Difference Equations Arising in Evolutionary Population Dynamics - - PowerPoint PPT Presentation

• J. M. Cushing

Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona Tucson, Arizona, USA

Difference Equations Arising in Evolutionary Population Dynamics

Simon Maccracken

Department of Ecology & Evolutionary Biology University of Arizona Tucson, Arizona, USA

Support by the National Science Foundation

ICDEA 2012 Barcelona

OUTLINE

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Semelparous Juvenile-Adult Models

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1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J R sb inherent net reproductive number

( expected offspring per individual per life time )

b s

inherent (low density) adult fertility inherent (low density) juvenile survival

1 2 2

: [0, ) : (0,1) (0,0) R C R are where

• pen

(0,0) (0,0) 1 Density dependence

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Iteroparous Juvenile-Adult Models

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1 1

( , ) ( , ) ( , )

t t t t t j j t t t a a t t t

J b J A A A s J A J s J A A

1 2 2

: [0, ) , : (0,1) (0,0)

j a

R C R are where

• pen

(0,0) (0,0) (0,0) 1

j a j a

b s s

inherent adult fertility inherent juvenile survival, inherent adult survival

1

j a

s R b s inherent net reproductive number

( expected offspring per individual per life time )

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Iteroparous Juvenile-Adult Models

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1 1

( , ) ( , ) ( , )

t t t t t j j t t t a a t t t

J b J A A A s J A J s J A A

Define :

1 1 1 1 1

j j j a a a a a a a

J j A A a J A j J a

s s s s s s s s s s

a Within-class

competitive effects Between-class competitive effects

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Iteroparous Juvenile-Adult Models

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1 1

( , ) ( , ) ( , )

t t t t t j j t t t a a t t t

J b J A A A s J A J s J A A

Fundamental Bifurcation Theorem

• 1. Origin is stable if R0 < 1 and unstable if R0 > 1.
• 2. Positive equilibria (transcritically) bifurcate from the origin at R0 = 1.

Assume a+ ≠ 0.

• 3. Stability depends on the direction of bifurcation:

Right (forward) bifurcating positive equilibria are stable. Left (backward) bifurcating positive equilibria are unstable.

• 4. a+ < 0 => right ( hence stable ) bifurcation

a+ > 0 => left ( hence unstable ) bifurcation

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Iteroparous Juvenile-Adult Models

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1 1

( , ) ( , ) ( , )

t t t t t j j t t t a a t t t

J b J A A A s J A J s J A A

Notes

• A right (stable) bifurcation occurs if there are

no positive feedback effects (at low density), i.e. if there are no positive derivatives

• Positive feedback terms (positive derivatives)

are called Allee effects.

, , ,

J A J A .

• A left (unstable) bifurcation can only occur in the

presence of strong Allee effects.

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Lloyd & Dybas (1966, 1974) Hoppensteadt & Keller (1976) Bulmer (1977) May (1979) Ebenman (1987, 1988) JC & Li (1989, 1992) Wikan & (1996) Nisbet & Onyiah (1994) Wikan & Mjølhus (1997) Behncke (2000) Davydova (2004) Davydova, Diekmann & van Gils (2003, 2005) Mjølhus, Wikan & Solberg (2005) JC (1991, 2003, 2006, 2010) Kon (2005, 2007) Diekmann & Yan (2008) JC & Henson (2012)

Semelparous Juvenile-Adult Models

1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J

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Plants

Monocarpic perennials

Vertebrates

Species of : Fish Lizards Amphibians Marsupials … even a mammal !

Invertebrates

Species of Insects Arachnids Molluscs 13, 17 year cycles

Periodical Insects

Magicicada spp. Melontha spp. 3, 4, 5 year cycles Lasiocampa quercus var 1, 2 year cycles Annuals

Semelparity

“Poster-child” species

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Semelparous Juvenile-Adult Models

1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J

Define :

J j A J j A

a s s

Within-class competitive effects Between-class competitive effects

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Semelparous Juvenile-Adult Models

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Fundamental bifurcation Theorem

JC & Jia Li (1989), JC (2006), JC & Henson (2012)

1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J

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Semelparous Juvenile-Adult Models

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Fundamental bifurcation Theorem

J A

s (or ) right (or left) bifurcation

• 3. Synchronous 2-cycles also bifurcate from the origin at R0 = 1.
• 2. Positive equilibria bifurcate from the origin at R0 = 1.
• 1. Origin is stable if R0 = sb < 1 and unstable if R0 > 1.

