quancol . ........ . . . ... ... ... ... ... ... ... - PowerPoint PPT Presentation

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu Time discr cont Aggr none state none state discr cont discr cont discr cont discr cont State popn diff DTMC ? DTMC eqn/ CTMC ODE DTMC discrete-time Markov chain CTMC continuous-time Markov chain SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu Time discr cont Aggr none state none state discr cont discr cont discr cont discr cont State popn diff DTMC ? DTMC eqn/ CTMC LMP ODE DTMC discrete-time Markov chain CTMC continuous-time Markov chain LMP labelled Markov process popn population SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu Time discr cont Aggr none state none state discr cont discr cont discr cont discr cont State popn diff popn DTMC ? DTMC eqn/ CTMC LMP CTMC ODE DTMC discrete-time Markov chain CTMC continuous-time Markov chain LMP labelled Markov process popn population SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu Time discr cont Aggr none state none state discr cont discr cont discr cont discr cont State popn diff popn DTMC ? DTMC eqn/ CTMC LMP CTMC ODE ODE DTMC discrete-time Markov chain CTMC continuous-time Markov chain LMP labelled Markov process popn population SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu cont Time Aggr none state discr cont discr cont State popn CTMC LMP CTMC ODE SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu cont Time Aggr none state discr cont discr cont State LMP popn CTMC CTMC ODE SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu cont Time Aggr none state discr cont discr cont State LMP popn CTMC CTMC ODE lumpability (exact) − − − − − − − − − − − − − − − − − − − → numerical vector form SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Non-spatial dimensions www.quanticol.eu cont Time Aggr none state discr cont discr cont State LMP popn CTMC CTMC ODE lumpability (exact) fluid − − − − − − − − − − − − − − − − − − − → − − − − − − − − → approx numerical vector form SFM-16 12 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Addition of space SFM-16 13 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling fluid approximation State-space − − − − − − − − − − − − − − → ODEs explosion SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling fluid approximation State-space − − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling fluid approximation State-space − − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? Add space SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling fluid approximation State-space − − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? fluid approximation Bigger − − − − − − − − − − − − − − → PDEs state-space explosion SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Scalable analysis applied to space www.quanticol.eu Current approach for scalable modelling fluid approximation State-space − − − − − − − − − − − − − − → ODEs explosion Does this work with discrete space? fluid approximation Bigger − − − − − − − − − − − − − − → PDEs state-space explosion Maybe but not the only approach SFM-16 14 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Quantitative modelling with space www.quanticol.eu � what is the main objective here? SFM-16 15 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Quantitative modelling with space www.quanticol.eu � what is the main objective here? � modelling collective adaptive systems (CAS) � quantitative: behaviour over time is important � aggregation: many agents lead to state space explosion � space: behaviour with respect to space relevant to many CAS SFM-16 15 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Quantitative modelling with space www.quanticol.eu � what is the main objective here? � modelling collective adaptive systems (CAS) � quantitative: behaviour over time is important � aggregation: many agents lead to state space explosion � space: behaviour with respect to space relevant