An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics

Transcription

1 An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics Barbara Boldin Department of Mathematics, Natural Sciences and Information Technologies University of Primorska Based on joint work with Odo Diekmann Barbara Boldin (University of Primorska) AD extension April / 29

2 Outline 1. Classification of evolutionary singularities in the standard setting Geritz, S. A. H., Kisdi, E., Meszéna, G., Metz, J. A. J.: Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology, 12, pp (1998). 2. Why extend the framework? The motivating example: superinfections and the adaptive dynamics of infectious agents 3. Dynamics of adaptation in an extended setting: Classification of singularities with respect to convergence stability, invadability and existence of nearby dimorphisms Dimorphic dynamics nearby monomorphic evolutionary singularities 4. Conclusions Barbara Boldin (University of Primorska) AD extension April / 29

3 The standard AD setting Classification of singularities The standard setting of Adaptive Dynamics Classification of evolutionary singularities Considers scalar traits and assumes that the invasion fitness s x (y) is differentiable twice as a function of resident (x) and invader strategy (y). A singular strategy x is a point where the selection gradient vanishes, i.e. s y = 0. y=x=x Classification of evolutionary singularities considers four properties: 1. Invadability 2. Convergence stability 3. Existence of nearby dimorphisms 4. Attainability All four properties can be expressed with a computable condition involving 2 s x 2 y=x=x and/or 2 s y 2 y=x=x. Barbara Boldin (University of Primorska) AD extension April / 29

4 The standard AD setting Classification of singularities Geritz, S. A. H., Kisdi, E., Meszéna, G., Metz, J. A. J.: Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology, 12, pp (1998). Barbara Boldin (University of Primorska) AD extension April / 29

5 The standard AD setting Classification of singularities If we focus on convergence stable singularities (i.e. assume 2 s 2 s > 0) x 2 y 2 and forget about attainability, there are only 3 local configurations: An ESS with An ESS with An invadable an empty MIR a nonempty MIR singularity 2 s y 2 < 0 & 2 s y 2 < 0 & 2 s y 2 > 0 2 s 2 s < 0 2 s 2 s > 0 x 2 y 2 x 2 y 2 Barbara Boldin (University of Primorska) AD extension April / 29

6 The standard AD setting Dimorphic dynamics The standard setting of Adaptive Dynamics Dimorphic dynamics nearby an evolutionary singularity s x1,x 2 (y) = 2 s y 2 y=x=x (y x 1)(y x 2 ) 1. Dimorphisms nearby ESS are converging: 2. Dimorphisms nearby an invading c. stable strategy are diverging (BP): Barbara Boldin (University of Primorska) AD extension April / 29

7 The standard AD setting Alternative notation The standard setting of Adaptive Dynamics Alternative notation Write x = x u, y = x v, take into account s z (z) = 0. Write ( ) s u (v) = (u v)l(u, v) = (u v) (a 0 a 1 )u a 1 v h.o.t.. Convergence stability: a 0 > 0 An ESS with An ESS with An invadable an empty MIR a nonempty MIR singularity a 1 > 0 & a 0 < 2a 1 a 1 > 0 & a 0 > 2a 1 a 1 < 0 Barbara Boldin (University of Primorska) AD extension April / 29

8 The standard AD setting Standard setting: conclusions The standard setting of Adaptive Dynamics Conclusion The type of the singularity, as well as the fate of dimorphisms nearby the singularity can be determined from 2 s x 2 y=x=x or, alternatively, from a 0 and a 1. and/or 2 s y 2 y=x=x Barbara Boldin (University of Primorska) AD extension April / 29

9 The standard AD setting Standard setting: conclusions The standard setting of Adaptive Dynamics Conclusion The type of the singularity, as well as the fate of dimorphisms nearby the singularity can be determined from 2 s x 2 y=x=x or, alternatively, from a 0 and a 1. and/or 2 s y 2 y=x=x Question: What if the invasion fitness s x (y) is not sufficiently smooth? Barbara Boldin (University of Primorska) AD extension April / 29

