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Adaptive Dynamics studying the change of community dynamical parameters through mutation and selection Hans (= J A J * ) Metz (formerly ADN ) IIASA VEOLIA- Ecole Poly- technique & Mathematical Institute, Leiden University

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context

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evolutionary scales micro-evolution : changes in gene frequencies on a population dynamical time scale, meso-evolution : evolutionary changes in the values of traits of representative individuals and concomitant patterns of taxonomic diversification (as result of multiple mutant substitutions), macro-evolution : changes, like anatomical innovations, that cannot be described in terms of a fixed set of traits. Goal: get a mathematical grip on meso-evolution.

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function trajectories form trajectories genome development selection (darwinian) (causal) demography physics almost faithful reproduction ecology (causal) fitness environment components of the evolutionary mechanism

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fitness function trajectories form trajectories genome development selection (darwinian) (causal) physics almost faithful reproduction ecology (causal) environment adaptive dynamics demography Stefan Geritz, me & various collaborators (1992, 1996, 1998,...)

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(pheno) type morph strategy trait vector ( trait value) point (in trait space) terminology corresponding terms: population genetics: evolutionary ecology: (meso-evolutionary statics) adaptive dynamics: (meso-evolutionary dynamics)

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(usual perspective) adaptive dynamics limit , ln( ) rescale time, only consider traits rescale numbers to densities = system size, = mutations / birth t

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from individual dynamics through community dynamics to adaptive dynamics (AD)

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community dynamics: residents Populations are represented as frequency distributions (measures) over a space of i(ndividual)-states (e.g. spanned by age and size). Environments ( E ) are delimited such that given their environment individuals are independent, and hence their mean numbers have linear dynamics. Resident populations are assumed to be so large that we can approximate their dynamics deterministically. These resident populations influence the environment so that they do not grow out of bounds. Therefore the community dynamics have attractors, which are assumed to produce ergodic environments.

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community dynamics: mutants Mutations are rare. They enter the population singly. Hence, initially their impact on the environment can be neglected. The initial growth of a mutant population can be approximated with a branching process. Invasion fitness is the (generalised) Malthusian parameter (= averaged long term exponential growth rate of the mean) of this proces: (Existence guaranteed by the multiplicative ergodic theorem.)

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fitness as dominant transversal eigenvalue resident population size i.a. population sizes mutant population size of other species

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resident population size i.a. population sizes mutant population size or, more generally, dominant transversal Lyapunov exponent fitness as dominant transversal eigenvalue of other species

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Fitnesses are not given quantities, but depend on (1) the traits of the individuals, X, Y, (2) the environment in which they live, E : (Y,E) | (Y | E) with E set by the resident community : E = E attr (C), C={X 1,...,X k ) Residents have fitness zero. implications

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fitness landscape perspective Evolution proceeds through uphill movements in a fitness landscape that keeps changing so as to keep the fitness of the resident types at exactly zero. Evolution proceeds through uphill movements in a fitness landscape resident trait value(s) x evolutionary time fitness landscape: (y,E(t)) mutant trait value y 0 0

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underlying simplifications i.e., separated population dynamical and mutational time scales: the population dynamics relaxes before the next mutant comes 1. mutation limited evolution 2. clonal reproduction 3. good local mixing 4. largish system sizes 5. “good” c (ommunity) -attractors 6. interior c-attractors unique 7. fitness smooth in traits 8. small mutational steps essential for most conclusions essential conceptuallly

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meso-evolution proceeds by the repeated substitution of novel mutations

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fate of novel mutations C : = {X 1,..,X k } : trait values of the residents E nvironment: E attr (C) Y : trait value of mutant Fitness (rate of exponential growth in numbers) of mutant s C (Y) : = ( Y | E attr (C)) Y has a positive probability to invade into a C community iff s C (Y) > 0. After invasion, X i can be ousted by Y only if s X 1,.., Y,.., X k ( X i ) ≤ 0. For small mutational steps Y takes over, except near so-called “ess”es.

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Invasion of a "good" c-attractor of X leads to a substitution such that this c-attractor is inherited by Y community dynamics: ousting the resident Proposition: Let = | Y – X | be sufficiently small, and let X not be close to an “evolutionarily singular strategy”, or to a c(ommunity)-dynamical bifurcation point. “ For small mutational steps invasion implies substitution.” Y and up to O( 2 ), s Y (X) = – s X (Y).

