Transcript Evolution models

Evolution models: competition
Lectures I-II
George Kampis
ETSU 2007 Spring
Lecture I: Verhulst and LV
Evolution
• Darwin: natural selection = differential survival/repr.
• Selection (competition) dynamics
• GA
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Malthus: population growth
Verhulst growth equation
Lotka-Volterra competition equations
Hypercompetition
• Coming together: selection, competition, dynamics, complex systems
modeling
Dynamical Systems
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dx/dt = f(x,t)
e.g. „simplest”:
equation
– dx/dt = a
– dx/dt = ax
– dx/dt = ax2
function
consant
linear
quadratic
growth
linear
exponential
hyperbolic
Real systems
„trees don’t grow to the sky”
sheep in Australia (1840-1930)
Selection is..
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Differential reproduction
Not all offspring can survive …
… or reproduce
Derived concept: fitness
This together with mutation and crossover
defines classic evolution (and GA)
Verhulst equation
• dx/dt = rx , r = „Malthusian parameter”
• dx/dt = rx (K-x)/K , K = limit
intraspecies competition (density dep.)
„logistic equation”, generates S-shape
see inset
• Effect of r and K?
see inset, interactive simulation
Lotka-Volterra competion
• dx/dt = r1 (K-y-x)/K x
• dy/dt = r2 (K-x-y)/K y
intraspecies competition (for K)
with interspecies coupling
• interspecies competition coefficient (how much room occupied)..
• dx/dt = r1 (K-ay-x) x
• dx/dt = r2 (K-bx-y) y
Phase plot
Phase plot concepts
• Trajectories, attractors, stability – fixed points, limit cyles, etc.
LV competition
Lotka-Volterra comp. (reminder)
• dx/dt = r1 x (K-y-x)/K
• dy/dt = r2 y (K-x-y)/K
intraspecies competition (for the K)
with interspecies coupling
• interspecies competition coefficient (how much room occupied)..
• dx/dt = r1 x (K-ay-x)/K
• dx/dt = r2 y (K-bx-y)/K
• Effect of various K-s for populations, e.g. efficiency (small vs. big)
• dx/dt = r1 x (K1-ay-x)/K1
• dx/dt = r2 y (K2-bx-y)/K2
LV competition, analysis
• Isoclyne, vector field: see inset,
• Analysis: see inset
Zero isoclynes
• 0 = r1 x (K1-ay-x)/K1;
• 0 = r2 y (K2-bx-y)/K2;
ay = K1-x line;
bx = K2-y line;
x=0 at y=K1/a, y=0 at x=K1
x=0 at y=K2, y=0 at x=K2/b
• Now suppose again K1=K2 for simplicity
• Relation of K, N = x + y, and a,b at the attractor (where growth is zero)
• For a=b=1
• For a,b general, either K = ay + x or K = bx + y
• if x = 0 then y =K (or if y=0 then x=K)
N = K (ie. x = K-y)
N = K (ie. x = K-y)
• BUT at coexistence of x and y: K = ay + x = bx + y (where lines cross),
• y/x = (b-1)/(a-1), x = K (a-1)/(ab-1), y = K (b-1)/(ab-1)
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N= K (a+b-2)/(ab-1)
Cexistence cont’d
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If the x and y population coexist, this can be where the lines cross, which in turn can be
anywhere on the plane (bw 0,0 and K,K), depending on the values of a and b. So N =
x+ y can be anything bw 0 and 2K.
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The geometric meaning of N: the coinciding "natural" K lines at 45 degrees mark the
"baseline" case N=K. For a,b >1 their cross point moves downwards, for a,b < 1
upwards, pushing N=x+y above or below the baseline, i.e. N above (or below) K.
Examples (interactive class)
Some are counterintuitive; here sp2 is smaller but faster, and seems to take over, yet…
Which illustrates the fact (to be disdcussed) that for a,b>1 the fixed point is highly unstable
and difficult to reach – here we are „close” to it -> density dependence can help, see there
Coexistence vs selection in LVc
• Coexistence at common points of K lines
• Because these are the only points, where dx/dt = 0 AND dy/dt =0
• So unless the two lines coincide (!),
• coexistence is at the cross point, if there is one (!)