to many CAS � consider mathematical representations of space and movement � results obtained by analysis techniques � semantic target for CAS modelling languages � understand relationships between representations SFM-16 15 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont grid, lattice, regular neighbourhood single individual at each node multiple individuals at each node SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont grid, lattice, regular neighbourhood single individual at each node multiple individuals at each node patches, irregular explicit adjacency relationship regions of cont space multiple individuals in each patch SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont grid, lattice, regular infinite neighbourhood location usually single individual at each node changes smoothly multiple individuals at each node patches, irregular explicit adjacency relationship regions of cont space multiple individuals in each patch SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont grid, lattice, regular infinite neighbourhood location usually single individual at each node changes smoothly multiple individuals at each node patches, irregular explicit adjacency relationship regions of cont space multiple individuals in each patch typically 2-dimensional or 3-dimensional SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial aspects of representations www.quanticol.eu Space discr cont grid, lattice, regular infinite neighbourhood location usually single individual at each node changes smoothly multiple individuals at each node patches, irregular Another approach: topological space explicit adjacency relationship regions of cont space will covered tomorrow in spatio-temporal logic multiple individuals in each patch typically 2-dimensional or 3-dimensional SFM-16 16 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial classification www.quanticol.eu Time continuous Aggr none state and/or space State discrete continuous discrete continuous Space B 2 B 2 B 2 A 1 A 1 A 1 B 1 A 1 A 1 B 1 A 1 B 1 A 1 B 3 B 3 A 1 B 3 A 2 A 2 A 2 B 3 B 3 B 3 B 3 B 3 B 3 B 3 discrete B 3 A 2 A 2 A 2 A 1 A 1 A 1 A 1 A 1 B 1 B 1 B 3 B 3 B 2 A 2 B 2 A 2 B 2 B 2 regular 100 50 A 1 B 1 A 1 0 0 A 1 B 2 A 2 2 B 3 B 3 4 4 2 continuous 0 SFM-16 17 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Discrete space SFM-16 18 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu discrete regular discrete assumption that vertices and edges are static but parameters as- sociated with edges (movement) and vertices (local interaction) may be time-dependent SFM-16 19 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu � set of locations: L � undirected graph over locations: ( L , E L ) � edges are two element sets: { ℓ 1 , ℓ 2 } ∈ P 2 ( L ) � graph is a skeleton which captures where movement or interaction is possible SFM-16 20 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu � set of locations: L � undirected graph over locations: ( L , E L ) � edges are two element sets: { ℓ 1 , ℓ 2 } ∈ P 2 ( L ) � graph is a skeleton which captures where movement or interaction is possible � spatial parameters (ranges remain abstract) � λ ( ℓ ) for all locations ℓ ∈ L , and � η ( ℓ 1 , ℓ 2 ) and η ( ℓ 2 , ℓ 1 ) for all edges { ℓ 1 , ℓ 2 } ∈ E L SFM-16 20 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu � set of locations: L � undirected graph over locations: ( L , E L ) � edges are two element sets: { ℓ 1 , ℓ 2 } ∈ P 2 ( L ) � graph is a skeleton which captures where movement or interaction is possible � spatial parameters (ranges remain abstract) � λ ( ℓ ) for all locations ℓ ∈ L , and � η ( ℓ 1 , ℓ 2 ) and η ( ℓ 2 , ℓ 1 ) for all edges { ℓ 1 , ℓ 2 } ∈ E L � locations � points in space: L � regions in space: f : R × R → L SFM-16 20 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Connectivity in discrete space www.quanticol.eu � full connectivity: ease of analysis SFM-16 21 