10 Why extend the existing framework? Why extend the existing framework? Superinfections and the AD of pathogens (B.Boldin, O.Diekmann. J. Math. Biol. (2008)) Basic model assumptions: Viruses compete at two levels: for uninfected cells within a host and for susceptible hosts at the population level. Strains are characterized by their withinhost reproduction rate p. WH reproduction rate affects two parameters that determine the spread at the population level: transmissibility and virulence. Barbara Boldin (University of Primorska) AD extension April / 29

11 Why extend the existing framework? Why extend the existing framework? Superinfections and the AD of pathogens (B.Boldin, O.Diekmann. J. Math. Biol. (2008)) Basic model assumptions: Viruses compete at two levels: for uninfected cells within a host and for susceptible hosts at the population level. Strains are characterized by their withinhost reproduction rate p. WH reproduction rate affects two parameters that determine the spread at the population level: transmissibility and virulence. Two levels of population dynamics and evolution: 1. Withinhost dynamics: viruses compete for uninfected target cells; optimal strain minimizes abundance of uninfected cells, ˆT (p). Barbara Boldin (University of Primorska) AD extension April / 29

12 Why extend the existing framework? 2. Epidemiological model, single strain: ds = b β(p)si ds dt di = β(p)si (d α(p))i dt Barbara Boldin (University of Primorska) AD extension April / 29

13 Why extend the existing framework? 2. Epidemiological model, two strains, superinfections ds dt = b β(p)si p β(q)si q ds di ( ) p dt = β(p)si p β(q)φ(p, q) β(p)φ(q, p) I p I q (d α(p))i p di ( ) q dt = β(q)si q β(q)φ(p, q) β(p)φ(q, p) I p I q (d α(q))i q The superinfection function φ(p, q) gives the probability that strain q takes over a host already infected by strain p. Barbara Boldin (University of Primorska) AD extension April / 29

14 Why extend the existing framework? 2. Epidemiological model, two strains, superinfections ds dt = b β(p)si p β(q)si q ds di ( ) p dt = β(p)si p β(q)φ(p, q) β(p)φ(q, p) I p I q (d α(p))i p di ( ) q dt = β(q)si q β(q)φ(p, q) β(p)φ(q, p) I p I q (d α(q))i q The superinfection function φ(p, q) gives the probability that strain q takes over a host already infected by strain p. Superinfection probability can be derived from the underlying branching process (n is the reinfection dose): ( 1 1 k ˆT (p) φ n (p, q) = c k ˆT (p) k ˆT (q) ) n c k ˆT ; ˆT (q) < ˆT (p) (q) 0 ; ˆT (q) ˆT (p) Barbara Boldin (University of Primorska) AD extension April / 29

15 Why extend the existing framework? The superinfection probability is 0 when ˆT (q) > ˆT (p) and grows away from 0 with a nonzero derivative on ˆT (q) < ˆT (p) ^T(p) ^T(q) Barbara Boldin (University of Primorska) AD extension April / 29

16 Why extend the existing framework? The superinfection probability is 0 when ˆT (q) > ˆT (p) and grows away from 0 with a nonzero derivative on ˆT (q) < ˆT (p) Invasion fitness: ^T(p) ^T(q) { β(q)φn (p, q) ; ˆT (q) < ˆT (p) s p (q) = β(q)ŝ(p) (α(q) d) Î p β(p)φ n (q, p) ; ˆT (q) > ˆT (p) Barbara Boldin (University of Primorska) AD extension April / 29

