/4IXD4p49Q9589+YtDlo64OBx3szzMwLYiksuu63s7a+sbm1Xdop7+7tHxxWjo7bNkoM4y0Wych0A2q5FJq3UKDk3dhwqgLJO8HkLvc7T9xYEelHnMbcV3SkRSgYxVzqI00Glapbc+cgq8QrSBUKNAeVr/4wYoniGpmk1vY8N0Y/pQYFk3xW7ieWx5RN6Ij3Mqqp4tZP57fOyHmmDEkYmaw0krn6eyKlytqpCrJORXFsl71c/M/rJRje+KnQcYJcs8WiMJEEI5I/TobCcIZymhHKjMhuJWxMDWWYxVPOQvCWX14l7XrNu6zVH66qjdsijhKcwhlcgAfX0IB7aEILGIzhGV7hzVHOi/PufCxa15xi5gT+wPn8ASJ7jkw=</latexit>
k <latexit sha1_base64="S3l8nnBLDerjgmOsR9KRP3N3+lk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5rhfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8f09OM8w==</latexit>
S<latexit sha1_base64="vFchJGr6Z+gyWiveB04FdIL/myI=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hyiOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaHSPJYPZpygH9GB5CFn1Fipft8rltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AK9zjNs=</latexit>
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Now let’s apply these ideas to savannas
Savannas are ecosystems defined by a mixture of trees and grass, where fire is an essential
mechanism for balancing the two populations.
Selfie with a lion from a trip to Kenya in August 2016
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Disclaimer: I don’t know a lot about savannas.
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Disclaimer: I don’t know a lot about savannas.
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G: grass S: saplings T: trees
Three simple deterministic ODE savanna models
All three of these models show some kind of bistability.
G: grass T: trees OR
Fire and grazing
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Model #1
dG
dt= r1
G
G+ c1T + c2� d1G� c3G� k1nG
dT
dt= r2
c1T
G+ c1T + c2� d2T � c4T � k2n↵GT
<latexit sha1_base64="Mfj+CDnFQ2VeEw1GRmO0lG1Kl08=">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</latexit>
growth mortality
grazing
fire
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Bistability between savanna and woodland
Model #1
From Figure 7
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Model #2
growth mortality
fire
G = µS + ⌫T � �GT
S = �GT � (!(G) + µ)S
T = !(G)S � ⌫T<latexit sha1_base64="dAeQj+8ishtT7NQHNaB670t6LxY=">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</latexit>
The (simplest) model in this paper
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Model #2
from Figure 1
Bistability between grass-dominated and tree-dominated
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Model #3
Applied mathematical modeling (2016)
dGdt = r1G(1�G/KG)� �TGdTdt = r2T (1� T/KT )
)for t 6= n⌧
<latexit sha1_base64="lrQnDGK9iMEvsSwdvtngxwZc8Pw=">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</latexit>
�G = G(n⌧+)�G(n⌧�) = �k1G
�T = T (n⌧+)� T (n⌧�) = �k2(k1G)2
22 + (k1G)2T
<latexit sha1_base64="vtgs4FSu5+IZij8hXQlBTYvp5Lw=">AAACVXicbVFbS8MwGE3rfd6mPvoSHMqGbLRV0JeBqDAfFToV1m2kWaphaVqSr8Io+5O+iP/EF8Fs1rsfBE7OhSQnYSq4Bsd5tuyZ2bn5hcWl0vLK6tp6eWPzWieZoqxNE5Go25BoJrhkbeAg2G2qGIlDwW7C4dlEv3lgSvNE+jBKWTcmd5JHnBIwVL8sgnMmgODWXrNVlQGQrLdfq3/Aeq1ZH/bdVhCUCp+/1/S/fP4Pn4eDSBGaVyeRWs8b517P2//cYb9frjgNZzr4L3ALUEHFXPbLj8EgoVnMJFBBtO64TgrdnCjgVLBxKcg0SwkdkjvWMVCSmOluPm1ljHcNM8BRosySgKfs90ROYq1HcWicMYF7/VubkP9pnQyi427OZZoBk/T9oCgTGBI8qRgPuGIUxMgAQhU3d8X0nphmwHxEyZTg/n7yX3DtNdyDhnd1WDk5LepYRNtoB1WRi47QCbpAl6iNKHpEL5Zl2daT9WrP2vPvVtsqMlvox9jrb6z5rOg=</latexit>
fire
competitiongrowth&mortality
![Page 18: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/18.jpg)
Model #3
Applied mathematical modeling (2016)
Figure 5c Figure 7a
Woodland&Savanna Bistability
Woodland&Grassland Bistability
![Page 19: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/19.jpg)
Model #3
Applied mathematical modeling (2016)
This model fits into our flow-kick resilience framework and we will base our model on this one!