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Connectivity in discrete space www.quanticol.eu � full connectivity: ease of analysis � neighbourhood: general location graph � one-hop neighbour: traverse a single edge � n -hop neighbour: traverse n edges SFM-16 21 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Connectivity in discrete space www.quanticol.eu � full connectivity: ease of analysis � neighbourhood: general location graph � one-hop neighbour: traverse a single edge � n -hop neighbour: traverse n edges � neighbourhood: spatially regular graph � von Neumann: N, E, S, W � Moore: N, NE, E, SE, S, SW, W, NW SFM-16 21 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Aspects of discrete space www.quanticol.eu � boundary conditions � avoid: graph defined over torus or sphere (0,Ymax) (Xmax,Ymax) Closed Coverage Area (0,0) (Xmax,0) � include: can be an accurate model of reality 600 500 400 300 200 100 0 0 50 100 150 200 250 300 Figure 6: Traveling pattern of an MN using the Random Direction Mobility Model. SFM-16 22 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and homogeneity www.quanticol.eu � location homogeneous : λ ( ℓ i ) = λ ( ℓ j ) for all locations ℓ i , ℓ j ∈ L SFM-16 23 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and homogeneity www.quanticol.eu � location homogeneous : λ ( ℓ i ) = λ ( ℓ j ) for all locations ℓ i , ℓ j ∈ L � transfer homogeneous (movement or interaction): η ( ℓ i , ℓ j ) = η ( ℓ j , ℓ i ) = η ( ℓ i ′ , ℓ j ′ ) = η ( ℓ j ′ , ℓ i ′ ) for all edges { ℓ i , ℓ j } , { ℓ i ′ , ℓ j ′ } ∈ E L SFM-16 23 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and homogeneity www.quanticol.eu � location homogeneous : λ ( ℓ i ) = λ ( ℓ j ) for all locations ℓ i , ℓ j ∈ L � transfer homogeneous (movement or interaction): η ( ℓ i , ℓ j ) = η ( ℓ j , ℓ i ) = η ( ℓ i ′ , ℓ j ′ ) = η ( ℓ j ′ , ℓ i ′ ) for all edges { ℓ i , ℓ j } , { ℓ i ′ , ℓ j ′ } ∈ E L � (spatially) parameter homogeneous : location and transfer homogeneous SFM-16 23 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and homogeneity www.quanticol.eu � location homogeneous : λ ( ℓ i ) = λ ( ℓ j ) for all locations ℓ i , ℓ j ∈ L � transfer homogeneous (movement or interaction): η ( ℓ i , ℓ j ) = η ( ℓ j , ℓ i ) = η ( ℓ i ′ , ℓ j ′ ) = η ( ℓ j ′ , ℓ i ′ ) for all edges { ℓ i , ℓ j } , { ℓ i ′ , ℓ j ′ } ∈ E L � (spatially) parameter homogeneous : location and transfer homogeneous � spatially homogeneous : parameter homogeneous and complete location graph (every location neighbours every other location) SFM-16 23 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and homogeneity www.quanticol.eu � location homogeneous : λ ( ℓ i ) = λ ( ℓ j ) for all locations ℓ i , ℓ j ∈ L � transfer homogeneous (movement or interaction): η ( ℓ i , ℓ j ) = η ( ℓ j , ℓ i ) = η ( ℓ i ′ , ℓ j ′ ) = η ( ℓ j ′ , ℓ i ′ ) for all edges { ℓ i , ℓ j } , { ℓ i ′ , ℓ j ′ } ∈ E L � (spatially) parameter homogeneous : location and transfer homogeneous � spatially homogeneous : parameter homogeneous and complete location graph (every location neighbours every other location) � spatial homogeneity may lead to analytic solutions rather than simulation of differential equations SFM-16 23 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Space and regularity www.quanticol.eu � spatially regular � maybe be parameter homogeneous � not spatially homogeneous � not easy to define from a graph but obvious to identify � two dimensions: triangles, rectangles, hexagons � one dimension: path � other possibilities � characterised by regular way to define neighbours SFM-16 24 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial classification www.quanticol.eu Time continuous Aggr none state and/or space State discrete continuous discrete continuous Space B 2 B 2 B 2 A 1 A 1 A 1 B 1 A 1 A 1 B 1 A 1 B 1 A 1 B 3 B 3 A 1 B 3 A 2 A 2 A 2 B 3 B 3 B 3 B 3 B 3 B 3 B 3 discrete B 3 A 2 A 2 A 2 A 1 A 1 A 1 A 1 A 1 B 1 B 1 B 3 B 3 B 2 A 2 B 2 A 2 B 2 B 2 