17 Why extend the existing framework? The superinfection probability is 0 when ˆT (q) > ˆT (p) and grows away from 0 with a nonzero derivative on ˆT (q) < ˆT (p) Invasion fitness: ^T(p) ^T(q) { β(q)φn (p, q) ; ˆT (q) < ˆT (p) s p (q) = β(q)ŝ(p) (α(q) d) Î p β(p)φ n (q, p) ; ˆT (q) > ˆT (p) Observation q s p (q) is differentiable once. The left and the right second derivatives in q = p exist, but they are unequal. Barbara Boldin (University of Primorska) AD extension April / 29

18 The extended framework The extended framework Let x be a monomorphic singularity, write u = x u, y = x v and { L (u, v), u v s u (v) = (u v) L (u, v), u v with and L (u, u) = L (u, u) L (0, 0) = 0 = L (0, 0). Barbara Boldin (University of Primorska) AD extension April / 29

19 The extended framework The extended framework Let x be a monomorphic singularity, write u = x u, y = x v and { L (u, v), u v s u (v) = (u v) L (u, v), u v with and Write L (u, u) = L (u, u) L (0, 0) = 0 = L (0, 0). L (u, v) = (a 0 a 1 )u a 1 v h.o.t. L (u, v) = (a 0 a 1 )u a 1 v h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

20 The extended framework Convergence stability Properties of evolutionary singularities Convergence stability with s u (v) = (u v) { L (u, v), u v L (u, v), u v L (u, v) = (a 0 a 1 )u a 1 v h.o.t. L (u, v) = (a 0 a 1 )u a 1 v h.o.t. Singularity convergence stable if L ± (u, u) < 0 for small u < 0 and L ± (u, u) > 0 for small u > 0. Barbara Boldin (University of Primorska) AD extension April / 29

21 The extended framework Convergence stability Properties of evolutionary singularities Convergence stability with s u (v) = (u v) { L (u, v), u v L (u, v), u v L (u, v) = (a 0 a 1 )u a 1 v h.o.t. L (u, v) = (a 0 a 1 )u a 1 v h.o.t. Singularity convergence stable if L ± (u, u) < 0 for small u < 0 and L ± (u, u) > 0 for small u > 0. Convergence stability Since L ± (u, u) = a 0 u, the singularity is convergence stable when a 0 > 0 and an evolutionary repeller when a 0 < 0. Barbara Boldin (University of Primorska) AD extension April / 29

22 The extended framework Invadability Properties of evolutionary singularities Invadability s 0 (v) = v { { L (0, v), u v a L (0, v), u v = 1 v 2, 0 v a1 v 2, 0 v Barbara Boldin (University of Primorska) AD extension April / 29

23 The extended framework Invadability Properties of evolutionary singularities Invadability s 0 (v) = v { { L (0, v), u v a L (0, v), u v = 1 v 2, 0 v a1 v 2, 0 v Invadability Singularity is: Invadable if a < 0 and a < 0, Uninvadable (ESS) if a > 0 and a > 0, Invadable from above, uninvadable from below if a < 0 and a > 0, Uninvadable from above, invadable from below if a > 0 and a < 0. We call the last two types a onesided ESS. Barbara Boldin (University of Primorska) AD extension April / 29

24 The extended framework Existence of nearby dimorphisms Properties of evolutionary singularities Existence of nearby dimorphisms Focus on u 1 u 2, define invadability subsets: IR 1 = {(u 1, u 2 ) : u 2 u 1 & s u2 (u 1 ) > 0} = {(u 1, u 2 ) : u 2 > u 1 & L (u 2, u 1 ) > 0} IR 2 = {(u 1, u 2 ) : u 2 u 1 & s u1 (u 2 ) > 0} Local mutual invadability region = {(u 1, u 2 ) : u 2 > u 1 & L (u 1, u 2 ) < 0} MIR(δ) = IR 1 IR 2 B δ. Barbara Boldin (University of Primorska) AD extension April / 29