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community dynamics: ousting the resident When an equilibrium point or a limit cycle is invaded, the relative frequency p of Y satisfies = s X (Y) p(1-p) + O( 2 ), while the convergence of the dynamics of the total population densities occurs O(1). dp dt Singular strategies X* are defined by s X* (Y) = O ( ), instead of O( ). Proof (sketch):

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Near where the mutant trait value y equals the resident trait value x there is a degenerate transcritical bifurcation: community dynamics: the bifurcation structure

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evolution will be towards increasing x evolution will be towards decreasing x Near where the mutant trait value y equals the resident trait value x there is a degenerate transcritical bifurcation: The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase relative frequency of mutant mutant trait value y

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community dynamics: invasion probabilities probability that mutant invades evolution will be towards increasing x evolution will be towards decreasing x The probability that the mutant invades changes as depicted below: The effective speeds of evolutionary change are proportional to the probabilities that invading mutants survive the initial stochastic phase

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graphical tools

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+ + - y x - fitness contour plot x: resident y: potential mutant Pairwise Invasibility Plot trait value x x0x0 x1x1 x1x1 x2x2 x PIP

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X X 1 2 Mutual Invasibility Plot MIP y x trait value X x Mutual Invasibility Plot Pairwise Invasibility Plot PIP x1x1 protection boundary ? ? substitution boundary

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X X 1 2 Mutual Invasibility Plot MIP y x trait value X x Mutual Invasibility Plot Pairwise Invasibility Plot PIP x2x2

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X X 1 2 trait value x Trait Evolution Plot TEP x2x2 Trait Evolution Plot y x Pairwise Invasibility Plot PIP

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evolutionarily singular strategies

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definition

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(monomorphic) linearisation around y = x = x* c 11 +2c 10 +c 00 =0 a=0 b 1 +b 0 =0 neutrality of resident s u (u)= 0

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local PIP classification

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the associated local MIPs

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dimorphic linearisation around y = x 1 = x 2 = x* Only directional derivatives (!)

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community dynamics: non-genericity strikes

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dimorphic linearisation around y = x 1 = x 2 = x* s u,u (v) = + 1 (w 1, w 2 u 0 v 11 (w 1, w 2 u 2 10 (w 1, w 2 uv 00 V 2 h.o.t. 12 ( * ) Only directional derivatives (!) : u 1 =uw 1, u 2 =uw 2

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dimorphic linearisation around y = x 1 = x 2 = x*

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local dimorphic evolution

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local TEP classification

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more about adaptive branching evolutionary time t i m e t r a i t fitness minimum population

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beyond clonality: thwarting the Mendelian mixer assortativeness

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extensions

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a toy example Lotka-Volterra all per capita growth rates are linear functions of the population densities Lotka-Volterra all per capita growth rates are linear functions of the population densities LV models are unrealistic, but useful since they have explicit expressions for the invasion fitnesses.

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a toy example viable range competition kernel carrying capacity width 1 –––––––––––– √2

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matryoshka galore isoclines correspond to loci of monomorphic singular points. interrupted : branching prone ( trimorphically repelling)

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of two lines about to merge one goes extinct

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more consistency conditions There also exist various global consistency relations. Use that on the boundaries of the coexistence set one type is extinct.

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a more realistic example

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a potential difficulty: heteroclinic loops ?

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? The larger the number of types, the larger the fraction of heteroclinic loops among the possible attractor structures !

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things that remain to be done (Many partial results are available, e.g. Dercole & Rinaldi 2008.) Analyse how to deal with the heteroclinic loop problem. Classify the geometries of the fitness landscapes, and coexistence sets near singular points in higher dimensions. Extend the collection of known global geometrical results. Develop a fullfledged bifurcation theory for AD. Develop analogous theories for less than fully smooth s-functions. Delineate to what extent, and in which manner, AD results stay intact for Mendelian populations. (Some recent results by Odo Diekmann and Barbara Boldin.) (Some results in next lecture.)

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The end Stefan Geritz Ulf Dieckmann in next lecture:

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The different spaces that play a role in adaptive dynamics: the trait space in which their evolution takes place ( = parameter space of their i- and therefore of their p-dynamics ) = the ‘state space’ of their adaptive dynamics the physical space inhabited by the organisms the state space of their i(ndividual)-dynamics the space of the influences that they undergo ( fluctuations in light, temperature, food, enemies, conspecifics ): their ‘environment’ the parameter spaces of families of adaptive dynamics the state space of their p(opulation)-dynamics subsequent levels of abstraction

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population dynamics: branching process results or "grow exponentially” either go extinct, mutant populations starting from single individuals In an a priori given ergodic environment E : (with a probability that to first order in | Y – X | is proportional to ( ( E, Y)) +, and with ( E, Y) as rate parameter).

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matryoshka galore polymorphisms are invariant under permutation of indices X2X2 the six purple volumes should be identified ! adjacent purple volumes are mirror symmetric around a diagonal plane X1X1 X3X3

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matryoshka galore the sets of trimorphisms connect to the isoclines of the dimorphisms X1X1 X2X2 X3X3 ( x 2 = x 3 )( x 2 = x 1 )

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