Difference Equations Arising in Evolutionary Population Dynamics - PowerPoint PPT Presentation

Difference Equations Arising in Evolutionary Population Dynamics J. M. Cushing Simon Maccracken Department of Mathematics Department of Ecology & Evolutionary Biology Interdisciplinary Program in Applied Mathematics University of Arizona University of Arizona Tucson, Arizona, USA Tucson, Arizona, USA Support by the National Science Foundation

OUTLINE 1/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Density dependence 1 : R [0, ) 2 are C : (0,1) 2 where open (0,0) R (0,0) (0,0) 1 b inherent (low density) adult fertility s inherent (low density) juvenile survival R sb 0 inherent net reproductive number ( expected offspring per individual per life time ) 2/30 ICDEA 2012 Barcelona

Iteroparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) s ( J A A , ) t 1 j j t t t a a t t t 1 : R [0, ) 2 are C , : (0,1) j a 2 where open (0,0) R (0,0) (0,0) (0,0) 1 j a b inherent adult fertility s inherent juvenile survival, 0 s inherent adult survival j a s j inherent net reproductive number R b 0 1 s a ( expected offspring per individual per life time ) 3/30 ICDEA 2012 Barcelona

Iteroparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) s ( J A A , ) t 1 j j t t t a a t t t Define : s s s s s 0 0 0 0 0 0 j j j a a a J j A A a J A j J a 1 s 1 s 1 s 1 s 1 s a a a a a Within-class Between-class competitive effects competitive effects 4/30 ICDEA 2012 Barcelona

Iteroparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) s ( J A A , ) t 1 j j t t t a a t t t Fundamental Bifurcation Theorem 1. Origin is stable if R 0 < 1 and unstable if R 0 > 1 . Assume a + ≠ 0 . 2. Positive equilibria (transcritically) bifurcate from the origin at R 0 = 1. 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 5/30 ICDEA 2012 Barcelona

Iteroparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) s ( J A A , ) t 1 j j t t t a a t t t Notes  A right (stable) bifurcation occurs if there are no positive feedback effects (at low density), i.e. if there are no positive derivatives 0 0 0 0 A . , , , J A J  Positive feedback terms (positive derivatives) are called Allee effects.  A left (unstable) bifurcation can only occur in the presence of strong Allee effects. 6/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Lloyd & Dybas (1966, 1974) Wikan & Mjølhus (1997) Hoppensteadt & Keller (1976) Behncke (2000) Bulmer (1977) Davydova (2004) May (1979) Davydova, Diekmann & van Gils (2003, 2005) Ebenman (1987, 1988) Mjølhus, Wikan & Solberg (2005) JC & Li (1989, 1992) JC (1991, 2003, 2006, 2010) Wikan & (1996) Kon (2005, 2007) Nisbet & Onyiah (1994) Diekmann & Yan (2008) JC & Henson (2012) 7/30 ICDEA 2012 Barcelona

Semelparity Plants Invertebrates Vertebrates Annuals Species of Insects Arachnids Molluscs Species of : Monocarpic perennials Fish Lizards Amphibians Periodical Insects Marsupials … even a mammal ! Magicicada spp. 13, 17 year cycles Lasiocampa “Poster - child” quercus var Melontha spp. species 1, 2 year cycles 3, 4, 5 year cycles 8/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Define : 0 0 0 0 a s s J j A J j A Within-class Between-class competitive effects competitive effects 10/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Fundamental bifurcation Theorem JC & Jia Li (1989), JC (2006), JC & Henson (2012) 11/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Fundamental bifurcation Theorem 1. Origin is stable if R 0 = sb < 1 and unstable if R 0 > 1 . Assume a + ≠ 0 . 2. Positive equilibria bifurcate from the origin at R 0 = 1. (or ) right (or left) bifurcation a 0 0 3. Synchronous 2-cycles also bifurcate from the origin at R 0 = 1. 0 0 (or ) right (or left) bifurcation s 0 0 J A 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 equilibria stabl e & 2 - cycles un stable a 0 Dynamic equilibria unstab le & 2 - cycles st able a 0 Dichotomy 11/30 ICDEA 2012 Barcelona

Notes  If there are no Allee effects, that is, no positive derivatives 0 0 0 0 , , , J A J A then Dynamic Dichotomy occurs.  Weak between-class competition gives stable equilibria 0 0 0 0 a s s 0 J j A J j A  Strong between-class competition gives stable synchronous 2-cycles 0 0 0 0 a s s 0 J j A J 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) 12/30 ICDEA 2012 Barcelona