25 The extended framework Existence of nearby dimorphisms Properties of evolutionary singularities Existence of nearby dimorphisms Focus on u 1 u 2, define invadability subsets: IR 1 = {(u 1, u 2 ) : u 2 u 1 & s u2 (u 1 ) > 0} = {(u 1, u 2 ) : u 2 > u 1 & L (u 2, u 1 ) > 0} IR 2 = {(u 1, u 2 ) : u 2 u 1 & s u1 (u 2 ) > 0} Local mutual invadability region = {(u 1, u 2 ) : u 2 > u 1 & L (u 1, u 2 ) < 0} MIR(δ) = IR 1 IR 2 B δ. Dimorphisms nearby evolutionary singularity If a 0 > a 1 a 1 If a 0 < a 1 a 1 then MIR(δ) is nonempty for δ > 0 then MIR(δ) is empty for small positive δ. Barbara Boldin (University of Primorska) AD extension April / 29

26 The extended framework Seven local configurations Seven local configurations nearby a convergence stable s.s. I. a 1 VII. II. IV. VI. V. III. a 1 a 1 = a0 a 1 Barbara Boldin (University of Primorska) AD extension April / 29

27 The dimorphic invasion fitness The extended framework The dimorphic invasion fitness (u 2 v)k (u 1, u 2, v) u 1 u 2 v s u1,u 2 (v) = (u 2 v)(u 1 v)k 0 (u 1, u 2, v) u 1 v u 2 (u 1 v)k (u 1, u 2, v) v u 1 u 2 Barbara Boldin (University of Primorska) AD extension April / 29

28 The dimorphic invasion fitness The extended framework The dimorphic invasion fitness (u 2 v)k (u 1, u 2, v) u 1 u 2 v s u1,u 2 (v) = (u 2 v)(u 1 v)k 0 (u 1, u 2, v) u 1 v u 2 (u 1 v)k (u 1, u 2, v) v u 1 u 2 Requirement I. K (0, 0, v) = L (0, v), v 0 K (0, 0, v) = L (0, v), v 0 Barbara Boldin (University of Primorska) AD extension April / 29

29 The dimorphic invasion fitness The extended framework The dimorphic invasion fitness (u 2 v)k (u 1, u 2, v) u 1 u 2 v s u1,u 2 (v) = (u 2 v)(u 1 v)k 0 (u 1, u 2, v) u 1 v u 2 (u 1 v)k (u 1, u 2, v) v u 1 u 2 Requirement I. Requirement II. K (0, 0, v) = L (0, v), v 0 K (0, 0, v) = L (0, v), v 0 s u1,u 2 (v) = s u1 (v) if s u1 (u 2 ) = 0 s u1,u 2 (v) = s u2 (v) if s u2 (u 1 ) = 0 Barbara Boldin (University of Primorska) AD extension April / 29

30 The dimorphic invasion fitness The extended framework The dimorphic invasion fitness (u 2 v)k (u 1, u 2, v) u 1 u 2 v s u1,u 2 (v) = (u 2 v)(u 1 v)k 0 (u 1, u 2, v) u 1 v u 2 (u 1 v)k (u 1, u 2, v) v u 1 u 2 Requirement I. Requirement II. K (0, 0, v) = L (0, v), v 0 K (0, 0, v) = L (0, v), v 0 s u1,u 2 (v) = s u1 (v) if s u1 (u 2 ) = 0 s u1,u 2 (v) = s u2 (v) if s u2 (u 1 ) = 0 Requirement III. K (u 1, u 2, u 2 ) = (u 1 u 2 )K 0 (u 1, u 2, u 2 ) K (u 1, u 2, u 1 ) = (u 2 u 1 )K 0 (u 1, u 2, u 1 ) Barbara Boldin (University of Primorska) AD extension April / 29

31 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 2 We know K (0, 0, v) = a 1 v h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

32 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 2 We know Ansatz K (0, 0, v) = a 1 v h.o.t. K (u 1, u 2, v) = k 1 u 1 k 2 u 2 a 1 v h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