dGdt = r1G(1�G/KG)� �TGdTdt = r2T (1� T/KT )
)for t 6= n⌧
<latexit sha1_base64="lrQnDGK9iMEvsSwdvtngxwZc8Pw=">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</latexit>
�G = G(n⌧+)�G(n⌧�) = �k1G
�T = T (n⌧+)� T (n⌧�) = �k2(k1G)2
22 + (k1G)2T
<latexit sha1_base64="vtgs4FSu5+IZij8hXQlBTYvp5Lw=">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</latexit>
![Page 20: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/20.jpg)
Underlying nondimensionalized one-sided inhibitory Lotka-Volterra system
x = x(1� x� ↵y)
y = �y(1� y)<latexit sha1_base64="zohXocSlF6b0Xyof7CBG/W9hG84=">AAACG3icbVDLSgMxFM3UV62vqks3waK0i5aZKuhGKLpxWcE+oDOUTJq2oZkHyR3pMPQ/3Pgrblwo4kpw4d+YTrvQ1gOBwznnJrnHDQVXYJrfRmZldW19I7uZ29re2d3L7x80VRBJyho0EIFsu0QxwX3WAA6CtUPJiOcK1nJHN1O/9cCk4oF/D3HIHI8MfN7nlICWuvmq3QsgGU9Or8ZFqzwu20SEQ4Ljkm3nUivWlt1jArSIdSQudfMFs2KmwMvEmpMCmqPezX/qm2jkMR+oIEp1LDMEJyESOBVskrMjxUJCR2TAOpr6xGPKSdLdJvhEKz3cD6Q+PuBU/T2REE+p2HN10iMwVIveVPzP60TQv3QS7ocRMJ/OHupHAkOAp0XhHpeMgog1IVRy/VdMh0QSCrrOnC7BWlx5mTSrFeusUr07L9Su53Vk0RE6RkVkoQtUQ7eojhqIokf0jF7Rm/FkvBjvxscsmjHmM4foD4yvH9IUn2A=</latexit>
↵ = 0.8
� = 0.6<latexit sha1_base64="tdjop1hk85xTtFyblq4ygU46Vwo=">AAACAHicbVDLSsNAFJ3UV42vqAsXbgaL4CokVbQboejGZQX7gCaUyWTSDp1MwsxEKKEbf8WNC0Xc+hnu/BsnbRbaeuDCmXPuZe49QcqoVI7zbVRWVtfWN6qb5tb2zu6etX/QkUkmMGnjhCWiFyBJGOWkrahipJcKguKAkW4wvi387iMRkib8QU1S4sdoyGlEMVJaGlhHHmLpCF07dsPzTC8kTBWPy4FVc2xnBrhM3JLUQInWwPrywgRnMeEKMyRl33VS5edIKIoZmZpeJkmK8BgNSV9TjmIi/Xx2wBSeaiWEUSJ0cQVn6u+JHMVSTuJAd8ZIjeSiV4j/ef1MRQ0/pzzNFOF4/lGUMagSWKQBQyoIVmyiCcKC6l0hHiGBsNKZmToEd/HkZdKp2+65Xb+/qDVvyjiq4BicgDPggivQBHegBdoAgyl4Bq/gzXgyXox342PeWjHKmUPwB8bnDw4JlMM=</latexit>
Our Model
x =G
KGand y =
T
KT<latexit sha1_base64="m1M945JXEzCquoLIuZjNipnShLE=">AAACFXicbVA9SwNBEN3zM8avqKXNYhAsJNxFQRshaBHBJkK+IAlhb7OXLNnbO3bnJOG4P2HjX7GxUMRWsPPfuEmu0MQHA4/3ZpiZ54aCa7Dtb2tpeWV1bT2zkd3c2t7Zze3t13UQKcpqNBCBarpEM8ElqwEHwZqhYsR3BWu4w5uJ33hgSvNAVmEcso5P+pJ7nBIwUjd3Orpqe4rQuJzEd91ygtvARhBjIns4yY5Tszoxq0k3l7cL9hR4kTgpyaMUlW7uq90LaOQzCVQQrVuOHUInJgo4FSzJtiPNQkKHpM9ahkriM92Jp18l+NgoPewFypQEPFV/T8TE13rsu6bTJzDQ895E/M9rReBddmIuwwiYpLNFXiQwBHgSEe5xxSiIsSGEKm5uxXRATA5ggsyaEJz5lxdJvVhwzgrF+/N86TqNI4MO0RE6QQ66QCV0iyqohih6RM/oFb1ZT9aL9W59zFqXrHTmAP2B9fkDRFWe4g==</latexit>
![Page 21: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/21.jpg)
Add fire as an impulse
�x = �k1x
�y = �k2y<latexit sha1_base64="ttUGMoq9OWxkknyMLoTjb10e9Dw=">AAACDHicbVDLSsNAFJ3UV42vqks3g0VwY0mqoBuhqAuXFewDmlAm00k7dPJg5kYaQj/Ajb/ixoUibv0Ad/6N0zYLbT0wcDjnXO7c48WCK7Csb6OwtLyyulZcNzc2t7Z3Srt7TRUlkrIGjUQk2x5RTPCQNYCDYO1YMhJ4grW84fXEbz0wqXgU3kMaMzcg/ZD7nBLQUrdUdm6YAIJH+BKfDLv2yHHMXEpnUjXVKatiTYEXiZ2TMspR75a+nF5Ek4CFQAVRqmNbMbgZkcCpYGPTSRSLCR2SPutoGpKAKTebHjPGR1rpYT+S+oWAp+rviYwESqWBp5MBgYGa9ybif14nAf/CzXgYJ8BCOlvkJwJDhCfN4B6XjIJINSFUcv1XTAdEEgq6P1OXYM+fvEia1Yp9WqnenZVrV3kdRXSADtExstE5qqFbVEcNRNEjekav6M14Ml6Md+NjFi0Y+cw++gPj8weUkJjI</latexit>