regular 100 50 A 1 B 1 A 1 0 0 A 1 B 2 A 2 2 B 3 B 3 4 4 2 continuous 0 SFM-16 25 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu discrete state without aggregation B 2 A 1 A 1 B 1 A 1 B 3 A 2 B 3 B 3 B 3 SFM-16 26 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: no state aggregation www.quanticol.eu � no state aggregation: modelling individuals SFM-16 27 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: no state aggregation www.quanticol.eu � no state aggregation: modelling individuals � consider J , named individual � loc( J , t ) ∈ L , location at time t � state( J , t ) = A i or state( J , t ) = Y � set of rules to describe behaviour SFM-16 27 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: no state aggregation www.quanticol.eu � no state aggregation: modelling individuals � consider J , named individual � loc( J , t ) ∈ L , location at time t � state( J , t ) = A i or state( J , t ) = Y � set of rules to describe behaviour � discrete state, discrete space and rates: CTMC with states of the form � � (loc( J 1 , t ) , state( J 1 , t )) , . . . , (loc( J N , t ) , state( J N , t )) SFM-16 27 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: no state aggregation www.quanticol.eu � no state aggregation: modelling individuals � consider J , named individual � loc( J , t ) ∈ L , location at time t � state( J , t ) = A i or state( J , t ) = Y � set of rules to describe behaviour � discrete state, discrete space and rates: CTMC with states of the form � � (loc( J 1 , t ) , state( J 1 , t )) , . . . , (loc( J N , t ) , state( J N , t )) � potential for ( n × p ) N states in CTMC where p is number of locations, n is number of states and N is number of individuals SFM-16 27 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu continuous state without aggregation SFM-16 28 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu discrete state with aggregation B 2 A 2 A 2 A 1 A 1 A 1 A 1 A 1 B 1 A 1 B 3 B 1 A 2 B 3 B 3 B 3 B 3 A 2 B 2 B 2 SFM-16 29 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Notation www.quanticol.eu A subpopulation is the subset of a population that is in a given state. Assuming n subpopulations and p locations, then X ( j ) ( t ) is the size i of the subpopulation i at location j at time t . ( X (1) , . . . , X ( p ) j =1 X ( j ) � p X i = ) X i = i i i ( X ( j ) 1 , . . . , X ( j ) i =1 X ( j ) � n X ( j ) X ( j ) = n ) = i � p j =1 X ( j ) ( X (1) , . . . , X ( n ) ) � n X = X = i =1 i i =1 X ( j ) � p � n = j =1 i The size of subpopulation X ( j ) at time t is N ( j ) ( t ). i i The total size of subpopulation X i at time t is N i ( t ). SFM-16 30 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: state aggregation www.quanticol.eu � discrete space, discrete aggregated state: population CTMC with states of the form X (1) 1 , . . . , X (1) n , . . . , X ( k ) , . . . , X ( k ) , . . . , X ( p ) , . . . , X ( p ) � � n n 1 1 � this CTMC is much smaller than that for discrete space and discrete state without aggregation � analysis provides same results at population level � potential for ( M + 1) n × p states in CTMC where p is number of locations, n is number of states and M is the maximum subpopulation size SFM-16 31 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space www.quanticol.eu continuous state with aggregation B 2 A 2 A 1 A 1 A 1 B 1 A 1 B 3 B 1 A 2 B 3 B 3 B 2 A 2 B 2 B 3 SFM-16 32 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: state aggregation www.quanticol.eu � continuous aggregated state gives population ODEs that approximate CTMC results (under certain conditions) dX ( j ) X ( j ) 1 , . . . , X ( j ) i � � = f i , j + n dt p g i , j , k ( X ( k ) , . . . , X ( k ) ) − h i , j , k ( X ( j ) 1 , . . . , X ( j ) � � � n ) n 1 k =1 , k � = j SFM-16 33 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: state aggregation www.quanticol.eu � continuous aggregated state gives population ODEs that approximate CTMC