Difference Equations Arising in Evolutionary Population Dynamics · 2012. 11. 20. · Difference Equations Arising in Evolutionary Population Dynamics ... Examples ICDEA 2012 Barcelona

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

1/30

Semelparous Juvenile-Adult Models


1

1

( , )

( , )

t t t t

t t t t

J b J A A

A s J A J

0R 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)

RC

R

are

where open

(0,0) (0,0) 1

Density dependence

2/30

Iteroparous Juvenile-Adult Models


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

RC

R

are

where open

(0,0) (0,0) (0,0) 1j a

0j a

b

s s

inherent adult fertility

inherent juvenile survival, inherent adult survival

01

j

a

sR b

s inherent net reproductive number

( expected offspring per individual per life time )

3/30



1

1

( , )

( , ) ( , )

t t t t


J b J A A

A s J A J s J A A

Define :

0 0 0 0 0 0

1 1 1 1 1

j j ja a

a a a a a

J j A A a J A j J a

s s ss s

s s s s sa

Within-class competitive effects

Between-class competitive effects

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1

1

( , )

( , ) ( , )

t 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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1

1

( , )

( , ) ( , )

t 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.

0 0 0 0, , ,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)


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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1

1

( , )

( , )

t t t t

t t t t

J b J A A

A s J A J

Define :

0 0 0 0

J j A J j Aa s s

Within-class competitive effects

Between-class competitive effects

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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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Fundamental bifurcation Theorem

0 0 0 0 J As (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

0 0 a (or ) right (or left) bifurcation

0 a equilibria stabl & 2 - cycles une stable

0 a equilibria unstab 2 - cycles le st & ableDynamic

Dichotomy

1

1

( , )

( , )

t t t t

t t t t

J b J A A

A s J A J

11/30

Notes

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

0 0 0 0 0J j A J j Aa s s

0 0 0 0 0J j A J j Aa s s

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

then Dynamic Dichotomy occurs.

0 0 0 0, , ,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)

ICDEA 2012 Barcelona 12/30



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 1J 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


u U trait interval

max ( ) 1.U u

Ub

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

14/30

u mean of a phenotypic trait subject to Darwinian evolution



1 0

0

1

2ln ( , , )

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

t t uu 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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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

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

2t t uu 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

16/30



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

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


(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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Synchronous 2-cycles

1 2

0

0

J

A

u u

1

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


(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

18/30

Main Theorem


(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)

19/30

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

0 0 0 0 0 00 0J j A J j A J Aa s s s 3. and

Then …

Examples


11 12 21 22

1 1( , ) , ( , )

1 1

0, 0 ij ii

J A J Ac J c A c J c A

c cat 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

20/30

Examples


Mathematically:

s(u) and b (u) have opposite monotonicity

( ) ( ) 2 ( ) ( ) ( ) ( ) 0u s u u s u u s u

Trade-offs are of fundamental biological interest:

Survival & fertility change in opposite manner as u increases.

Dynamic Dichotomy criterion:

Critical trait equation: 0

0(0,0, ) 0uR u

( ) ( ) ( ) ( ) 0u s u u s u

0(0,0, ) ( ) ( )R u b u s u

Net reproductive number:

0

0(0,0, ) 0uuR u

21/30

1

[ ( ) ( )]1

2 ( ) ( )

u t tt t

t t

u s uu 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


( ) ( ) 2 ( ) ( ) ( ) ( ) 0u s u u s u u s u



0(0,0, ) 0uR u

( ) ( ) ( ) ( ) 0u s u u s u

0(0,0, ) ( ) ( )R u b u s u


0

0(0,0, ) 0uuR u

Stability critieria:

21 12 11 22[ ( *) ] [ ( *) ] 0

a c s u c c s u c

stable equilibria

21 12

11 22

( *)

( *)

c s u cc

c s u c

competition ratio

22/30

1

[ ( ) ( )]1

2 ( ) ( )

u t tt t

t t

u s uu 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


( ) ( ) 2 ( ) ( ) ( ) ( ) 0u s u u s u u s u



0(0,0, ) 0uR u

( ) ( ) ( ) ( ) 0u s u u s u

0(0,0, ) ( ) ( )R u b 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

0

0(0,0, ) 0uuR u

Stability critieria:

21 12

11 22

( *)

( *)

c s u cc

c s u c

competition ratio

1 c stable equilibria

1 c stable 2 - cycles

22/30

1

[ ( ) ( )]1

2 ( ) ( )

u t tt t

t t

u s uu u v

u s u


b (u)

s(u)

R0(0,0,u)

u u

Dynamical Dichotomy holds at critical trait

u* = 1

b/4

0 2

1( ) , ( )

1 1

[0, )

(0,0, )(1 )

us u u

u u

U

uR 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

11

2 (1 )

tt t

t t

uu u v

u u

23/30

Example 1


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

24/30

Example 2


b (u)

s(u)

R0(0,0,u)

u u

Large a > a0

a = s′′(0)

2

2 2

1

2

0 2 2

2 2( ) , ( )

1 2

0, 0 , 1

2 2(0,0, )

1 2

m us u u

au u

m

U R

m uR 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 AJ b A

mA A

au c J c A

2

2 21

1 2 2

(1 )(1 )

t

t t

t t

u

u uu u vu

25/30

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 AJ b A

mA A

au c J c A

2

2 21

1 2 2

(1 )(1 )

t

t t

t t

u

u uu u vu

Example 2


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

0 2 2

2 2( ) , ( )

1 2

0, 0 , 1

2 2(0,0, )

1 2

m us u u

au u

m

U R

m uR u b

au u

trait interval :

Large a > a0

25/30

Example 1


Sample Time Series

R0(0,0,0) = 1.5

c < 1

Stable Equilibrium

R0(0,0,0) = 1.5

c > 1

Stable 2-cycle

26/30

Example 2


b (u)

s(u)

u u

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

2

2 2

1

2

0 2 2

2 2( ) , ( )

1 2

0, 0 , 1

2 2(0,0, )

1 2

m us u u

au u

m

U R

m uR 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 AJ b A

mA A

au c J c A

2

2 21

1 2 2

(1 )(1 )

t

t t

t t

u

u uu u vu

27/30

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 AJ b A

mA A

au c J c A

2

2 21

1 2 2

(1 )(1 )

t

t t

t t

u

u uu u vu

Example 2


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

0 2 2

2 2( ) , ( )

1 2

0, 0 , 1

2 2(0,0, )

1 2

m us u u

au u

m

U R

m uR u b

au u

trait interval :a = s′′(0)

27/30

Example 2


Sample Time Series

R0(0,0, u*) = 1.5

c < 1

Stable Equilibrium

R0(0,0,u*) = 1.5

c > 1

Stable 2-cycle

28/30

u will equilibrate at negative critical trait

for other initial

conditions


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 :

29/30


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

30/30

Documents

Difference Equations Arising in Evolutionary Population Dynamics · 2012. 11. 20. · Difference Equations Arising in Evolutionary Population Dynamics ... Examples ICDEA 2012 Barcelona