• Role of (x0, y0) and (r1,r2): determine if this point on the phase
plane is actually accessible
• For a,b < 1 there is cross, and always accessible
• For a,b > 1 there is cross, difficult to access
• Sensitive switching behavior (test interactively)
Lessons from LV competition
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the „big trick” is with the K lines.
if they are different, there can be states where species1 cannot grow
but species2 still has growth reserves (or the way around).
so one species will grow and take over the other, no matter what.
competition coeff. has similar effects (in reality perhaps K and a,b
are related) so there would be one single parameter (eg efficiency)
• Malthusian parameter has (often) no effect!
• Initial conditions have (often) no effect!
• Many, perhaps most (realistic) combinations yield coexistence!!
Lecture II: Hyperbolic growth
Can this really happen?
A small fluctuation or any advantage will give monopoly. Density matters!
Hyperbolic competition
• Hypercycle
• (Eigen 1971)
Explanation of terms!
Hyperbolic dynamics
Spatial effects of hyperbolic
dynamics
• Tree growth, crystal growth, etc.
Voronoi polygons
A polygon whose interior consists of all points in the plane which
are closer to a particular lattice point than to any other. The
generalization to
dimensions is called a Dirichlet region,
Thiessen polytope, or Voronoi cell.
Voronoi, nonlin. growth examples
Voronoi, different examples (!)
Descartes 1644
Hay, beecomb… etc.
http://www.igg.tu-berlin.de/fileadmin/Daten_FMG/GeoTech/geoinfo_technology_lect11.ppt
Applications
• Biology, Ecology, Forestry -- Model and
analyze plant competition ("Area
potentially available to a tree", "Plant
polygons")
• Cartography -- Piece together satellite
photographs into large "mosaic" maps
• Crystallography and Chemistry -- Study
chemical properties of metallic sodium
("Wigner-Seitz regions");
The temporal dynamics of
nonlinear growth
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Once-and for all selection
Frozen accidents (QWERTY phenomenon)
Path dependence
„first come first served”, „winner takes all”
„founder effect” etc.
Path dependence
• Not (!) just state dependence (i.e. the past
matters)
• But order of events, especially initial
events have an amplified, persistent effect
• Path = order of events
• Testable by
– Counterfactuals
– Random models
Pólya urn model
A special case of „balls-and-bins”
http://www-stat.stanford.edu/%7Esusan/surprise/Polya.html
Important: urns, balls, restaurants etc. are all equivalent in this model! And yet…
Fancy words to summarize
• Unpredictability: whatever leads to whatever else,
fluctuations drive outcomes
• Inflexibility: the deeper into, the more rigid
• Nonergodicity: small changes dont cancel out each other
– Def or ergodicity: temporal and statistical averages coincide
• Potential inefficiency: better candidates cant take over
after an initial period
– Similar to ESS, evolutionary stable strategy
Hyperbolic selection
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Brian Arthur, Michael Mitzenmacher: „law of increasing returns”
Theory of monopoly
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But: it is not the growth phase that selects! In that phase everybody grows as can
Limited organization (market) effects cause selection as in Verhulst and LV
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Thus, to achieve selection (ie extinction) we need the minus terms too…
…which will eliminate the less abundant that produces the more loss
Note that the losses are distributed proportionally in this model
So, „proportional is not proportional”
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Unless loss is quadratic too, higher numbers win
A small advantage becomes a big advantage
Hyperbolic (non)selection
Conclusions
• Evolutionary dynamics is often counterintuitive
• Selection may be difficult to achieve when you think
• Outcome usually depends on factors other than Malthusian
Malthusian parameter is a poor predictor of success
Intial population value is a good predictor of success
• Question1: [how] can these factors be grasped in an agent based
model (ie. is there a reality behind the equations?)
• Question2: if any of the above is true [under realistic conditions] then
how can evolution happen, i.e. how can small new populations win
over?