[PPT] - Coevolution of habitat use in stochastic environments Sebastian J. PowerPoint Presentation

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Coevolution of habitat use in stochastic environments

Sebastian J. Schreiber

Department of Evolution & Ecology University of California, Davis http://schreiber.faculty.ucdavis.edu In collaboration with Alex Hening (Tufts) and Dang Nguyen (University of Alabama)

February 13, 2020

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How might species distribute themselves across spatially heterogeneous environments?

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How might species distribute themselves across spatially heterogeneous environments? A suggestion came from someone who wrote I was actually studying earthworm brains for my doctoral dissertation . . . but the formal research was not going well. . .

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How might species distribute themselves across spatially heterogeneous environments? A suggestion came from someone who wrote I was actually studying earthworm brains for my doctoral dissertation . . . but the formal research was not going well. . . I was irritated by Lack’s dogmatic position. . . that territorial behavior did not affect habitat selection. . .

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How might species distribute themselves across spatially heterogeneous environments? A suggestion came from someone who wrote I was actually studying earthworm brains for my doctoral dissertation . . . but the formal research was not going well. . . I was irritated by Lack’s dogmatic position. . . that territorial behavior did not affect habitat selection. . . in desperation,. . . I put it all into . . . mathematical models...[and] made several wondrous discoveries. . .

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How might species distribute themselves across spatially heterogeneous environments? A suggestion came from someone who wrote I was actually studying earthworm brains for my doctoral dissertation . . . but the formal research was not going well. . . I was irritated by Lack’s dogmatic position. . . that territorial behavior did not affect habitat selection. . . in desperation,. . . I put it all into . . . mathematical models...[and] made several wondrous discoveries. . . I soon dropped the earthworm research; both the worms and I were having nervous breakdowns and getting nowhere.

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How might species distribute themselves across spatially heterogeneous environments? A suggestion came from someone who wrote I was actually studying earthworm brains for my doctoral dissertation . . . but the formal research was not going well. . . I was irritated by Lack’s dogmatic position. . . that territorial behavior did not affect habitat selection. . . in desperation,. . . I put it all into . . . mathematical models...[and] made several wondrous discoveries. . . I soon dropped the earthworm research; both the worms and I were having nervous breakdowns and getting nowhere.

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A species exhibits an ideal free distribution (IFD) if the per-capita growth rates in all occupied patches are equal and individuals moving to an unoccupied patch would lower their per-capita growth rate.

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A species exhibits an ideal free distribution (IFD) if the per-capita growth rates in all occupied patches are equal and individuals moving to an unoccupied patch would lower their per-capita growth rate. Properties include evolutionarily stability [Kˇ

rivan et al., 2008]

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A species exhibits an ideal free distribution (IFD) if the per-capita growth rates in all occupied patches are equal and individuals moving to an unoccupied patch would lower their per-capita growth rate. Properties include evolutionarily stability [Kˇ

rivan et al., 2008]

At equilibrium, the per-capita growth rate are zero in occupied patches ⇒ no sink populations (i.e. local populations that decline without

immigration)

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A species exhibits an ideal free distribution (IFD) if the per-capita growth rates in all occupied patches are equal and individuals moving to an unoccupied patch would lower their per-capita growth rate. Properties include evolutionarily stability [Kˇ

rivan et al., 2008]

At equilibrium, the per-capita growth rate are zero in occupied patches ⇒ no sink populations (i.e. local populations that decline without

immigration)

Can lead to...

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input matching for resource consumption [Parker, 1978]

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input matching for resource consumption [Parker, 1978]

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Ghost of competition past: Competing species evolve to

nly select habitats where

they are competitively

superior. Connell [1980]

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Enemy-free space: prey use low quality habitat to avoid natural enemies [Jeffries

and Lawton, 1984, Schreiber et al., 2000]

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal...

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

vermatching and undermatching of resource

availability is the norm

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

vermatching and undermatching of resource

availability is the norm many competing species have overlapping geo- graphical ranges

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

vermatching and undermatching of resource

availability is the norm many competing species have overlapping geo- graphical ranges Main questions: How should habitat selection of interacting species coevolve when environmental conditions vary in space and time?