Possible impact of fire
Not good enough: Intensity of fire depends on grass biomass that burns, influencing the level of tree mortality.
![Page 22: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/22.jpg)
Add fire as an impulse
Option 2
where is an increasing sigmoidal function!<latexit sha1_base64="tlCosEe7acB15WG/y1yAvFPNxVY=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw==</latexit>
�x = �k1x
�y = �k2!(k1x)y<latexit sha1_base64="N88fNSS2J1whp41sAG4D29kmDO0=">AAACGnicbZC7SgNBFIZn4y2ut1VLm8GgxMKwGwVthKAWlhHMBbIhzE5OkiGzF2ZmJUvIc9j4KjYWitiJjW/jbLKFRg8M/Hz/OZw5vxdxJpVtfxm5hcWl5ZX8qrm2vrG5ZW3v1GUYCwo1GvJQND0igbMAaoopDs1IAPE9Dg1veJX6jXsQkoXBnUoiaPukH7Aeo0Rp1LEc9xq4IniEDy/w8bDjjFzXzFiSsbIb+tAnxdQ9SjpWwS7Z08J/hZOJAsqq2rE+3G5IYx8CRTmRsuXYkWqPiVCMcpiYbiwhInRI+tDSMiA+yPZ4etoEH2jSxb1Q6BcoPKU/J8bElzLxPd3pEzWQ814K//Naseqdt8csiGIFAZ0t6sUcqxCnOeEuE0AVT7QgVDD9V0wHRBCqdJqmDsGZP/mvqJdLzkmpfHtaqFxmceTRHtpHReSgM1RBN6iKaoiiB/SEXtCr8Wg8G2/G+6w1Z2Qzu+hXGZ/fDXKdyQ==</latexit>
!(k1x) =(k1x)2
a
2 + (k1x)2<latexit sha1_base64="AkuH8uR2ZtsBtw3lJk/KRxzOmCY=">AAACE3icbVDLSsNAFJ3UV62vqEs3wSJUhZJEQTdC0Y3LCvYBTRom00k7dGYSZiZiCf0HN/6KGxeKuHXjzr9x+kC09cDAuefcy517woQSqWz7y8gtLC4tr+RXC2vrG5tb5vZOXcapQLiGYhqLZgglpoTjmiKK4mYiMGQhxY2wfzXyG3dYSBLzWzVIsM9gl5OIIKi0FJhHXsxwF5b6gXN/eOFFAqJsUrTdYQbb7vFPFZhFu2yPYc0TZ0qKYIpqYH56nRilDHOFKJSy5diJ8jMoFEEUDwteKnECUR92cUtTDhmWfja+aWgdaKVjRbHQjytrrP6eyCCTcsBC3cmg6slZbyT+57VSFZ37GeFJqjBHk0VRSi0VW6OArA4RGCk60AQiQfRfLdSDOhelYyzoEJzZk+dJ3S07J2X35rRYuZzGkQd7YB+UgAPOQAVcgyqoAQQewBN4Aa/Go/FsvBnvk9acMZ3ZBX9gfHwDGkCcZQ==</latexit>
a = 0.08<latexit sha1_base64="tls9tb9ZyIKVyucFVMMsWmzsCEU=">AAAB7XicbVDLSgMxFL1TX7W+qi7dBIvgqsxUwW6EohuXFewD2qFk0kwbm0mGJCOUof/gxoUibv0fd/6N6XQW2nog5HDOvdx7TxBzpo3rfjuFtfWNza3idmlnd2//oHx41NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJrdzv/NElWZSPJhpTP0IjwQLGcHGSm187Vbd+qBcsV8GtEq8nFQgR3NQ/uoPJUkiKgzhWOue58bGT7EyjHA6K/UTTWNMJnhEe5YKHFHtp9m2M3RmlSEKpbJPGJSpvztSHGk9jQJbGWEz1sveXPzP6yUmrPspE3FiqCCLQWHCkZFofjoaMkWJ4VNLMFHM7orIGCtMjA2oZEPwlk9eJe1a1buo1u4vK42bPI4inMApnIMHV9CAO2hCCwg8wjO8wpsjnRfn3flYlBacvOcY/sD5/AEJ4Y4e</latexit>
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x = x(1� x� ↵y)
y = �y(1� y)
�for t 6= n⌧
<latexit sha1_base64="pYt+rpT6mO+vKoX30QpQc0vprW8=">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</latexit>
Running our model
�x(n⌧) = �k1x
