Spatial coupling of patch dynamics x i ( t ) = � k ℓ =1 x ℓ i ( t ) total density of species i Assume Cov [ x ℓ i ( t + ∆t ) − x ℓ i ( t ) | x ( t )] ≈ x ℓ i ( t ) σ ℓ m i ( t ) , x m i ( t + ∆t ) − x m i ( t ) x m ∆t i 10 / 20
Spatial coupling of patch dynamics x i ( t ) = � k ℓ =1 x ℓ i ( t ) total density of species i Assume Cov [ x ℓ i ( t + ∆t ) − x ℓ i ( t ) | x ( t )] ≈ x ℓ i ( t ) σ ℓ m i ( t ) , x m i ( t + ∆t ) − x m i ( t ) x m ∆t i In limit ∆ t ↓ 0, get the Itˆ o stochastic differential equations (SDEs) n � � x ℓ a ℓ ij x ℓ j ( t ) + b ℓ dt + dE ℓ dx i ( t ) = i ( t ) i ( t ) ( ⋆ ) i j =1 ℓ where E ℓ i ( t ) are Brownian motions satisfying Cov [ E ℓ i ( t )] = σ ℓ m i ( t ) , E m t i 10 / 20
Spatial coupling of patch dynamics p ℓ i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k x i ( t ) = � k ℓ =1 x ℓ i ( t ) total density of species i Assume Cov [ x ℓ i ( t + ∆t ) − x ℓ i ( t ) | x ( t )] ≈ x ℓ i ( t ) σ ℓ m i ( t ) , x m i ( t + ∆t ) − x m i ( t ) x m ∆t i In limit ∆ t ↓ 0, get the Itˆ o stochastic differential equations (SDEs) n � � x ℓ a ℓ ij x ℓ j ( t ) + b ℓ dt + dE ℓ dx i ( t ) = i ( t ) i ( t ) ( ⋆ ) i j =1 ℓ where E ℓ i ( t ) are Brownian motions satisfying Cov [ E ℓ i ( t )] = σ ℓ m i ( t ) , E m t i 10 / 20
Spatial coupling of patch dynamics p ℓ i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k x i ( t ) = � k ℓ =1 x ℓ i ( t ) total density of species i Assume Cov [ x ℓ i ( t + ∆t ) − x ℓ i ( t ) | x ( t )] ≈ x ℓ i ( t ) σ ℓ m i ( t ) , x m i ( t + ∆t ) − x m i ( t ) x m ∆t i In limit ∆ t ↓ 0, get the Itˆ o stochastic differential equations (SDEs) n � � p ℓ a ℓ ij p ℓ j x j ( t ) + b ℓ dt + dE ℓ dx i ( t ) = i x i ( t ) i ( t ) ( ⋆ ) i j =1 ℓ where E ℓ i ( t ) are Brownian motions satisfying Cov [ E ℓ i ( t )] = σ ℓ m i ( t ) , E m t i 10 / 20
Spatial coupling of patch dynamics p ℓ i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k x i ( t ) = � k ℓ =1 x ℓ i ( t ) total density of species i Assume Cov [ x ℓ i ( t + ∆t ) − x ℓ i ( t ) | x ( t )] ≈ x ℓ i ( t ) σ ℓ m i ( t ) , x m i ( t + ∆t ) − x m i ( t ) x m ∆t i In limit ∆ t ↓ 0, get the Itˆ o stochastic differential equations (SDEs) f i ( x ( t )) √ V i � �� � � �� � �� n � � p ℓ a ℓ ij p ℓ j x j ( t ) + b ℓ dt + i p m i σ ℓ m p ℓ dx i ( t ) = x i ( t ) dB i ( t ) i i i j =1 ℓ ℓ, m where B i ( t ) are Brownian motions satisfying Var [ B i ( t )] = t 10 / 20
11 / 20
To persist or not to persist, that is the question 11 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ When ( ⋆ ) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: � � x ) − V i ( � x ) f i ( � ( ♣ ) E 2 stationary � x = ( � x 1 , . . . , � x n ) s.t. P [min i � x i = 0] = 1 . 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ When ( ⋆ ) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: � � x ) − V i ( � x ) max f i ( � > 0 ( ♣ ) E 2 i for all stationary � x = ( � x 1 , . . . , � x n ) s.t. P [min i � x i = 0] = 1 . 12 / 20
Consider � � � dx i ( t ) = x i ( t ) f i ( x ( t )) dt + V i ( x ( t )) dE i ( t ) ( ⋆ ) where E i ( t ) a multivariate Brownian motion with Var [ E i ( t )] = t . ( ⋆ ) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. # { 1 ≤ τ ≤ t : min i x i ( t ) ≤ δ } lim sup ≤ ε a.s. whenever min x i (0) > 0 t i t →∞ When ( ⋆ ) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: � � x ) − V i ( � x ) max f i ( � > 0 ( ♣ ) E 2 i for all stationary � x = ( � x 1 , . . . , � x n ) s.t. P [min i � x i = 0] = 1 . ım [2018] extended ( ♣ ) to non-compact domains Hening and Nguyen [2018], Bena¨ (e.g. LV system) with additional condition to ensure tightness 12 / 20