33 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 2 We know Ansatz K (0, 0, v) = a 1 v h.o.t. K (u 1, u 2, v) = k 1 u 1 k 2 u 2 a 1 v h.o.t. Along we have Along C 1 = {(u 1, u 2 ) : u 2 u 1 & s u2 (u 1 ) = 0} = {(u 1, u 2 ) : u 2 > u 1 & L (u 2, u 1 ) = 0} k 1 u 1 k 2 u 2 a 1 v = (a 0 a 1 )u 2 a 1 v h.o.t. C 2 = {(u 1, u 2 ) : u 2 u 1 & s u1 (u 2 ) = 0} = {(u 1, u 2 ) : u 2 > u 1 & L (u 1, u 2 ) = 0} we have (u 2 v)(k 1 u 1 k 2 u 2 a 1 v) = (u 1 v)((a 0 a 1 )u 1 a 1 v) h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

34 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 2 We know Ansatz K (0, 0, v) = a 1 v h.o.t. K (u 1, u 2, v) = k 1 u 1 k 2 u 2 a 1 v h.o.t. Along C 1 we have Along C 2 we have k 1 u 1 k 2 u 2 = (a 0 a 1 )u 2 k 1 u 1u 2 k 2 u2 2 = (a 0 a 1 )u2 1 a 1 u 2 k 1 u 1 k 2 u 2 = (2a 1 a 0)u 1 Barbara Boldin (University of Primorska) AD extension April / 29

35 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 2 We know Ansatz K (0, 0, v) = a 1 v h.o.t. K (u 1, u 2, v) = k 1 u 1 k 2 u 2 a 1 v h.o.t. The tangent lines to: C 1 : (a 0 a 1 )u 2 a 1 u 1 = 0, C 2 : (a 0 a 1 )u 1 a 1 u 2 = 0. This gives k 1 = a 0a 1 2a 1 a 1 a 1 a 1 a 0 k 2 = a 1 a 1 (a 1 )2 a 1 a 1 a. 0 Barbara Boldin (University of Primorska) AD extension April / 29

36 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: v u 1 We know Ansatz K (0, 0, v) = a 1 v h.o.t. K (u 1, u 2, v) = k 1 u 1 k 2 u 2 a 1 v h.o.t. k 1 = a 1 a 1 (a 1 )2 a 1 a 1 a 0 k2 = a 0a 1 2a 1 a 1 a 1 a 1 a. 0 Barbara Boldin (University of Primorska) AD extension April / 29

37 The extended framework The dimorphic invasion fitness Dimorphic invasion fitness: u 1 v u 2 Equalities K (u 1, u 2, u 2 ) = (u 1 u 2 )K 0 (u 1, u 2, u 2 ) K (u 1, u 2, u 1 ) = (u 2 u 1 )K 0 (u 1, u 2, u 1 ) link first order terms of K ± to zeroth order terms of K 0. The lowest order term of K 0 is a function of just u 1 and u 2, J 0 (u 1, u 2 ). Since both equalities give k 1 a 1 = k 1 k 1 a 1 = k 2, J 0 (u 1, u 2 ) = k 1 u 1 k 2 u 2 u 1 u 2. Barbara Boldin (University of Primorska) AD extension April / 29

38 Dimorphic dynamics The extended framework Dimorphic dynamics Along the 2 extinction boundary we have s u1,u 2 (v) = s u1 (v) and so s u1,u 2 (v) v s u1,u 2 (v) v v=u1 = a 0 u 1 h.o.t. v=u2 = a 1 (u 1 u 2 ) h.o.t.. Barbara Boldin (University of Primorska) AD extension April / 29

39 Dimorphic dynamics The extended framework Dimorphic dynamics Along the 2 extinction boundary we have s u1,u 2 (v) = s u1 (v) and so s u1,u 2 (v) v s u1,u 2 (v) v v=u1 = a 0 u 1 h.o.t. v=u2 = a 1 (u 1 u 2 ) h.o.t.. Along the 1 extinction boundary we have s u1,u 2 (v) = s u2 (v) and so s u1,u 2 (v) v s u1,u 2 (v) v v=u1 = a 1 (u 2 u 1 ) h.o.t. v=u2 = a 0 u 2 h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