�y(n⌧) = �k2!(k1x)y<latexit sha1_base64="pS02LT5qq4xAEgxIyirEnIP9yUs=">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</latexit>
↵ = 0.8, � = 0.6
k1 = 0.16, k2 = 0.8, ⌧ = 1.75<latexit sha1_base64="vmnuuO3jnuQW7Pc/1gfUCAQUo3U=">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</latexit>
![Page 24: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/24.jpg)
Running our model
![Page 25: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/25.jpg)
For what set of disturbances is this bistability maintained?
Savanna-Savanna Bistability! woodlandsavanna woodlandsavanna grasslandgrassland
![Page 26: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/26.jpg)
Bifurcations in tau using MatContM
3D bifurcation diagram in tau
![Page 27: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/27.jpg)
A slice in tau gives the phase portrait.
![Page 28: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/28.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 29: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/29.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 30: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/30.jpg)
Grass branchForest branch
Limit point (saddle node)
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Grass branchForest branch
Limit point (saddle node)
![Page 32: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/32.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 33: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/33.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 34: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/34.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 35: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/35.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 36: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/36.jpg)
Grass branchForest branch
Limit point (saddle node)
![Page 37: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/37.jpg)
using MatContM
![Page 38: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/38.jpg)
There are 8 regions in the stability diagram.Region 1 Region 2
Region 3 Region 4 Region 5
Region 6 Region 7 Region 8
![Page 39: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/39.jpg)
woodlandsavanna woodlandsavanna grasslandgrassland
There are 8 regions in the stability diagram.
Region 2
Region 4
![Page 40: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/40.jpg)
woodlandsavanna woodlandsavanna grasslandgrassland
There are 8 regions in the stability diagram.
Region 5
![Page 41: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/41.jpg)
woodlandsavanna woodlandsavanna grasslandgrassland
There are 8 regions in the stability diagram.