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

vermatching and undermatching of resource

availability is the norm many competing species have overlapping geo- graphical ranges Main questions: How should habitat selection of interacting species coevolve when environmental conditions vary in space and time? When is there selection for sink populations?

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Hylocichla ¡ mustelina ¡

However, populations often aren’t ideal... sink populations are common [Pulliam, 1988, Holt,

1997, Schreiber, 2012, Furrer and Pasinelli, 2016]

vermatching and undermatching of resource

availability is the norm many competing species have overlapping geo- graphical ranges Main questions: How should habitat selection of interacting species coevolve when environmental conditions vary in space and time? When is there selection for sink populations? What effect does spatio-temporal variation have on the ghost of competition past or enemy-free space?

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Implicit space!

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Implicit space! Mass action!!

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Implicit space! Mass action!! Diffusion approximations!!!

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Dynamics within patch ℓ

xℓ

i (t) density of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

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Dynamics within patch ℓ

xℓ

i (t) density of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

bℓ

i intrinsic per-capita growth rate of spp. i in patch ℓ

aℓ

ij per-capita interaction rate of spp i with spp j in patch ℓ.

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Dynamics within patch ℓ

xℓ

i (t) density of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

bℓ

i intrinsic per-capita growth rate of spp. i in patch ℓ

aℓ

ij per-capita interaction rate of spp i with spp j in patch ℓ.

Assume E[xℓ

i (t + ∆t) − xℓ i (t)|xℓ(t)] ≈ xℓ i (t)

 

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  ∆t,

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Dynamics within patch ℓ

xℓ

i (t) density of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

bℓ

i intrinsic per-capita growth rate of spp. i in patch ℓ

aℓ

ij per-capita interaction rate of spp i with spp j in patch ℓ.

Assume E[xℓ

i (t + ∆t) − xℓ i (t)|xℓ(t)] ≈ xℓ i (t)

 

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  ∆t,

and Var[xℓ

i (t + ∆t) − xℓ i (t) | xℓ(t)] ≈ σℓℓ i

xℓ

i (t)

2 ∆t

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Dynamics within patch ℓ

xℓ

i (t) density of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

bℓ

i intrinsic per-capita growth rate of spp. i in patch ℓ

aℓ

ij per-capita interaction rate of spp i with spp j in patch ℓ.

Assume E[xℓ

i (t + ∆t) − xℓ i (t)|xℓ(t)] ≈ xℓ i (t)

 

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  ∆t,

and Var[xℓ

i (t + ∆t) − xℓ i (t) | xℓ(t)] ≈ σℓℓ i

xℓ

i (t)

2 ∆t

In limit ∆t ↓ 0, get the Itˆ

stochastic differential equations (SDEs)

dxℓ

i (t) = xℓ i (t)

   

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  dt + dE ℓ

i (t)

 

where E ℓ

i (t) is a Brownian motion with mean 0 and variance σℓℓ i t

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Spatial coupling of patch dynamics

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t

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Spatial coupling of patch dynamics

xi(t) = k

ℓ=1 xℓ i (t) total density of species i

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t

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Spatial coupling of patch dynamics

xi(t) = k

ℓ=1 xℓ i (t) total density of species i

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t In limit ∆t ↓ 0, get the Itˆ

stochastic differential equations (SDEs)

dxi(t) =

ℓ

 xℓ

i (t)

   

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  dt + dE ℓ

i (t)

   

(⋆) where E ℓ

i (t) are Brownian motions satisfying Cov[Eℓ i(t), Em i (t)] = σℓm i

t

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Spatial coupling of patch dynamics

pℓ

i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

xi(t) = k

ℓ=1 xℓ i (t) total density of species i

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t In limit ∆t ↓ 0, get the Itˆ

stochastic differential equations (SDEs)

dxi(t) =

ℓ

 xℓ

i (t)