Assume a+ ≠ 0.

• 4. (a) Left bifurcations are unstable.

(b) A right bifurcation is stable if the other bifurcation is to the left. (c) If both bifurcations are to the right, then

a (or ) right (or left) bifurcation a equilibria stabl & 2 - cycles un e stable a equilibria unstab 2 - cycles le st & able

Dynamic Dichotomy

1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J

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Notes

• Weak between-class competition gives stable equilibria
• Strong between-class competition gives stable synchronous 2-cycles

J j A J j A

a s s

J j A J j A

a s s

• If there are no Allee effects, that is, no positive derivatives

then Dynamic Dichotomy occurs.

, , ,

J A J A

• Between-class (nymph) competition is leading hypothesis for periodical

cicada dynamics

• Dichotomy observed in experiments with Tribolium castaneum

JC et al., Chaos in Ecology, Academic Press (2003) King, Costantino, JC, Henson, Desharnais & Dennis, Proc Nat Acad Sci (2003) Dennis, Desharnais, JC, Henson & Costantino, Ecol Monogr (2001) Costantino, JC, Dennis & Desharnais, Science (1997)

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Semelparous Juvenile-Adult Models

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Typical example for which the Dynamic Dichotomy occurs:

1 1

( , ) ( , )

t t t t t t t t

J b J A A A s J A J

11 12 21 22

1 1 ( , ) , ( , ) 1 1 J A J A c J c A c J c A Leslie-Gower-type competition interaction functionals … a natural extension of discrete logistic ( or Beverton-Holt ) equation for an unstructured population.

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Evolutionary Semelparous Juvenile-Adult Models

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u U trait interval

max ( ) 1.

U

u U

b

Assume Then maximum adult fertility ( over the trait interval ) (0,0, ) 1 (0,0, ) 1 u u

1 1

( ) ( , , ) ( ) ( , , )

t t t t t t t t

J b u J A u A A s u J A u J

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u

mean of a phenotypic trait subject to Darwinian evolution

Evolutionary Semelparous Juvenile-Adult Models

ICDEA 2012 Barcelona

1

1 2

ln ( , , ) ( , , ) ( ) ( ) ( , , ) ( , , )

t t u

u u v R J A u R J A u b u s u J A u J A u v

trait variance ( assumed constant in time )

Using Evolutionary Game Theory Methodology

Vincent & Brown 2005

1 1

( ) ( , , ) ( ) ( , , )

t t t t t t t t t t t t

J b u J A u A A s u J A u J (0,0, ) 1 (0,0, ) 1 u u

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Evolutionary Semelparous Juvenile-Adult Models

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• What are the dynamics of this model ?
• When v = 0 the Fundamental Bifurcation Theorem applies

and the Dynamics Dichotomy is a possibility.

• What happens when v > 0 (i.e. when evolution is present)?

1

1 ln[ ( ) ( ) ( , , ) ( , , )] 2

t t u

u u v b u s u J A u J A u (0,0, ) 1 (0,0, ) 1 u u

1 1

( ) ( , , ) ( ) ( , , )

t t t t t t t t t t t t

J b u J A u A A s u J A u J

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Evolutionary Semelparous Juvenile-Adult Models

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Extinction Equilibria & Critical Traits

An extinction equilibrium (J, A, u) is an equilibrium with J = A = 0. A critical trait u satisfies

0(0,0, )

0.

uR

u

Easy to see that …

(0, 0, u) is an extinction equilibrium if and only if u is a critical trait.