Semelparous Juvenile-Adult Models J b ( J A A , ) t 1 t t t A s ( J A J , ) t 1 t t t Typical example for which the Dynamic Dichotomy occurs: 1 1 ( , ) J A , ( , ) J A 1 c J c A 1 c J c A 11 12 21 22 Leslie-Gower-type competition interaction functionals … a natural extension of discrete logistic ( or Beverton-Holt ) equation for an unstructured population. 13/30 ICDEA 2012 Barcelona

Evolutionary Semelparous Juvenile-Adult Models J b ( ) ( u J A u A , , ) (0,0, ) u 1 t 1 t t t A s u ( ) ( J A u J , , ) (0,0, ) u 1 t 1 t t t u mean of a phenotypic trait subject to Darwinian evolution trait interval u U Assume max ( ) u 1. U Then maximum adult fertility ( over the trait interval ) b U 14/30 ICDEA 2012 Barcelona

Evolutionary Semelparous Juvenile-Adult Models J b ( ) ( u J A u A , , ) (0,0, ) u 1 t 1 t t t t t A s u ( ) ( J A u J , , ) (0,0, ) u 1 t 1 t t t t t Using Evolutionary Game Theory Methodology 1 u u v ln R J A u ( , , ) t 1 t u 0 2 R J A u ( , , ) b ( ) ( ) ( , , ) ( , , ) u s u J A u J A u 0 v trait variance ( assumed constant in time ) Vincent & Brown 2005 15/30 ICDEA 2012 Barcelona

Evolutionary Semelparous Juvenile-Adult Models J b ( ) ( u J A u A , , ) (0,0, ) u 1 t 1 t t t t t A s u ( ) ( J A u J , , ) (0,0, ) u 1 t 1 t t t t t 1 u u v ln[ b ( ) ( ) ( , , ) ( , , )] u s u J A u J A u t 1 t u 2  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)? 16/30 ICDEA 2012 Barcelona

Evolutionary Semelparous Juvenile-Adult Models J b ( ) ( u J A u A , , ) (0,0, ) u 1 t 1 t t t t t A s u ( ) ( J A u J , , ) (0,0, ) u 1 t 1 t t t t t 1 u u v ln[ b ( ) ( ) ( , , ) ( , , )] u s u J A u J A u t 1 t u 2 Extinction Equilibria & Critical Traits An extinction equilibrium ( J, A, u ) is an equilibrium with J = A = 0. A critical trait u satisfies u R 0 (0,0, ) u 0. Easy to see that … (0 , 0 , u ) is an extinction equilibrium if and only if u is a critical trait. 17/30 ICDEA 2012 Barcelona

Evolutionary Semelparous Juvenile-Adult Models J b ( ) ( u J A u A , , ) (0,0, ) u 1 t 1 t t t t t A s u ( ) ( J A u J , , ) (0,0, ) u 1 t 1 t t t t t 1 u u v ln[ b ( ) ( ) ( , , ) ( , , )] u s u J A u J A u t 1 t u 2 Synchronous 2-cycles J 0 0 A u u 1 2 18/30 ICDEA 2012 Barcelona

Main Theorem Assume … 1. There exists a critical trait u * : ∂ u R 0 (0, 0, u *) = 0 2. ∂ uu R 0 (0 ,0, u *) ≠ 0 0 0 0 0 0 0 3. and a s s 0 s 0 J j A J j A J A Then … (a) If ∂ uu R 0 (0 ,0, u *) < 0 then the Fundamental Bifurcation Theorem and the Dynamic Dichotomy hold … using R 0 (0 ,0, u *) as a bifurcation parameter. (b) If ∂ uu R 0 (0 ,0, u *) > 0 then the equilibria (extinction & positive) equilibria and the synchronous 2-cycles are all unstable . NOTE: R 0 (0, 0, u ) = b b ( u ) s ( u ) 19/30 ICDEA 2012 Barcelona

Examples Leslie-Gower Nonlinearities 1 1 ( , ) J A , ( , ) J A 1 c J c A 1 c J c A 11 12 21 22 at least one c 0, c 0 ij ii • 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 ∂ uu R 0 (0 ,0, u *) < 0 20/30 ICDEA 2012 Barcelona