   

n

j=1

aℓ

ijxℓ j (t) + bℓ i

  dt + dE ℓ

i (t)

   

(⋆) where E ℓ

i (t) are Brownian motions satisfying Cov[Eℓ i(t), Em i (t)] = σℓm i

t

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Spatial coupling of patch dynamics

pℓ

i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

xi(t) = k

ℓ=1 xℓ i (t) total density of species i

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t In limit ∆t ↓ 0, get the Itˆ

stochastic differential equations (SDEs)

dxi(t) =

ℓ

 pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t) + bℓ i

  dt + dE ℓ

i (t)

   

(⋆) where E ℓ

i (t) are Brownian motions satisfying Cov[Eℓ i(t), Em i (t)] = σℓm i

t

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Spatial coupling of patch dynamics

pℓ

i fraction of species i in patch ℓ where 1 ≤ i ≤ n and 1 ≤ ℓ ≤ k

xi(t) = k

ℓ=1 xℓ i (t) total density of species i

Assume Cov[xℓ

i (t + ∆t) − xℓ i (t), xm i (t + ∆t) − xm i (t) | x(t)] ≈ xℓ i (t)xm i (t)σℓm i

∆t In limit ∆t ↓ 0, get the Itˆ

stochastic differential equations (SDEs)

dxi(t) = xi(t)

      

fi(x(t))

ℓ

pℓ

i

 

n

j=1

aℓ

ijpℓ j xj(t) + bℓ i

  dt +

√Vi

ℓ,m

pℓ

i pm i σℓm i

dBi(t)

      

where Bi(t) are Brownian motions satisfying Var[Bi(t)] = t

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To persist or not to persist, that is the question

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t.

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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SLIDE 48

Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0

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Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0 When (⋆) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: E

fi(

x) − Vi( x) 2

(♣)

stationary x = ( x1, . . . , xn) s.t. P [mini xi = 0] = 1.

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SLIDE 52

Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0 When (⋆) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: max

i

E

fi(

x) − Vi( x) 2

> 0

(♣) for all stationary x = ( x1, . . . , xn) s.t. P [mini xi = 0] = 1.

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SLIDE 53

Consider dxi(t) = xi(t)

fi(x(t))dt +
Vi(x(t))dEi(t)
(⋆)

where Ei(t) a multivariate Brownian motion with Var[Ei(t)] = t. (⋆) is stochastically persistent if for all ε > 0 there is δ > 0 s.t. lim sup

t→∞

#{1 ≤ τ ≤ t : mini xi(t) ≤ δ} t ≤ ε a.s. whenever min

i

xi(0) > 0 When (⋆) compactly supported, Schreiber et al. [2011] introduced the sufficient condition: max

i

E

fi(

x) − Vi( x) 2

> 0

(♣) for all stationary x = ( x1, . . . , xn) s.t. P [mini xi = 0] = 1.

Hening and Nguyen [2018], Bena¨ ım [2018] extended (♣) to non-compact domains

(e.g. LV system) with additional condition to ensure tightness

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Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

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Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

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Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t) + bℓ i

  dt + dE ℓ

i (t)

 

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Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t)+aii′qℓ i′yi′(t) + bℓ i

  dt + dE ℓ

i (t)

 

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SLIDE 58

Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t)+aii′qℓ i′yi′(t) + bℓ i

  dt + dE ℓ

i (t)

 

dyi′(t) =

k

ℓ=1

qℓ

i′yi′(t)

   

n

j=1

aℓ

i′jpℓ j xj(t)+aℓ i′i′qℓ i′yi′(t)) + bℓ i′

  dt + dE ℓ

i′(t)

 

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SLIDE 59

Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t)+aii′qℓ i′yi′(t) + bℓ i

  dt + dE ℓ

i (t)

 

dyi′(t) =

k

ℓ=1

qℓ

i′yi′(t)

   

n

j=1

aℓ

i′jpℓ j xj(t)+aℓ i′i′qℓ i′yi′(t)) + bℓ i′

  dt + dE ℓ

i′(t)

 

Theorem. Assume residents satisfy (♣) (i.e. persistence) with a positive

stationary distribution ˆ x = (ˆ x1, . . . , ˆ xn).