Region 7
Region 6
Region 7
![Page 42: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/42.jpg)
Stability diagram for different values of k2<latexit sha1_base64="L0pxogbG3u7cVl46PjHJ9DiDAgE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0MO7X+uWKW3XnIKvEy0kFcjT65a/eIGZphNIwQbXuem5i/Iwqw5nAaamXakwoG9Mhdi2VNELtZ/NTp+TMKgMSxsqWNGSu/p7IaKT1JApsZ0TNSC97M/E/r5ua8NrPuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbRqVe+iWru/rNRv8jiKcAKncA4eXEEd7qABTWAwhGd4hTdHOC/Ou/OxaC04+cwx/IHz+QP6pY2Y</latexit>
k2 = 0.8<latexit sha1_base64="z/KeNiSBgpp22xMJccRoCnA6/ds=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4CkkV7EUoevFYwX5AG8pmu2mXbHbD7kYooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1POtPG8b2dtfWNza7u0U97d2z84rBwdt7XMFKEtIrlU3RBrypmgLcMMp91UUZyEnHbC+G7md56o0kyKRzNJaZDgkWARI9hYqRMPajeeWx9Uqp7rzYFWiV+QKhRoDipf/aEkWUKFIRxr3fO91AQ5VoYRTqflfqZpikmMR7RnqcAJ1UE+P3eKzq0yRJFUtoRBc/X3RI4TrSdJaDsTbMZ62ZuJ/3m9zET1IGcizQwVZLEoyjgyEs1+R0OmKDF8YgkmitlbERljhYmxCZVtCP7yy6ukXXP9S7f2cFVt3BZxlOAUzuACfLiGBtxDE1pAIIZneIU3J3VenHfnY9G65hQzJ/AHzucP03COkw==</latexit>
k2 = 0.6<latexit sha1_base64="FH11Jenia/Q7kuLIq+kEABHzysI=">AAAB7nicbVBNS8NAEJ31s9avqkcvi0XwFJIq6kUoevFYwX5AG8pmu2mXbDZhdyOU0B/hxYMiXv093vw3btsctPXBwOO9GWbmBang2rjuN1pZXVvf2Cxtlbd3dvf2KweHLZ1kirImTUSiOgHRTHDJmoYbwTqpYiQOBGsH0d3Ubz8xpXkiH804ZX5MhpKHnBJjpXbUr924zmW/UnUddwa8TLyCVKFAo1/56g0SmsVMGiqI1l3PTY2fE2U4FWxS7mWapYRGZMi6lkoSM+3ns3Mn+NQqAxwmypY0eKb+nshJrPU4DmxnTMxIL3pT8T+vm5nw2s+5TDPDJJ0vCjOBTYKnv+MBV4waMbaEUMXtrZiOiCLU2ITKNgRv8eVl0qo53rlTe7io1m+LOEpwDCdwBh5cQR3uoQFNoBDBM7zCG0rRC3pHH/PWFVTMHMEfoM8f0GiOkQ==</latexit>
![Page 43: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/43.jpg)
Stability diagram for different values of (the switching value in )
a<latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSrlW9i2qteVmp3+RxFOEETuEcPLiCOtxBA1rAAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q==</latexit>
!<latexit sha1_base64="tlCosEe7acB15WG/y1yAvFPNxVY=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw==</latexit>
a = 0.08<latexit sha1_base64="tls9tb9ZyIKVyucFVMMsWmzsCEU=">AAAB7XicbVDLSgMxFL1TX7W+qi7dBIvgqsxUwW6EohuXFewD2qFk0kwbm0mGJCOUof/gxoUibv0fd/6N6XQW2nog5HDOvdx7TxBzpo3rfjuFtfWNza3idmlnd2//oHx41NYyUYS2iORSdQOsKWeCtgwznHZjRXEUcNoJJrdzv/NElWZSPJhpTP0IjwQLGcHGSm187Vbd+qBcsV8GtEq8nFQgR3NQ/uoPJUkiKgzhWOue58bGT7EyjHA6K/UTTWNMJnhEe5YKHFHtp9m2M3RmlSEKpbJPGJSpvztSHGk9jQJbGWEz1sveXPzP6yUmrPspE3FiqCCLQWHCkZFofjoaMkWJ4VNLMFHM7orIGCtMjA2oZEPwlk9eJe1a1buo1u4vK42bPI4inMApnIMHV9CAO2hCCwg8wjO8wpsjnRfn3flYlBacvOcY/sD5/AEJ4Y4e</latexit>
a = 0.05<latexit sha1_base64="uPEqCWkHaHmrad+lK2DGvTwmF2M=">AAAB7XicbVDLSgMxFL1TX7W+Rl26CRbBVZmpim6EohuXFewD2qFk0kwbm0mGJCOUof/gxoUibv0fd/6NaTsLbT0QcjjnXu69J0w408bzvp3Cyura+kZxs7S1vbO75+4fNLVMFaENIrlU7RBrypmgDcMMp+1EURyHnLbC0e3Ubz1RpZkUD2ac0CDGA8EiRrCxUhNfexXvoueW7TcDWiZ+TsqQo95zv7p9SdKYCkM41rrje4kJMqwMI5xOSt1U0wSTER7QjqUCx1QH2WzbCTqxSh9FUtknDJqpvzsyHGs9jkNbGWMz1IveVPzP66QmugoyJpLUUEHmg6KUIyPR9HTUZ4oSw8eWYKKY3RWRIVaYGBtQyYbgL568TJrVin9Wqd6fl2s3eRxFOIJjOAUfLqEGd1CHBhB4hGd4hTdHOi/Ou/MxLy04ec8h/IHz+QMFVY4b</latexit>
![Page 44: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/44.jpg)
-stability diagramk2k1<latexit sha1_base64="c4fWjY+d8PHZ48KqjwrBfpHcsuI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0IWy2m3bNZjfsboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBvGtzO/+0SVZlI8mElK/QSPBIsYwcZKnThoxIEXVGtu3Z0DrRKvIDUo0AqqX4OhJFlChSEca9333NT4OVaGEU6nlUGmaYpJjEe0b6nACdV+Pr92is6sMkSRVLaEQXP190SOE60nSWg7E2zGetmbif95/cxE137ORJoZKshiUZRxZCSavY6GTFFi+MQSTBSztyIyxgoTYwOq2BC85ZdXSadR9y7qjfvLWvOmiKMMJ3AK5+DBFTThDlrQBgKP8Ayv8OZI58V5dz4WrSWnmDmGP3A+fwDqFo6x</latexit>
![Page 45: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/45.jpg)
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
0.8
1.0
1.2
x
y
Back to Resilience!
