Minimizing Landscape Resistance for Habitat Conservation CPAIOR - PowerPoint PPT Presentation

Minimizing Landscape Resistance for Habitat Conservation CPAIOR 2017 na , Graeme Gange, Peter Schachte and Peter J. Stuckey Diego de U˜ June 6, 2017 University of Melbourne — CIS Department

Table of Contents 1. Motivation 2. Isolation by Resistance 3. Modeling the Problem 4. Solving the Problem Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 2

Motivation

Why did the chicken cross the road? • Repeatedly shown that isolation leads to extinction . • Direct perturbation may not lead to extinction, but isolation does (Brook et al., 2008) . • Landscape influences genetic variation (Manel et al., 2003) , (Rosenberg et al., 1997) , (Pimm et al., 1988) • Action needed to restore to movement/spread of species Fragmentation of habitats Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 4

Gene Flow Matters • The Great Prairie Chicken (Westemeier et al., 1998) • Threatened Species • Individuals: 1933 �− → 25000, 1993 �− → 50 • Hatching success rate: 1960 �− → 90%, 1990 �− → 74% • Genetic variation declined 30% Great Prairie Chicken (1960-1990) • Introducing groups from other areas: Hatching success rate 94% • Deer Mouse (Schwartz et al., 2005) • Study compare survival rates: • Control group �− → 88% Deer Mouse • Migrant group introduced �− → 98% Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 5

Isolation by Resistance

Modeling Movement of Species via Genetic Distance • Model for genetic distance (a.k.a. population differentiation, “fixation index”, F ST ). • Better understanding of movement of species • Better action planning to improve survival • Metrics ( R 2 “fit the data”): • IBD: Geographic distance (0.24) • LCP/LCC: “Shortest” path (0.37) • IBR: Effective resistance in an electrical circuit (0.68) Fig. 1 : Comparison of models and how the fit the data (McRae et al., 2007) . Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 7

Modeling Movement of Species via Genetic Distance • Model for genetic distance (a.k.a. population differentiation, “fixation index”, F ST ). • Better understanding of movement of species • Better action planning to improve survival • Metrics ( R 2 “fit the data”): • IBD: Geographic distance (0.24) • LCP/LCC: “Shortest” path (0.37) • IBR: Effective resistance in an electrical circuit (0.68) Fig. 1 : Comparison of models and how the fit the data (McRae et al., 2007) . Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 8

Optimization Problem Isolation by Resistance: • Used by experts e.g. (Amos et al., 2012) , (Amos et al., 2014) • Justified: commute time between nodes ∝ to effective resistance (Lov´ asz, 1993) , (Doyle et al., 1984) • Justified: current intensity at a node ∝ probability of random walker passing through that node ⇒ The more resistance, the more isolation ⇒ The more isolation, the least chance of survival Problem Given a maximum budget, where to improve the habitat (i.e. invest) to minimize landscape resistance? We are trying to decide where to reforest , build an animal bridge , eradicate invasive species , improve soil , etc Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 9

Modeling the Problem

Modeling landscape as an electrical circuit (1) • Split landscape into “patches” s • Land patches ← → nodes 1 A • Adjacent patches ← → resistor • The value of each resistor is the “difficulty” of movement between patches • Eff. res. depends on all the resistors t • Decrease the value of resistors = ⇒ Eff. res. decreases Fig. 2 : Modeling landscape as a resistor = ⇒ Which resistors matter most? network (habitats are s and t ) Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 11

Modeling landscape as an electrical circuit (2) Fig. 3 : Example of landscape: current (left) and conductivity (right) Definitions Effective Resistance between s and t : R st Conductance = 1/Resistance (G = 1/R) Intensity at a node n = � b ∈ branches ( n ) | i b | / 2 Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 12

Modeling landscape as an electrical circuit (3) Fig. 4 : Example of landscape: current (left) and conductivity (right) More than 2 habitats We define the total resistance as the sum of all-pairs resistances. Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 13