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SLIDE 60

Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t)+aii′qℓ i′yi′(t) + bℓ i

  dt + dE ℓ

i (t)

 

dyi′(t) =

k

ℓ=1

qℓ

i′yi′(t)

   

n

j=1

aℓ

i′jpℓ j xj(t)+aℓ i′i′qℓ i′yi′(t)) + bℓ i′

  dt + dE ℓ

i′(t)

 

Theorem. Assume residents satisfy (♣) (i.e. persistence) with a positive

stationary distribution ˆ x = (ˆ x1, . . . , ˆ xn). If I(p, qi′) :=

ℓ

qℓ

i′

 

n

j=1

aℓ

i′jpℓ j E[

xj] + bℓ

i′

  − 1

2

k

ℓ,m=1

qℓ

i′qm i′ σℓm i′

< 0

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SLIDE 61

Resident-Mutant dynamics

x(t) = (x1(t), . . . , xn(t)) w/ pℓ

i

mutant yi′(t) in spp i′ w/ qℓ

i′

dxi(t) =

k

ℓ=1

pℓ

i xi(t)

   

n

j=1

aℓ

ijpℓ j xj(t)+aii′qℓ i′yi′(t) + bℓ i

  dt + dE ℓ

i (t)

 

dyi′(t) =

k

ℓ=1

qℓ

i′yi′(t)

   

n

j=1

aℓ

i′jpℓ j xj(t)+aℓ i′i′qℓ i′yi′(t)) + bℓ i′

  dt + dE ℓ

i′(t)

 

Theorem. Assume residents satisfy (♣) (i.e. persistence) with a positive

stationary distribution ˆ x = (ˆ x1, . . . , ˆ xn). If I(p, qi′) :=

ℓ

qℓ

i′

 

n

j=1

aℓ

i′jpℓ j E[

xj] + bℓ

i′

  − 1

2

k

ℓ,m=1

qℓ

i′qm i′ σℓm i′

< 0 then lim

δ→0 P

lim sup

t→∞

1 t log yi′(t) < 0|yi′(0) = δ

= 1

13 / 20

SLIDE 62

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

SLIDE 63

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

SLIDE 64

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

SLIDE 65

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

15 / 20

SLIDE 66

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

The resident strategy p is a coevolutionary stable strategy (coESS) if I(p, qi) < 0 for all 1 ≤ i ≤ n and qi = pi

15 / 20

SLIDE 67

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

The resident strategy p is a coevolutionary stable strategy (coESS) if I(p, qi) < 0 for all 1 ≤ i ≤ n and qi = pi

Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n

f ℓ

i (p) −

m

pm

i σmℓ i

= − 1 2Vi(p) in patches ℓ occupied by species i Note: f ℓ

i (p) are solutions to a system of linear equations

15 / 20

SLIDE 68

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

The resident strategy p is a coevolutionary stable strategy (coESS) if I(p, qi) < 0 for all 1 ≤ i ≤ n and qi = pi

Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n

f ℓ

i (p) −

m

pm

i σmℓ i

= − 1 2Vi(p) in patches ℓ occupied by species i f ℓ

i (p) −

m

pm

i σmℓ i

≤ − 1 2Vi(p) in patches ℓ not occupied by species i Note: f ℓ

i (p) are solutions to a system of linear equations

15 / 20

SLIDE 69

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

The resident strategy p is a coevolutionary stable strategy (coESS) if I(p, qi) < 0 for all 1 ≤ i ≤ n and qi = pi

Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n

f ℓ

i (p) −

m

pm

i σmℓ i

= − 1 2Vi(p) in patches ℓ occupied by species i f ℓ

i (p) −

m

pm

i σmℓ i

≤ − 1 2Vi(p) in patches ℓ not occupied by species i Note: f ℓ

i (p) are solutions to a system of linear equations

perfectly correlated fluctuations ⇒ ideal free distribution

15 / 20

SLIDE 70

Invasion rates I(p, qi) :=

ℓ

qℓ

i =:f ℓ

i (p)





n

j=1

aℓ

ijpℓ j E[

xj] + bℓ

i

  −1

2

=:Vi(q)

j,ℓ

qℓ

i qm i σℓm i

The resident strategy p is a coevolutionary stable strategy (coESS) if I(p, qi) < 0 for all 1 ≤ i ≤ n and qi = pi

Proposition. A necessary condition for p to be a coESS is: for all 1 ≤ i ≤ n

f ℓ

i (p) −

m

pm

i σmℓ i

= − 1 2Vi(p) in patches ℓ occupied by species i f ℓ

i (p) −

m

pm

i σmℓ i

≤ − 1 2Vi(p) in patches ℓ not occupied by species i Note: f ℓ

i (p) are solutions to a system of linear equations

perfectly correlated fluctuations ⇒ ideal free distribution imperfectly correlated fluctuations ⇒ local growth rate f ℓ

i (p) − σℓℓ

i

2 < 0 in

all occupied patches!!!!

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SLIDE 71

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

i (t)

i = 1, 2

where E ℓ

i (t) independent Brownian motions s.t. Var[E ℓ i (t)] = vt.

16 / 20

SLIDE 72

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 73

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 74

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 75

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 76

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 77

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 78

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0

16 / 20

SLIDE 79

Application: Competing species

dxℓ

i (t) = xℓ i (t)

r ℓ

i − xℓ 1(t) − xℓ 2(t)

dt + dE ℓ

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 lim sup

t→∞

1 t log X ℓ

1(t) < 0 a.s.

whenever X1(0)X2(0) > 0 Now, spatially couple patches with X ℓ

i = pℓ i Xi...

16 / 20

SLIDE 80

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

17 / 20

SLIDE 81

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

patch # intrinsic growth rate 5 10 15 20 r r + ∆r species 1 species 2

17 / 20

SLIDE 82

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

patch # intrinsic growth rate 5 10 15 20 r r + ∆r species 1 species 2

∆r > 2v

k ⇒ ghost of competition past

i.e. pℓ

1pℓ 2 = 0 for all ℓ

17 / 20

SLIDE 83

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

patch # intrinsic growth rate 5 10 15 20 r r + ∆r species 1 species 2

∆r > 2v

k ⇒ ghost of competition past

i.e. pℓ

1pℓ 2 = 0 for all ℓ

∆r < 2v

k ⇒ exorcism of the ghost

i.e. pℓ

1pℓ 2 > 0 for all ℓ

17 / 20

SLIDE 84

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

patch # intrinsic growth rate 5 10 15 20 r r + ∆r species 1 species 2

∆r > 2v

k ⇒ ghost of competition past

i.e. pℓ

1pℓ 2 = 0 for all ℓ

∆r < 2v

k ⇒ exorcism of the ghost

i.e. pℓ

1pℓ 2 > 0 for all ℓ 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 variation v fraction in sink habitat

k=20

0.0 0.5 1.0 1.5 2.0 8.0 9.0 10.0 variation v mean density

17 / 20

SLIDE 85

Application: Competing species

dxi(t) = xi(t)

ℓ

pℓ

i

r ℓ

i − pℓ 1x1(t) − pℓ 2x2(t)

dt + dE ℓ

i (t)

i = 1, 2

patch # intrinsic growth rate 5 10 15 20 r r + ∆r species 1 species 2

∆r > 2v

k ⇒ ghost of competition past

i.e. pℓ

1pℓ 2 = 0 for all ℓ

∆r < 2v

k ⇒ exorcism of the ghost

i.e. pℓ

1pℓ 2 > 0 for all ℓ 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 variation v fraction in sink habitat

k=2 k=20

0.0 0.5 1.0 1.5 2.0 2 4 6 8 10 variation v mean density

17 / 20

SLIDE 86

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

18 / 20

SLIDE 87

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

a source and sink habitat r 1 > 0 > r 2, weak competition ε ≈ 0

prey experiences temporal variation v in source habitat

18 / 20

SLIDE 88

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

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. p2

1 = p2 2 = 0

18 / 20

SLIDE 89

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

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. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