Resident-Mutant dynamics x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t )+ a ii ′ q ℓ i ′ y i ′ ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t )+ a ii ′ q ℓ i ′ y i ′ ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 k n � � dt + dE ℓ q ℓ a ℓ i ′ j p ℓ j x j ( t )+ a ℓ i ′ i ′ q ℓ i ′ y i ′ ( t )) + b ℓ dy i ′ ( t ) = i ′ y i ′ ( t ) i ′ ( t ) i ′ ℓ =1 j =1 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t )+ a ii ′ q ℓ i ′ y i ′ ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 k n � � dt + dE ℓ q ℓ a ℓ i ′ j p ℓ j x j ( t )+ a ℓ i ′ i ′ q ℓ i ′ y i ′ ( t )) + b ℓ dy i ′ ( t ) = i ′ y i ′ ( t ) i ′ ( t ) i ′ ℓ =1 j =1 Theorem. Assume residents satisfy ( ♣ ) (i.e. persistence) with a positive stationary distribution ˆ x = (ˆ x 1 , . . . , ˆ x n ). 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t )+ a ii ′ q ℓ i ′ y i ′ ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 k n � � dt + dE ℓ q ℓ a ℓ i ′ j p ℓ j x j ( t )+ a ℓ i ′ i ′ q ℓ i ′ y i ′ ( t )) + b ℓ dy i ′ ( t ) = i ′ y i ′ ( t ) i ′ ( t ) i ′ ℓ =1 j =1 Theorem. Assume residents satisfy ( ♣ ) (i.e. persistence) with a positive stationary distribution ˆ x = (ˆ x 1 , . . . , ˆ x n ). If n k � � � − 1 q ℓ a ℓ i ′ j p ℓ x j ] + b ℓ q ℓ i ′ q m i ′ σ ℓ m I ( p , q i ′ ) := j E [ � < 0 i ′ i ′ i ′ 2 ℓ j =1 ℓ, m =1 13 / 20
Resident-Mutant dynamics mutant y i ′ ( t ) in spp i ′ w/ q ℓ x ( t ) = ( x 1 ( t ) , . . . , x n ( t )) w/ p ℓ i i ′ k n � � dt + dE ℓ p ℓ a ℓ ij p ℓ j x j ( t )+ a ii ′ q ℓ i ′ y i ′ ( t ) + b ℓ dx i ( t ) = i x i ( t ) i ( t ) i ℓ =1 j =1 k n � � dt + dE ℓ q ℓ a ℓ i ′ j p ℓ j x j ( t )+ a ℓ i ′ i ′ q ℓ i ′ y i ′ ( t )) + b ℓ dy i ′ ( t ) = i ′ y i ′ ( t ) i ′ ( t ) i ′ ℓ =1 j =1 Theorem. Assume residents satisfy ( ♣ ) (i.e. persistence) with a positive stationary distribution ˆ x = (ˆ x 1 , . . . , ˆ x n ). If n k � � � − 1 q ℓ a ℓ i ′ j p ℓ x j ] + b ℓ q ℓ i ′ q m i ′ σ ℓ m I ( p , q i ′ ) := j E [ � < 0 i ′ i ′ i ′ 2 ℓ j =1 ℓ, m =1 then � � 1 lim lim sup t log y i ′ ( t ) < 0 | y i ′ (0) = δ = 1 δ → 0 P t →∞ 13 / 20
unbeatable strategy [Hamilton, 1967] “This word was applied in just the same sense in which it could be applied to the ‘minimax’ strategy of a zero-sum two-person game. 14 / 20
unbeatable strategy [Hamilton, 1967] “This word was applied in just the same sense in which it could be applied to the ‘minimax’ strategy of a zero-sum two-person game. Such a strategy should not, without qualification be called optimum because it is not optimum against - although unbeaten by - any strategy differing from itself.” 14 / 20
unbeatable strategy [Hamilton, 1967] “This word was applied in just the same sense in which it could be applied to the ‘minimax’ strategy of a zero-sum two-person game. Such a strategy should not, without qualification be called optimum because it is not optimum against - although unbeaten by - any strategy differing from itself.” Evolutionarily stable strategy [Smith and Price, 1973] - a strategy that cannot be invaded by any other strategy that is initially rare 14 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ 15 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ The resident strategy p is a coevolutionary stable strategy (coESS) if I ( p , q i ) < 0 for all 1 ≤ i ≤ n and q i � = p i 15 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ The resident strategy p is a coevolutionary stable strategy (coESS) if I ( p , q i ) < 0 for all 1 ≤ i ≤ n and q i � = p i Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n � = − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ occupied by species i i m Note: f ℓ i ( p ) are solutions to a system of linear equations 15 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ The resident strategy p is a coevolutionary stable strategy (coESS) if I ( p , q i ) < 0 for all 1 ≤ i ≤ n and q i � = p i Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n � = − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ occupied by species i i m � ≤ − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ not occupied by species i i m Note: f ℓ i ( p ) are solutions to a system of linear equations 15 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ The resident strategy p is a coevolutionary stable strategy (coESS) if I ( p , q i ) < 0 for all 1 ≤ i ≤ n and q i � = p i Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n � = − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ occupied by species i i m � ≤ − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ not occupied by species i i m Note: f ℓ i ( p ) are solutions to a system of linear equations perfectly correlated fluctuations ⇒ ideal free distribution 15 / 20