Problem Formulation Given: • Graph G = ( V , E ). • Set of pairs of “focal” nodes F ⊂ V 2 . • Conductance function g ∅ : E �→ R for original resistors. • Conductance function g E : E �→ R for improved resistors. • Budget B . • Cost function c : E �→ R for the cost of improving a resistor e . Minimize the Total Effective Resistance, subject to: • � � � b e ∗ c ( e ) ≤ B e ∈ E ���� Boolean decisions for investment Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 14

Computing Effective Resistance between s and t 1. Define Laplacian matrix ( | V | × | V | ):  − g ij if ( i , j ) ∈ E    � L i , j = if i = j k ∈ adj ( i ) g ik    0 otherwise 2. Remove t th row and column of L . , ..., 0) T 3. Solve system: Lv = i , where i = (0 , ..., 1 ���� i s 4. Effective resistance: v s . (5. Voltages of all nodes are in v = ⇒ We can compute currents.) Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 15

Mathematical Problem Formulation Two habitats: 1. Define Laplacian matrix:  ij + (1 − b e ) ∗ g ∅ − ( b e ∗ g E ij ) if ( i , j ) ∈ E    � ik + (1 − b e ) ∗ g ∅ L i , j = k ∈ adj ( i ) ( b e ∗ g E ik ) if i = j    0 otherwise 2. Remove t th row and column of L . 3. Minimize v s subject to the system: , ..., 0) T . Lv = i , where i = (0 , ..., 1 ���� i s Three or more habitats: • Define a system Lv = i for each pair of � s i , t i � ∈ F . • Minimize � � s i , t i �∈ F v � s i , t i � s i Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 16

Solving the Problem

Solving as a MIP problem • IBM CPLEx 12.4 • Use indicator constraints of the form: ⇒ g e = g E ( e ) b e = ⇒ g e = g ∅ ( e ) ¬ b e = Problem: • Weak relaxation • Optimality gap above 40% even afte 5 hours for 10 × 10 grids • Could not find better model (model is “imposed” limited by the definition of effective resistance) Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 18

Greedy algorithm 1. Define the Laplacians for each pair � s i , t i � ∈ F using g ∅ . 2. Solve all the systems (No decisions) 3. Compute currents of all branches: � s i , t i �∈ F ( v � s i , t i � − v � s i , t i � ab ∗ � i ab = g ∅ ) a b 4. Invest in the branches with highest current (more animals walking through there!) until exhausting the budget. • Surprisingly good results • Total resistance dropped between 40% and 60% (for small landscapes)! Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 19

Local Search 1. Begin with the solution s given by the Greedy algorithm. 2. Repeat: 2.1 Choose some “invested resistors” to disinvest (get some budget back). 2.2 Choose some “uninvested resistors” to invest (exhaust the budget). 2.3 Compute objective function. − ( snew − s ) 2.4 Accept new solution s new with probability e (simulated T annealing). Update s consequently. 2.5 Update temperature T : less and less likely to accept a worse solution. Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 20

Choosing where we don’t want to invest anymore • InvRand : Random resistors. • InvLC : Resistors with least current through them. • InvLCP : Probability that favors choosing resistors with low current: 1 Pr ( e ) = current ( e ) Intuition: places where animals aren’t walking anyway. Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 21

Choosing new places to invest • WilRand : Random resistors • WilBFS : Choose a node with a probability distribution that favors high current nodes. Perform BFS around that node until exhausted budget. ⇒ Investments are near each other. = ⇒ Stronger effect in dropping resistance in that region • WilHC : Choose resistors with highest current. • WilHCP : Probability that favors choosing high current resistors. Intuition: places where animals are walking and it could be made easier. Catch: WilHC + InvLC can get stuck in local minimum = ⇒ do one iteration of WilHCP . Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 22

Experimental setting • 500 artificial instances • Realistically generated instance: • Resistance 1Ω to 100Ω • Beta distribution of resistance: much more likely to have high resistance value (like in the real world) • Added oasis with low resistance (e.g. small tree areas) • Instances with homogeneous and heterogeneous cost functions • 200 Local Search iterations Diego de U˜ na Minimizing Landscape Resistance CPAIOR2017 23