1

1 ln[ ( ) ( ) ( , , ) ( , , )] 2

t t u

u u v b u s u J A u J A u (0,0, ) 1 (0,0, ) 1 u u

1 1

( ) ( , , ) ( ) ( , , )

t t t t t t t t t t t t

J b u J A u A A s u J A u J

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Evolutionary Semelparous Juvenile-Adult Models

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Synchronous 2-cycles

1 2

J A u u

1

1 ln[ ( ) ( ) ( , , ) ( , , )] 2

t t u

u u v b u s u J A u J A u (0,0, ) 1 (0,0, ) 1 u u

1 1

( ) ( , , ) ( ) ( , , )

t t t t t t t t t t t t

J b u J A u A A s u J A u J

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Main Theorem

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(b) If ∂uuR0(0 ,0, u*) > 0 then the equilibria (extinction & positive) equilibria and the synchronous 2-cycles are all unstable.

• 2. ∂uuR0(0 ,0, u*) ≠ 0

(a) If ∂uuR0(0 ,0, u*) < 0 then the Fundamental Bifurcation Theorem and the Dynamic Dichotomy hold … using R0(0 ,0, u*) as a bifurcation parameter. NOTE: R0(0, 0, u) = bb(u)s(u)

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Assume …

• 1. There exists a critical trait u* : ∂uR0(0, 0, u*) = 0

J j A J j A J A

a s s s

3. and

Then …

Examples

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11 12 21 22

1 1 ( , ) , ( , ) 1 1 0,

ij ii

J A J A c J c A c J c A c c at least one

• No trait dependence in these density feedback terms
• No Allee effects
• Dynamic Dichotomy holds in absence of evolution
• Dynamic dichotomy holds occurs in the presence of evolution if

∂uuR0(0 ,0, u*) < 0 Leslie-Gower Nonlinearities

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Examples

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Mathematically: s(u) and b (u) have

• pposite monotonicity

( ) ( ) 2 ( ) ( ) ( ) ( ) u s u u s u u s u

Trade-offs are of fundamental biological interest: Survival & fertility change in

• pposite manner as u increases.

Dynamic Dichotomy criterion: Critical trait equation:

0(0,0, ) uR

u

( ) ( ) ( ) ( ) u s u u s u

0(0,0, )

( ) ( ) R u b u s u

Net reproductive number:

0(0,0, ) uuR

u

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1

[ ( ) ( )] 1 2 ( ) ( )

u t t t t t t

u s u u u v u s u

21 22 11 12

1 1

1 1 1 1

( ) ( )

t

t t t t t t t t t

c J c A c J c A

J b u A A s u J

Examples

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( ) ( ) 2 ( ) ( ) ( ) ( ) u s u u s u u s u

Dynamic Dichotomy criterion: Critical trait equation:

0(0,0, ) uR

u

( ) ( ) ( ) ( ) u s u u s u

0(0,0, )

( ) ( ) R u b u s u

Net reproductive number:

0(0,0, ) uuR

u Stability critieria:

21 12 11 22

[ ( *) ] [ ( *) ] a c s u c c s u c stable equilibria

21 12 11 22

( *) ( *) c s u c c c s u c competition ratio

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1

[ ( ) ( )] 1 2 ( ) ( )

u t t t t t t

u s u u u v u s u

21 22 11 12

1 1

1 1 1 1

( ) ( )

t

t t t t t t t t t

c J c A c J c A

J b u A A s u J

Examples

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( ) ( ) 2 ( ) ( ) ( ) ( ) u s u u s u u s u

Dynamic Dichotomy criterion: Critical trait equation:

0(0,0, ) uR

u

( ) ( ) ( ) ( ) u s u u s u

0(0,0, )

( ) ( ) R u b u s u

Net reproductive number:

21 22 11 12

1 1

1 1 1 1

( ) ( )

t

t t t t t t t t t

c J c A c J c A

J b u A A s u J

0(0,0, ) uuR

u Stability critieria:

21 12 11 22

( *) ( *) c s u c c c s u c competition ratio

1 c stable equilibria 1 c stable 2-cycles

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1

[ ( ) ( )] 1 2 ( ) ( )

u t t t t t t

u s u u u v u s u

   

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   

b (u) s(u) R0(0,0,u) u u

Dynamical Dichotomy holds at critical trait u* = 1

b/4

2

1 ( ) , ( ) 1 1 [0, ) (0,0, ) (1 ) u s u u u u U u R u b u trait interval:

Example 1

21 22 11 12

1 1

1 1 1 1 1 1 1

t t t t

t t t t t t t

u c J c A u u c J c A

J b A A J

1

1 1 2 (1 )

t t t t t

u u u v u u

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Example 1

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Sample Time Series R0(0,0,1) = 0.75 Extinction R0(0,0,1) = 2

c < 1 Stable Equilibrium

R0(0,0,1) = 2

c > 1 Stable 2-cycle

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   

Example 2

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    

b (u) s(u) R0(0,0,u) u u Large a > a0 a = s′′(0)

2 2 2 1 2 2 2

2 2 ( ) , ( ) 1 2 0, , 1 2 2 (0,0, ) 1 2 m u s u u au u m U R m u R u b au u trait interval:

2 2 21 22

1 1 2 11 12

2 1 2 1

2 1 1 1

t t t t

t t t t t t t

u u c J c A

J b A m A A au c J c A

2 2 2

1

1 2 2 (1 )(1 )

t t t

t t

u u u

u u vu

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2 2 21 22

1 1 2 11 12

2 1 2 1

2 1 1 1

t t t t

t t t t t t t

u u c J c A

J b A m A A au c J c A

2 2 2

1

1 2 2 (1 )(1 )

t t t

t t

u u u

u u vu

   

Example 2

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    

b (u) s(u) R0(0,0,u) u u

Dynamical Dichotomy holds at critical trait u* = 0

a = s′′(0)

2 2 2 1 2 2 2

2 2 ( ) , ( ) 1 2 0, , 1 2 2 (0,0, ) 1 2 m u s u u au u m U R m u R u b au u trait interval:

Large a > a0

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Example 1

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Sample Time Series R0(0,0,0) = 1.5

c < 1 Stable Equilibrium

R0(0,0,0) = 1.5

c > 1 Stable 2-cycle

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   

    

Example 2

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b (u) s(u) u u Small a < a0 R0(0,0,u)

2 2 2 1 2 2 2

2 2 ( ) , ( ) 1 2 0, , 1 2 2 (0,0, ) 1 2 m u s u u au u m U R m u R u b au u trait interval:

a = s′′(0)

2 2 21 22

1 1 2 11 12

2 1 2 1

2 1 1 1

t t t t

t t t t t t t

u u c J c A

J b A m A A au c J c A

2 2 2

1

1 2 2 (1 )(1 )

t t t

t t

u u u

u u vu

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2 2 21 22

1 1 2 11 12

2 1 2 1

2 1 1 1

t t t t

t t t t t t t

u u c J c A

J b A m A A au c J c A

2 2 2

1

1 2 2 (1 )(1 )

t t t

t t

u u u

u u vu

   

    

Example 2

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b (u) s(u) u u Small a < a0 R0(0,0,u)

Dynamical Dichotomy holds at 2 critical traits u*

2 2 2 1 2 2 2

2 2 ( ) , ( ) 1 2 0, , 1 2 2 (0,0, ) 1 2 m u s u u au u m U R m u R u b au u trait interval:

a = s′′(0)

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Example 2

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Sample Time Series R0(0,0, u*) = 1.5

c < 1 Stable Equilibrium

R0(0,0,u*) = 1.5

c > 1 Stable 2-cycle

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u will equilibrate at negative critical trait for other initial conditions

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Summary

• The Fundamental Bifurcation Theorem holds at critical traits u*

where inherent (low density) fitness has a local maximum : ∂uR0(0,0,u*) = 0, ∂uuR0(0,0,u*) < 0

• Positive equilibria & synchronous 2-cycles bifurcate at R0(0,0,u*) = 1.
• The Dynamic Dichotomy holds.

For example, when no Allee effects are present. For the EGT semelparous juvenile-adult model :

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Open Problems

• What are the properties of the bifurcation at R0 = 1 for Leslie

semelparous of dimensions 3 and higher without evolution ? with evolution?

• Global stability results? Especially for Leslie-Gower nonlinearities.
• What happens to bifurcating branches for large R0 >> 1 ?
• EGT Leslie semelparous models with 2 or more phenotypic traits

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