bifurcation structure and resilience in impulsive models...

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

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?

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.

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

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=">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</latexit>

}

periodic equilibrium

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

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>

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>

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

Disclaimer: I don’t know a lot about savannas.

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

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

Bistability between savanna and woodland

Model #1

From Figure 7

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

Model #2

from Figure 1

Bistability between grass-dominated and tree-dominated

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=">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</latexit>

fire

competitiongrowth&mortality

Model #3


Figure 5c Figure 7a

Woodland&Savanna Bistability

Woodland&Grassland Bistability

Model #3


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>

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>

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.

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>

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>

Running our model

For what set of disturbances is this bistability maintained?

Savanna-Savanna Bistability! woodlandsavanna woodlandsavanna grasslandgrassland

Bifurcations in tau using MatContM

3D bifurcation diagram in tau

A slice in tau gives the phase portrait.

Grass branchForest branch

Limit point (saddle node)

using MatContM

There are 8 regions in the stability diagram.Region 1 Region 2

Region 3 Region 4 Region 5

Region 6 Region 7 Region 8

woodlandsavanna woodlandsavanna grasslandgrassland

There are 8 regions in the stability diagram.

Region 2

Region 4



Region 5



Region 7

Region 6

Region 7

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>

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>

-stability diagramk2k1<latexit sha1_base64="c4fWjY+d8PHZ48KqjwrBfpHcsuI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0IWy2m3bNZjfsboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8MOVMG9f9dkpr6xubW+Xtys7u3v5B9fCoo2WmCG0TyaXqhVhTzgRtG2Y47aWK4iTktBvGtzO/+0SVZlI8mElK/QSPBIsYwcZKnThoxIEXVGtu3Z0DrRKvIDUo0AqqX4OhJFlChSEca9333NT4OVaGEU6nlUGmaYpJjEe0b6nACdV+Pr92is6sMkSRVLaEQXP190SOE60nSWg7E2zGetmbif95/cxE137ORJoZKshiUZRxZCSavY6GTFFi+MQSTBSztyIyxgoTYwOq2BC85ZdXSadR9y7qjfvLWvOmiKMMJ3AK5+DBFTThDlrQBgKP8Ayv8OZI58V5dz4WrSWnmDmGP3A+fwDqFo6x</latexit>

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

Any region with a stable FP that is not on an axis.

woodlandsavanna woodlandsavannasavanna grasslandgrasslanddesert

Interpret structure and function as a stable FP in a socially valued region.


Social value preimage sets in disturbance space.

bistability

Savanna regions


Savanna regions (just to compare to bifurcation structure)


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

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


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!

bifurcation structure and resilience in impulsive models...

Documents