analysis of multi-species ecological and evolutionary dynamics ecole normale supérieure, paris...
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ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS
Ecole Normale Supérieure, ParisDecember 9-13, 2013
Adaptive Dynamics (AD) and its canonical equation (F. Dercole)Introduction to evolutionary dynamics with examples within and beyond biology. Modeling approaches to evolutionary dynamics. The AD approach through a representative example: the evolution driven by the competition for resources.The AD canonical equation.Further readingsAnalysis of Evolutionary Processes, Princeton Univ. Press, 2008, Chaps. 1-3 and Appx. B, CTechnovation (2008) 28:335-348 J. Theor. Biol. (1999) 197:149-162
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A naive introduction to innovation and competition processesInnovations and competition
Evolution
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stationary non-stationary(Red Queen Dynamics)
multiple
Evolutionary attractors
Evolutionary branching Evolutionary extinction
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Evolution in biology
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Evolution outside biology
(see f.r. 1 Chap. 1)
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Modeling approaches to innovation and competition processes
(see f.r. 1 Chap. 2)
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Each individual is characterized by 0, 1, or more inheritable traits (phenotypes/strategies)
Adaptive Dynamics – Basic assumptions
Reproduction is clonal (asexual)thus offspring are either characterized by the trait of the parent or are mutants
Mutations in different traits of the same individual are independent
Mutations are rare on the ecological time scale
Mutations are small
The coexistence of populations is stationary
The (abiotic) environment is isolated, uniform, and invariant
Traits are quantitative characteristicsdescribed as continuous variables (symbol ), possibly through a scaling
See f.r. 1. See also the original contributionsMetz et al. (in Stochastic and Spatial Structures of Dynamical Systems, Elsevier 1999)Geritz et al. (Phys. Rev. Lett. 78, 2024-2027, 1997; Evol. Ecol. 12, 35-57, 1998)Dieckmann & Law (J. Math. Biol. 34, 579-612, 1996)
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The AD working scheme
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The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The resident (ecological) equilibrium
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The AD working scheme
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The carrying capacity
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The AD working scheme
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The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The competition function
The resident-mutant model
The resident (ecological) equilibrium
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The competition function
The carrying capacity
symmetric competition asymmetric competition
The model parameters :
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The AD canonical equation through a simple exampleQuestion: Does the competition for resources optimize a morphological phenotype,e.g. body size, or promote genetic diversity? (see f.r. 2 and 3)
Let’s start with a single resident population: the resident model is the logistic one!
The competition function
The resident-mutant model
The resident (ecological) equilibrium
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The AD working scheme
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The mutant invasion fitness
The pairwise invasibility plot: the sign of the fitness
It is the initial per-capita rate of growth of the mutant population
moreover, invasion implies substitution(see f.r. 1 Appx. B)
The selection derivative :
We expect to to have the same sign of
Technically, it is the eigenvalue determining invasion
and have opposite sign
if then
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where is the probability of a mutation at birth, is the standard deviation of mutations,and in the limit of extremely rare and small mutations
The AD canonical equation
(see f.r. 1 Chap. 3 and Appx. C)
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The AD working scheme
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where is the probability of a mutation at birth, is the standard deviation of mutations,and in the limit of extremely rare and small mutations
The AD canonical equation
The evolutionary equilibrium
such that . . It results
Stability via linearization
eigenvalue
is stable for all parameter settings, so there are no bifurcations
(see f.r. 1 Chap. 3 and Appx. C)
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At , , so that invasion does not necessarily implysubstitution. Can residents and mutants coexist and undergoevolutionary branching?
?
And what if we have a large mutation (or, most likely, the introduction of an alien species)?The resident-mutant model (or a suitable resident model) gives the resulting ecological attractor(an equilibrium?) for which we can derive the corresponding canonical equation