=: f ℓ i ( p ) � �� � =: V i ( q ) � �� � n � � � − 1 q ℓ a ℓ ij p ℓ x j ] + b ℓ q ℓ i q m i σ ℓ m Invasion rates I ( p , q i ) := j E [ � i i i 2 ℓ j =1 j ,ℓ The resident strategy p is a coevolutionary stable strategy (coESS) if I ( p , q i ) < 0 for all 1 ≤ i ≤ n and q i � = p i Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n � = − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ occupied by species i i m � ≤ − 1 f ℓ p m i σ m ℓ i ( p ) − 2 V i ( p ) in patches ℓ not occupied by species i i m Note: f ℓ i ( p ) are solutions to a system of linear equations perfectly correlated fluctuations ⇒ ideal free distribution i ( p ) − σ ℓℓ imperfectly correlated fluctuations ⇒ local growth rate f ℓ 2 < 0 in i all occupied patches!!!! 15 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 16 / 20
Application: Competing species �� � � dx ℓ i ( t ) = x ℓ r ℓ i − x ℓ 1 ( t ) − x ℓ dt + dE ℓ i ( t ) 2 ( t ) i ( t ) i = 1 , 2 where E ℓ i ( t ) independent Brownian motions s.t. Var[ E ℓ i ( t )] = vt . If ℓ is the only patch and r ℓ 2 > r ℓ 1 , then 1 t log X ℓ lim sup 1 ( t ) < 0 a.s. t →∞ whenever X 1 (0) X 2 (0) > 0 Now, spatially couple patches with X ℓ i = p ℓ i X i ... 16 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ 17 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ r + ∆ r intrinsic growth rate r species 1 species 2 5 10 15 20 patch # 17 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ r + ∆ r intrinsic growth rate ∆ r > 2 v k ⇒ ghost of competition past i.e. p ℓ 1 p ℓ 2 = 0 for all ℓ r species 1 species 2 5 10 15 20 patch # 17 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ r + ∆ r intrinsic growth rate ∆ r > 2 v k ⇒ ghost of competition past i.e. p ℓ 1 p ℓ 2 = 0 for all ℓ ∆ r < 2 v r k ⇒ exorcism of the ghost species 1 species 2 i.e. p ℓ 1 p ℓ 2 > 0 for all ℓ 5 10 15 20 patch # 17 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ r + ∆ r intrinsic growth rate ∆ r > 2 v k ⇒ ghost of competition past i.e. p ℓ 1 p ℓ 2 = 0 for all ℓ ∆ r < 2 v r k ⇒ exorcism of the ghost species 1 species 2 i.e. p ℓ 1 p ℓ 2 > 0 for all ℓ 5 10 15 20 patch # 10.0 fraction in sink habitat 0.4 mean density 9.0 0.2 k=20 8.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 variation v variation v 17 / 20
Application: Competing species �� � � � p ℓ r ℓ i − p ℓ 1 x 1 ( t ) − p ℓ dt + dE ℓ dx i ( t ) = x i ( t ) 2 x 2 ( t ) i ( t ) i = 1 , 2 i ℓ r + ∆ r intrinsic growth rate ∆ r > 2 v k ⇒ ghost of competition past i.e. p ℓ 1 p ℓ 2 = 0 for all ℓ ∆ r < 2 v r k ⇒ exorcism of the ghost species 1 species 2 i.e. p ℓ 1 p ℓ 2 > 0 for all ℓ 5 10 15 20 patch # fraction in sink habitat 10 k=2 0.4 mean density 8 6 0.2 4 k=20 2 0.0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 variation v variation v 17 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ a source and sink habitat r 1 > 0 > r 2 , weak competition ε ≈ 0 prey experiences temporal variation v in source habitat 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ a source and sink habitat r 1 > 0 > r 2 , weak competition ε ≈ 0 prey experiences temporal variation v in source habitat There exist 0 < v ∗ < v ∗∗ < v ∗∗∗ s.t. v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ a source and sink habitat r 1 > 0 > r 2 , weak competition ε ≈ 0 prey experiences temporal variation v in source habitat There exist 0 < v ∗ < v ∗∗ < v ∗∗∗ s.t. v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ a source and sink habitat r 1 > 0 > r 2 , weak competition ε ≈ 0 prey experiences temporal variation v in source habitat There exist 0 < v ∗ < v ∗∗ < v ∗∗∗ s.t. v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 v ∗∗ < v < v ∗∗∗ ⇒ both species use both habitats 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ a source and sink habitat r 1 > 0 > r 2 , weak competition ε ≈ 0 prey experiences temporal variation v in source habitat There exist 0 < v ∗ < v ∗∗ < v ∗∗∗ s.t. v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 v ∗∗ < v < v ∗∗∗ ⇒ both species use both habitats v ∗∗∗ < v ⇒ predator only uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 18 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 v ∗∗ < v < v ∗∗∗ ⇒ both species use both habitats v ∗∗∗ < v ⇒ only predator uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 19 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 v ∗∗ < v < v ∗∗∗ ⇒ both species use both habitats v ∗∗∗ < v ⇒ only predator uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 prey fraction in sink 0.8 predator 0.4 0.0 0.0 0.5 1.0 1.5 source variance v 19 / 20
Application: Predator-prey interactions �� � � � r ℓ − ε p ℓ p ℓ 1 x 1 − ap ℓ dt + dE ℓ prey dx 1 ( t ) = x 1 ( t ) 2 x 2 ( t ) 1 ( t ) 1 ℓ �� � � � p ℓ cap ℓ dt + dE ℓ predator dx 2 ( t ) = x 2 ( t ) 1 x 1 ( t ) − d 2 ( t ) 2 ℓ v < v ∗ ⇒ no sink populations i.e. p 2 1 = p 2 2 = 0 v ∗ < v < v ∗∗ ⇒ only prey uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 v ∗∗ < v < v ∗∗∗ ⇒ both species use both habitats v ∗∗∗ < v ⇒ only predator uses sink habitat i.e. p 2 1 > 0, p 2 2 = 0 prey fraction in sink 0.8 total density 40 predator 0.4 20 0.0 0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 source variance v source variance v 19 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere 20 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere Partially correlated fluctuations select for negative local per-capita growth rates in all occupied patches that are, generically, unequal. 20 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere Partially correlated fluctuations select for negative local per-capita growth rates in all occupied patches that are, generically, unequal. For competing species, the ghost of competition past only excorcised if fitness differences are sufficiently small relative to temporal fluctuations 20 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere Partially correlated fluctuations select for negative local per-capita growth rates in all occupied patches that are, generically, unequal. For competing species, the ghost of competition past only excorcised if fitness differences are sufficiently small relative to temporal fluctuations For enemy-victim interactions, environmental fluctutations can select for enemy-free sink and enemy-free source populations 20 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere Partially correlated fluctuations select for negative local per-capita growth rates in all occupied patches that are, generically, unequal. For competing species, the ghost of competition past only excorcised if fitness differences are sufficiently small relative to temporal fluctuations For enemy-victim interactions, environmental fluctutations can select for enemy-free sink and enemy-free source populations Thanks to U.S. National Science Foundation for funding and CIRM for hosting. 20 / 20
Finale Perfectly correlated fluctuations across space select for an ideal free distribution whereby local per-capita growth rates = 0 in occupied patches, < 0 elsewhere Partially correlated fluctuations select for negative local per-capita growth rates in all occupied patches that are, generically, unequal. For competing species, the ghost of competition past only excorcised if fitness differences are sufficiently small relative to temporal fluctuations For enemy-victim interactions, environmental fluctutations can select for enemy-free sink and enemy-free source populations Thanks to U.S. National Science Foundation for funding and CIRM for hosting. You for listening! 20 / 20
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