Interpret structure and function as a stable FP in same basin of attraction
![Page 46: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/46.jpg)
Any region with a stable FP that is not on an axis.
![Page 47: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/47.jpg)
woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert
Interpret structure and function as a stable FP in a socially valued region.
![Page 48: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/48.jpg)
woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert
Social value preimage sets in disturbance space.
bistability
![Page 49: Bifurcation structure and resilience in impulsive models ...ahoyerle/links/SavannaBrandeisJan2020.pdf · Resilience is the capacity of a system to absorb disturbance and still retain](https://reader035.fdocuments.us/reader035/viewer/2022070816/5f0f54477e708231d4439ee7/html5/thumbnails/49.jpg)
Savanna regions
woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert
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Savanna regions (just to compare to bifurcation structure)
woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert
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1. We’re surprised by the range of behaviors in a simple model with complex dynamics.
• Proof of concept by giving the simplest interpretations of the mechanisms of a complicated ecosystem.
Summary
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1. We’re surprised by the range of behaviors in a simple model with complex dynamics.
• Proof of concept by giving the simplest interpretations of the mechanisms of a complicated ecosystem.
2. The savanna we value is delicate and exists only for a small subset of disturbances.
• Illustrates what decision support question involve.
Summary
woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert
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References 1. Walker, Brian, and David Salt. Resilience thinking: sustaining ecosystems and people in a changing world.
Island press, 2012.
2. Carpenter, Steve, Brian Walker, J. Marty Anderies, and Nick Abel. "From metaphor to measurement: resilience of what to what?." Ecosystems 4, no. 8 (2001): 765-781.
3. Zeeman, Mary Lou, Katherine Meyer, Erika Bussmann, Alanna Hoyer‐Leitzel, Sarah Iams, Ian J. Klasky, Victoria Lee, and Stephen Ligtenberg. "Resilience of socially valued properties of natural systems to repeated disturbance: A framework to support value‐laden management decisions." Natural Resource Modeling 31, no. 3 (2018): e12170.
4. Meyer, Katherine, Alanna Hoyer-Leitzel, Sarah Iams, Ian Klasky, Victoria Lee, Stephen Ligtenberg, Erika Bussmann, and Mary Lou Zeeman. "Quantifying resilience to recurrent ecosystem disturbances using flow–kick dynamics." Nature Sustainability 1, no. 11 (2018): 671.
5. Van Langevelde, Frank, Claudius ADM Van De Vijver, Lalit Kumar, Johan Van De Koppel, Nico De Ridder, Jelte Van Andel, Andrew K. Skidmore et al. "Effects of fire and herbivory on the stability of savanna ecosystems." Ecology 84, no. 2 (2003): 337-350.
6. Touboul, Jonathan David, Ann Carla Staver, and Simon Asher Levin. "On the complex dynamics of savanna landscapes." Proceedings of the National Academy of Sciences 115, no. 7 (2018): E1336-E1345.
7. Tamen, A.T., Dumont, Y., Tewa, J.J., Bowong, S. and Couteron, P., 2016. Tree–grass interaction dynamics and pulsed fires: Mathematical and numerical studies. Applied Mathematical Modelling, 40(11-12), pp.6165-6197.
Thank you!