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Integrodifference equations for invasive species in heterogeneous environments F . Lutscher University of Ottawa Ottawa, Canada Integrodifference equations for invasive species in heterogeneous environments – p. 1/5

Invasive Species Economic and ecological damage from non-native invasive species (forest insect pests, plants, mussels...) Theory for homogeneous landscapes (S KELLAM , W EINBERGER , K OT ET AL , L IEBHOLD ...) Sources of Heterogeneity Landscape features (S HIGESADA ) Population structure (C ASWELL ) Temporal variation (N EUBERT , C ASWELL , S CHREIBER ) Integrodifference equations for invasive species in heterogeneous environments – p. 2/5

Questions How does habitat heterogeneity impact species persistence and spread? How can we manage biological invasions? What are the important spatial scales? How does fragmentation affect diversity? Integrodifference equations for invasive species in heterogeneous environments – p. 3/5

Outline Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal Integrodifference equations for invasive species in heterogeneous environments – p. 4/5

Discrete-time models and spatial spread u t ( x ) : Density of individuals in generation t. dynamics dispersal u t − → f ( u t ) − → u t +1 � u t +1 ( x ) = K ( x − y ) f ( u t ( y )) dy =: K ∗ f ( u t )( x ) f ( u ) : growth function K ( z ) : Dispersal kernel K OT , L EWIS AND VAN DEN D RIESSCHE , 1996, Ecology Integrodifference equations for invasive species in heterogeneous environments – p. 5/5

Population growth Discrete time, non-overlapping generations u t + τ = f ( u t ) Beverton-Holt, Ricker Allee N a 1 1 density f(u t ) density f(u t ) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 density u t density u t Integrodifference equations for invasive species in heterogeneous environments – p. 6/5

Dispersal kernel Probability of moving from y to x : K ( z ) = K ( x − y ) Exponential root 0.6 0.5 Laplace density Gaussian 0.4 0.3 Tent Tophat 0.2 0.1 0 −4 −2 0 2 4 space Integrodifference equations for invasive species in heterogeneous environments – p. 7/5

Spread in homogeneous landscape Asymptotic speed of spread ( W EINBERGER , 1982 ) 1 c ∗ = inf s ln( RM ( s )) s> 0 f linearly bounded, R = f ′ (0) ≥ f ( u ) /u f monotone � K ( x ) e sx dx Moment generating function, M ( s ) = 2 σ 2 ln( R ) � For Gaussian kernel: c G = Integrodifference equations for invasive species in heterogeneous environments – p. 8/5

Example Constant speed and accelerating traveling waves K OT , L EWIS AND VAN DEN D RIESSCHE , 1996, Ecology Integrodifference equations for invasive species in heterogeneous environments – p. 9/5

Outline Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging Competing species Allee effect Density-Dependent dispersal Integrodifference equations for invasive species in heterogeneous environments – p. 10/5

Fragmented landscape type 1 "good" p 1−p type 2 "bad" space � u t +1 ( x ) = K ( x, y ) f ( u t ( y ) , y ) dy Integrodifference equations for invasive species in heterogeneous environments – p. 11/5

Single species � u t +1 ( x ) = K ( x, y ) f ( u t ( y ) , y ) dy homogeneous, i.e., K ( x − y ) , K AWASAKI AND S HIGESADA , 2006 diffusive dispersal submodel P OWELL AND Z IMMERMANN , 2004; R OBBINS AND L EWIS K ( x + L, y + L ) = K ( x, y ) W EINBERGER , 2002; W EINBERGER ET AL , 2008 origin dependent, e.g., variance: σ 2 ( y ) write K ( x − y, y ) . D EWHIRST AND L UTSCHER , 2009 Integrodifference equations for invasive species in heterogeneous environments – p. 12/5

Spread Linearization: r ( y ) = ∂f � u t +1 ( x ) = K ( x − y, y ) r ( y ) u t ( y ) dy, ∂u (0 , y ) Traveling profile: u t +1 ( x ) = u t ( x − c ) = e − s ( x − c ) v ( x ) � ∞ K ( x − y, y ) e s ( x − y ) r ( y ) v ( y ) dy e sc v ( x ) = −∞ K AWASAKI AND S HIGESADA , 2006 Integrodifference equations for invasive species in heterogeneous environments – p. 13/5

Exact implicit formula Laplace kernel K = 1 2 d exp( −| x − y | /d ) Piecewise constant r ( x ) cosh( µL ) = cosh( q 1 L 1 ) cosh( q 2 L 2 )+ q 2 1 + q 2 2 sinh( q 1 L 1 )+sinh( q 2 L 2 ) , 2 q 1 q 2 where q 1 = 1 q 2 = 1 � � 1 − r 1 e − µc , 1 − r 2 e − µc . d d Integrodifference equations for invasive species in heterogeneous environments – p. 14/5

Global averaging Assumption: variance(s) ≫ habitat period L � 1 �� � LK ( L ( x − y − n ) , y ) e sL ( x − y − n ) e sc v ( x ) = r ( y ) v ( y ) dy 0 n Riemann sum approximation LK ( L ( x − y − n ) , y ) e sL ( x − y − n ) ≈ M ( s, y ) � n In the limit L → 0 �� 1 � 1 c = min s ln M ( s, y ) r ( y ) dy s> 0 0 Integrodifference equations for invasive species in heterogeneous environments – p. 15/5