18 / 20

SLIDE 90

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

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. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

v∗∗ < v < v∗∗∗ ⇒ both species use both habitats

18 / 20

SLIDE 91

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

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. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

v∗∗ < v < v∗∗∗ ⇒ both species use both habitats v∗∗∗ < v ⇒ predator only uses sink habitat i.e. p2

1 > 0, p2 2 = 0

18 / 20

SLIDE 92

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

v < v∗ ⇒ no sink populations i.e. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

v∗∗ < v < v∗∗∗ ⇒ both species use both habitats v∗∗∗ < v ⇒ only predator uses sink habitat i.e. p2

1 > 0, p2 2 = 0

19 / 20

SLIDE 93

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

v < v∗ ⇒ no sink populations i.e. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

v∗∗ < v < v∗∗∗ ⇒ both species use both habitats v∗∗∗ < v ⇒ only predator uses sink habitat i.e. p2

1 > 0, p2 2 = 0 0.0 0.5 1.0 1.5 0.0 0.4 0.8 source variance v fraction in sink

prey predator

19 / 20

SLIDE 94

Application: Predator-prey interactions

prey dx1(t) =x1(t)

ℓ

pℓ

1

r ℓ − εpℓ

1x1 − apℓ 2x2(t)

dt + dE ℓ

1(t)

predator

dx2(t) =x2(t)

ℓ

pℓ

2

capℓ

1x1(t) − d

dt + dE ℓ

2(t)

v < v∗ ⇒ no sink populations i.e. p2

1 = p2 2 = 0

v∗ < v < v∗∗ ⇒ only prey uses sink habitat i.e. p2

1 > 0, p2 2 = 0

v∗∗ < v < v∗∗∗ ⇒ both species use both habitats v∗∗∗ < v ⇒ only predator uses sink habitat i.e. p2

1 > 0, p2 2 = 0 0.0 0.5 1.0 1.5 0.0 0.4 0.8 source variance v fraction in sink

prey predator

0.0 0.5 1.0 1.5 20 40 source variance v total density

19 / 20

SLIDE 95

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

SLIDE 96

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

SLIDE 97

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

SLIDE 98

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

SLIDE 99

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

SLIDE 100

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

SLIDE 101

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! Questions?

20 / 20

SLIDE 102

M. Bena¨

ım. Stochastic persistence. arXiv preprint arXiv:1806.08450, 2018. J.H. Connell. Diversity and the coevolution of competitors, or the ghost of competition past. Oikos, pages 131–138, 1980. R.D. Furrer and G. Pasinelli. Empirical evidence for source–sink populations: a review on occurrence, assessments and implications. Biological Reviews, 91(3):782–795, 2016. W.D. Hamilton. Extraordinary sex ratios. Science, 156(3774):477–488, 1967.

A. Hening and D.H. Nguyen. Coexistence and extinction for stochastic

kolmogorov systems. The Annals of Applied Probability, 28:1893–1942, 2018. R.D. Holt. On the evolutionary stability of sink populations. Evolutionary Ecology, 11(6):723–731, 1997. M.J. Jeffries and J.H. Lawton. Enemy free space and the structure of ecological communities. Biological Journal of the Linnean Society, 23 (4):269–286, 1984.

V. Kˇ

rivan, R. Cressman, and C. Schneider. The ideal free distribution: a review and synthesis of the game-theoretic perspective. Theoretical Population Biology, 73(3):403–425, 2008.

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SLIDE 103

G.A. Parker. Searching for mates. In “Behavioural Ecology: An evolutionary approach” (JR Krebs and NB Davies, eds), 1978. H.R. Pulliam. Sources, sinks, and population regulation. The American Naturalist, 132(5):652–661, 1988.

S. J. Schreiber, M. Bena¨

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