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**Adaptive Dynamics studying the change**

of community dynamical parameters through mutation and selection Hans (= J A J *) Metz VEOLIA- Ecole Poly- technique & Mathematical Institute, Leiden University (formerly ADN) IIASA 1

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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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**components of the evolutionary mechanism**

ecology environment (causal) ( causal ) demography function trajectories physics form selection fitness trajectories development ( darwinian ) genome almost faithful reproduction

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**Stefan Geritz, me & various collaborators**

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

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**terminology trait vector (trait value) corresponding terms:**

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

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**adaptive dynamics limit**

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

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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**

mutant population size i.a. population sizes of other species resident population size

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**fitness as dominant transversal eigenvalue**

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

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**implications 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 = Eattr(C), C={X1,...,Xk) Residents have fitness zero.

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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 r e s i d n t a v l u ( ) x evolutionary time f c p : y , E m

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**underlying simplifications**

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 conceptuallly essential essential for most conclusions i.e., separated population dynamical and mutational time scales: the population dynamics relaxes before the next mutant comes

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**repeated substitution**

meso-evolution proceeds by the repeated substitution of novel mutations

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**fate of novel mutations**

C := {X1,..,Xk}: trait values of the residents Environment: Eattr(C) Y: trait value of mutant Fitness (rate of exponential growth in numbers) of mutant sC(Y) := (Y | Eattr(C)) • Y has a positive probability to invade into a C community iff sC(Y) > 0. • After invasion, Xi can be ousted by Y only if sX1,..,Y,.., Xk(Xi) ≤ 0. • For small mutational steps Y takes over, except near so-called “ess”es.

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**“For small mutational steps invasion implies substitution.”**

community dynamics: ousting the resident Proposition: Let e = | Y – X | be sufficiently small, and let X not be close to an “evolutionarily singular strategy”, or to a c(ommunity)-dynamical bifurcation point. Invasion of a "good" c-attractor of X leads to a substitution such that this c-attractor is inherited by Y Y and up to O(e2), sY(X) = – sX(Y). “For small mutational steps invasion implies substitution.”

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**community dynamics: ousting the resident**

Proof (sketch): When an equilibrium point or a limit cycle is invaded, the relative frequency p of Y satisfies = sX(Y) p(1-p) + O(e2), while the convergence of the dynamics of the total population densities occurs O(1). dp dt Singular strategies X* are defined by sX*(Y) = O(e2 ), instead of O(e).

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**community dynamics: the bifurcation structure**

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

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**community dynamics: the bifurcation structure**

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

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**community dynamics: invasion probabilities**

The probability that the mutant invades changes as depicted below: probability that mutant invades evolution will be towards increasing x evolution will be towards decreasing x 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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**Pairwise Invasibility Plot**

fitness contour plot x: resident y: potential mutant Pairwise Invasibility Plot PIP + - y x x1 x x x0 x1 x2 t r a i t v a l u e

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**Mutual Invasibility Plot**

Pairwise Invasibility Plot PIP Mutual Invasibility Plot MIP X 1 2 - ? + y X + - x1 x x protection boundary t r a i t v a l u e substitution boundary

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**Mutual Invasibility Plot**

Pairwise Invasibility Plot PIP Mutual Invasibility Plot MIP X 1 2 - + y x2 X + - x x t r a i t v a l u e

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**Pairwise Invasibility Plot**

Trait Evolution Plot Pairwise Invasibility Plot PIP Trait Evolution Plot TEP y x + - X 2 1 X 1 2 x2 x t r a i t v a l u e

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**evolutionarily singular strategies**

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definition

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**neutrality of resident**

(monomorphic) linearisation around y = x = x* c 1 + 2 = a=0 b +b =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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**Only directional derivatives (!)**

dimorphic linearisation around y = x1 = x2 = x* Only directional derivatives (!)

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**community dynamics: non-genericity strikes**

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**Only directional derivatives (!)**

dimorphic linearisation around y = x1 = x2 = x* Only directional derivatives (!) : u1=uw1, u2=uw2 s u ,u (v) = a + b 1 (w , w 2 ) + b v g 11 + g 10 uv 00 V h.o.t. ( *

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**dimorphic linearisation around y = x1 = x2 = x***

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**local dimorphic evolution**

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**local TEP classification**

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**more about adaptive branching**

population t i m e t r a i t fitness evolutionary time fitness minimum

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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 1 √2 width competition kernel carrying capacity**

–––––––––––– √2 competition kernel carrying capacity viable range

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**interrupted: branching prone ( trimorphically repelling)**

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**

1 2 3 ?

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**? a potential difficulty: heteroclinic loops**

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**

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. (Many partial results are available, e.g. Dercole & Rinaldi 2008.) (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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**subsequent levels of abstraction**

The different spaces that play a role in adaptive dynamics: the physical space inhabited by the organisms the state space of their i(ndividual)-dynamics the state space of their p(opulation)-dynamics the space of the influences that they undergo (fluctuations in light, temperature, food, enemies, conspecifics): their ‘environment’ 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 parameter spaces of families of adaptive dynamics

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**population dynamics: branching process results**

In an a priori given ergodic environment E : mutant populations starting from single individuals either go extinct, or "grow exponentially” (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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**adjacent purple volumes are mirror symmetric around a diagonal plane**

matryoshka galore polymorphisms are invariant under permutation of indices X3 the six purple volumes should be identified ! X2 X1 adjacent purple volumes are mirror symmetric around a diagonal plane

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matryoshka galore the sets of trimorphisms connect to the isoclines of the dimorphisms X3 X1 X2 (x2 = x3) (x2 = x1)

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