Piecewise constant habitat r ( y ) = r 1 , 2 , r 1 > 1 , r 1 ≥ r 2 ≥ 0 K ( x − y, y ) = K 1 , 2 ( x − y ) Approximate spreading speed 1 c = inf ˆ s ln { r 1 pM 1 ( s ) + r 2 (1 − p ) M 2 ( s ) } s> 0 Averaged growth rate ¯ r = r 1 p + r 2 (1 − p ) Invasion threshold (rule of thumb) ¯ r = 1 : p min = 1 − r 2 r 1 − r 2 Integrodifference equations for invasive species in heterogeneous environments – p. 16/5

Simulations: Gauss 1.8 1.6 (a) 1.4 spread rate 1.2 (c) 1 0.8 (b) 0.6 0.4 0.2 0 0 0.2 0.4 0.5 0.6 0.8 1 percentage of good habitat p (a) r 1 = 2 , r 2 = 0 , σ 2 = 2 (b) r 1 = 2 , r 2 = 0 , σ 2 = 0 . 25 (c) r 1 = r 2 = 1 . 5 , σ 2 1 = 2 , σ 2 2 = 0 . 5 Integrodifference equations for invasive species in heterogeneous environments – p. 17/5

Random landscapes Gaussian kernel 1 0.8 spread rate 0.6 0.4 0.2 0 0.5 0.6 0.7 0.8 0.9 1 percentage of good habitat r 1 = 1 . 8 , r 2 = 0 . 2 , σ 2 1 = 1 , σ 2 2 = 3 Integrodifference equations for invasive species in heterogeneous environments – p. 18/5

Simulations: fat tails The exponential square root kernel 70 60 front location x f (t) p=1 50 40 p=0.75 30 20 p=0.5 10 0 0 5 10 15 20 25 30 generation t r 1 = 2 , r 2 = 0 , σ 2 1 = 1 Integrodifference equations for invasive species in heterogeneous environments – p. 19/5

Simulations: Laplace 2 (a) 1.5 spread rate (c) 1 (b) true 0.5 approximate 0 0 0.2 0.4 0.5 0.6 0.8 1 percentage of good habitat p (a) r 1 = 2 , r 2 = 0 , σ 2 = 2 (b) r 1 = 2 , r 2 = 0 , σ 2 = 0 . 25 (c) r 1 = r 2 = 1 . 5 , σ 2 1 = 2 , σ 2 2 = 0 . 5 Integrodifference equations for invasive species in heterogeneous environments – p. 20/5

Outline Model and homogeneous theory Averaging in fragmented landscapes Global averaging Patch averaging - persistence Competing species Allee effect Density-Dependent dispersal Integrodifference equations for invasive species in heterogeneous environments – p. 21/5

Patch-level averaging 2 u(x) 1.8 patch 1.6 type II u 1 1.4 1.2 <u> 1 0.8 0.6 u 2 patch 0.4 type I 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Integrodifference equations for invasive species in heterogeneous environments – p. 22/5

Single patch: persistence Linearized equation: � L R = f ′ (0) u t +1 ( x ) = R K ( x − y ) u t ( y ) dy, 0 Persistence condition (critical patch size): λ ( L ) > 1 /R λ ( L ) : leading eigenvalue of the integral operator (K OT AND S CHAFFER , 1986) Integrodifference equations for invasive species in heterogeneous environments – p. 23/5

Average dispersal success Spatially averaged probability of successful settlement inside the patch (V AN K IRK AND L EWIS , 1997) Point -release experiments � L � L ADS ( L ) = 1 K ( x, y ) dxdy L 0 0 Approximate persistence condition: ADS ( L ) > 1 /R L UTSCHER AND L EWIS , 2004, F AGAN AND L UTSCHER , 2006 Integrodifference equations for invasive species in heterogeneous environments – p. 24/5

Examples Average dispersal success of different kernels 1.5 1 0.8 1 density 0.6 ADS 0.4 0.5 0.2 0 0 −1 0 1 0 1 2 3 4 space length Integrodifference equations for invasive species in heterogeneous environments – p. 25/5

Average dispersal success in fragmentation Assume hostile bad habitat: r 2 = 0 ADS ( p ) = average probability to land in a good patch Invasion threshold: ADS ( p min ) = 1 /r 1 Gaussian kernel invasion threshold p min 0.5 Laplace kernel 0.4 0.3 0.2 Exponential square 0.1 root kernel 0 0 0.5 1 1.5 2 variance σ 2 1 Persistence and spread harder in fine grain landscapes D EWHIRST AND L UTSCHER 2009 Integrodifference equations for invasive species in heterogeneous environments – p. 26/5

Comparison: Trees Simulation model: Collingham and Huntley (2000) r 1 = 1 . 02 ... 1 . 09 gives invasion threshold p ≈ 1 . Integrodifference equations for invasive species in heterogeneous environments – p. 27/5

Partial dispersal only � u t +1 ( x ) = (1 − q ) f ( u t ( x )) + q K ( x, y ) f ( u t ( y ) , y ) dy f = 0 on bad patches, f ′ (0) = r > 1 on good patches p min = 1 − (1 − q ) r qr 1 c = inf ˆ s ln { (1 − q ) r + pqrM ( s ) } s> 0 Example: r = 1 . 04 , q = 0 . 05 , σ 2 = 94 gives p min ≈ 0 . 23 . Integrodifference equations for invasive species in heterogeneous environments – p. 28/5