results (under certain conditions) dX ( j ) X ( j ) 1 , . . . , X ( j ) i � � = f i , j + n dt p g i , j , k ( X ( k ) , . . . , X ( k ) ) − h i , j , k ( X ( j ) 1 , . . . , X ( j ) � � � n ) n 1 k =1 , k � = j � a simpler form with parameter homogeneity p dX ( j ) X ( j ) 1 , . . . , X ( j ) � g ( X ( j ) 1 ) − h ( X ( k ) i � � � = f + )) n 1 dt j = k , j � = j SFM-16 33 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space: state aggregation www.quanticol.eu � continuous aggregated state gives population ODEs that approximate CTMC results (under certain conditions) dX ( j ) X ( j ) 1 , . . . , X ( j ) i � � = f i , j + n dt p g i , j , k ( X ( k ) , . . . , X ( k ) ) − h i , j , k ( X ( j ) 1 , . . . , X ( j ) � � � n ) n 1 k =1 , k � = j � a simpler form with parameter homogeneity p dX ( j ) X ( j ) 1 , . . . , X ( j ) � g ( X ( j ) 1 ) − h ( X ( k ) i � � � = f + )) n 1 dt j = k , j � = j � n × p ODEs where p is number of locations and n is number of states SFM-16 33 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Movement in discrete space SFM-16 34 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space and movement www.quanticol.eu B 2 B 2 A 1 A 1 A 1 B 1 A 1 A 1 B 1 A 1 B 3 B 3 A 2 A 2 B 3 B 3 B 3 B 3 B 3 B 3 � discrete space with state-based aggregation � interaction between and within populations at locations � movement between locations � patch population models � population CTMCs and ODEs with locations SFM-16 35 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Discrete space and movement www.quanticol.eu � single occupancy of locations � discrete space without state-based aggregation � graph transformation rules, change at a location � cellular automata � interacting particle systems SFM-16 36 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Continuous space SFM-16 37 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Continuous space www.quanticol.eu without aggregation A 1 B 1 A 1 A 1 B 2 A 2 B 3 B 3 SFM-16 38 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Spatial classification www.quanticol.eu Time continuous Aggr none state and/or space State discrete continuous discrete continuous Space B 2 B 2 B 2 A 1 A 1 A 1 B 1 A 1 A 1 B 1 A 1 B 1 A 1 B 3 B 3 A 1 B 3 A 2 A 2 A 2 B 3 B 3 B 3 B 3 B 3 B 3 B 3 discrete B 3 A 2 A 2 A 2 A 1 A 1 A 1 A 1 A 1 B 1 B 1 B 3 B 3 B 2 A 2 B 2 A 2 B 2 B 2 regular 100 50 A 1 B 1 A 1 0 0 A 1 B 2 A 2 2 B 3 B 3 4 4 2 continuous 0 SFM-16 39 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Continuous space www.quanticol.eu � R × R or contiguous subset � radius to define neighbourhood � boundary conditions: similar to discrete space � no aggregation, consider individual J � loc( J , t ) = ( x , y ) which is its location at time t � state( J , t ) = A i or state( J , t ) = Y � interaction rules � movement description: random walk, etc � agent model � continuous space and state: continuous-time Markov processes � continuous space and discrete state: hybrid SFM-16 40 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Continuous space www.quanticol.eu with aggregation 100 50 0 0 2 4 4 2 0 SFM-16 41 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Continuous space www.quanticol.eu � discrete aggregated state: spatio-temporal point processes X i (( x , y ) , t ) ∈ N and λ (( x , y ) , t ) describes behaviour dE [ X i ] describes change in average density, global measure dt SFM-16 42 / 78

quan�col . ........ . . . ... ... ... ... ... ... ... Continuous space www.quanticol.eu � discrete aggregated state: spatio-temporal point processes X i (( x , y ) , t ) ∈ N and λ (( x , y ) , t ) describes behaviour dE [ X i ] describes change in average density, global measure dt � continuous aggregated state: partial differential equations ∂ X i f i ( X 1 , . . . , X ()) = n ∂ t ∂ � D ( X i , ( x , y )) ∂ X i � + ∂ x ∂ x � � ∂ D ( X i , ( x , y )) ∂ X i + ∂ y ∂ y D can depend on more than current population size and location SFM-16 42 / 78