40 The extended framework Dimorphic dynamics Mutant trait Mutant trait Mutant trait Mutant trait Resident trait Resident trait Resident trait Resident trait Barbara Boldin (University of Primorska) AD extension April / 29

41 The extended framework Dimorphic dynamics Mutant trait Mutant trait Mutant trait Mutant trait Resident trait Resident trait Resident trait Resident trait Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 1 Resident trait 1 Resident trait 1 Resident trait 1 (a) (b) (c) (d) Barbara Boldin (University of Primorska) AD extension April / 29

42 The extended framework Dimorphic dynamics Mutant trait Mutant trait Mutant trait Mutant trait Resident trait Resident trait Resident trait Resident trait Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 1 Resident trait 1 Resident trait 1 Resident trait 1 (a) (b) (c) (d) Observations: (a) is an ESS, (c) is a BP. In (b) and (d) both traits change direction of evolution, isoclines connect to the singularity. Barbara Boldin (University of Primorska) AD extension April / 29

43 The extended framework Dimorphic dynamics Mutant trait Mutant trait Mutant trait Mutant trait Resident trait Resident trait Resident trait Resident trait Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 2 Resident trait 1 Resident trait 1 Resident trait 1 Resident trait 1 (a) (b) (c) (d) Observations: (a) is an ESS, (c) is a BP. In (b) and (d) both traits change direction of evolution, isoclines connect to the singularity. Since s u1,u 2 (v) = (u 1 v)(u 2 v)(j 0 (u 1, u 2 ) h.o.t.) when u 1 v u 2, both isoclines given locally by J 0 = 0 (i.e., are tangent). Barbara Boldin (University of Primorska) AD extension April / 29

44 The extended framework Unfolding of the isoclines (I) Dimorphic dynamics s u2 (u 1)=0 s u1 (u 2)=0 Resident trait 2 1isocline 2isocline Resident trait 1 (a) (i) (ii) (iii) (iv) (v) (b) Barbara Boldin (University of Primorska) AD extension April / 29

45 The extended framework Unfolding of the isoclines (I) Dimorphic dynamics s u2 (u 1)=0 s u1 (u 2)=0 Resident trait 2 1isocline 2isocline Resident trait 1 (a) (i) (ii) (iii) (iv) (v) (b) s u2 (u 1)=0 s u1 (u 2)=0 Resident trait 2 2isocline 1isocline Resident trait 1 (c) (i) (ii) (iii) (iv) (v) (d) Barbara Boldin (University of Primorska) AD extension April / 29

46 The extended framework Unfolding of the isoclines (II) Dimorphic dynamics s u1 (u 2)=0 s u2 (u 1)=0 Resident trait 2 1isocline 2isocline Resident trait 1 (a) (i) (ii) (iii) (iv) (v) (b) s u1 (u 2)=0 s u2 (u 1)=0 Resident trait 2 2isocline 1isocline Resident trait 1 (c) (i) (ii) (iii) (iv) (v) (d) Barbara Boldin (University of Primorska) AD extension April / 29

47 The extended framework Dimorphic dynamics Are both unfoldings possible? Yes! Example: superinfection model (I) Resident trait 2 Mutant trait Resident trait Virulence Virulence Resident trait Barbara Boldin (University of Primorska) # mutations 0 AD extension # mutations April / 29

48 The extended framework Dimorphic dynamics Are both unfoldings possible? Yes! Example: superinfection model (II) Resident trait 2 Mutant trait Resident trait Resident trait Virulence Virulence # mutations Barbara Boldin (University of Primorska) AD extension # mutations April / 29

49 The extended framework Dimorphic dynamics Branching or ESS behaviour nearby a onesided ESS? Both isoclines can be determined from s u1,u 2 (v) in the region u 1 v u 2. On this interval, s u1,u 2 (v) = (u 1 v)(u 2 v)k 0 (u 1, u 2, v), so 1isocline: (u 2 u 1 )K 0 (u 1, u 2, u 1 ) = 0 2isocline: (u 1 u 2 )K 0 (u 1, u 2, u 2 ) = 0. Barbara Boldin (University of Primorska) AD extension April / 29

50 The extended framework Dimorphic dynamics Branching or ESS behaviour nearby a onesided ESS? Both isoclines can be determined from s u1,u 2 (v) in the region u 1 v u 2. On this interval, s u1,u 2 (v) = (u 1 v)(u 2 v)k 0 (u 1, u 2, v), so 1isocline: (u 2 u 1 )K 0 (u 1, u 2, u 1 ) = 0 2isocline: (u 1 u 2 )K 0 (u 1, u 2, u 2 ) = 0. Ansatz K 0 (u 1, u 2, v) = J 0 (u 1, u 2 ) γ 1 u 1 γ 2 u 2 γ 3 v h.o.t. Barbara Boldin (University of Primorska) AD extension April / 29

51 The extended framework Dimorphic dynamics Branching or ESS behaviour nearby a onesided ESS? Both isoclines can be determined from s u1,u 2 (v) in the region u 1 v u 2. On this interval, s u1,u 2 (v) = (u 1 v)(u 2 v)k 0 (u 1, u 2, v), so 1isocline: (u 2 u 1 )K 0 (u 1, u 2, u 1 ) = 0 2isocline: (u 1 u 2 )K 0 (u 1, u 2, u 2 ) = 0. Ansatz K 0 (u 1, u 2, v) = J 0 (u 1, u 2 ) γ 1 u 1 γ 2 u 2 γ 3 v h.o.t. By using the formula for the signed curvature of an implicit curve (or by parametrization u 2 = u 1 t with t 0) we find that the order of the isoclines is determined by the sign of Conclusion λ := γ 3 (k 1 k 2 ). A one sided ESS acts as a BP when λ < 0 and as an ESS when λ > 0. Barbara Boldin (University of Primorska) AD extension April / 29

52 Conclusions Conclusion We extend the existing framework by considering AD of scalar traits in the case where selection gradient exists, but lack of additional smoothess of invasion fitness precludes classification in terms of 2 s and 2 s. x 2 y 2 Main findings: Invadability, convergence stability and (non)existence of nearby dimorphisms can be determined from three parameters: a 0, a 1 and a 1 In addition to ESS and invadable strategies, we observe onesided ESS Tangent isoclines emerge from a onesided ESS on the diagonal of the MIP; dichotomy of CSS behaviour and branching based on the precise way in which these isoclines unfold. If branching occurs, we may observe divergence or evolutionary pursuits (evolutionary arms races). Barbara Boldin (University of Primorska) AD extension April / 29

53 Question. How common is the type of limited smoothness considered here? Competitive asymmetries (superinfections, body size advantage in competition for resources, competition for suitable territories etc.) Interplay between morphology and feeding behaviour Rueffler, C., Van Dooren, T.J., Metz, J.A.: The interplay between behavior and morphology in the evolutionary dynamics of resource specialization. The American Naturalist 169(2), E34 E52 (2007) Barbara Boldin (University of Primorska) AD extension April / 29

54 Question. How common is the type of limited smoothness considered here? Competitive asymmetries (superinfections, body size advantage in competition for resources, competition for suitable territories etc.) Interplay between morphology and feeding behaviour Rueffler, C., Van Dooren, T.J., Metz, J.A.: The interplay between behavior and morphology in the evolutionary dynamics of resource specialization. The American Naturalist 169(2), E34 E52 (2007) Thank you for your attention! Barbara Boldin (University of Primorska) AD extension April / 29