Evolution models: competitionLectures I-II

George Kampis

ETSU 2007 Spring

Lecture I: Verhulst and LV

Evolution• Darwin: natural selection = differential survival/repr.• Selection (competition) dynamics• GA

• Malthus: population growth• Verhulst growth equation • Lotka-Volterra competition equations• Hypercompetition

• Coming together: selection, competition, dynamics, complex systems modeling

Dynamical Systems

• dx/dt = f(x,t)

• e.g. „simplest”:

equation function growth

– dx/dt = a consant linear

– dx/dt = ax linearexponential

– dx/dt = ax2 quadratic hyperbolic

Real systems

„trees don’t grow to the sky” sheep in Australia (1840-1930)

Selection is..

• 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       intraspecies competition (for K)• dy/dt = r2 (K-x-y)/K y      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        intraspecies competition (for the K)• dy/dt = r2 y (K-x-y)/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; ay = K1-x line; x=0 at y=K1/a, y=0 at x=K1• 0 = r2 y (K2-bx-y)/K2; bx = K2-y line; 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 N = K (ie. x = K-y)• 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)

• 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) • N= K (a+b-2)/(ab-1)

Cexistence cont’d• 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.

 

• 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

• 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

• 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

http://www-stat.stanford.edu/%7Esusan/surprise/Polya.html

Important: urns, balls, restaurants etc. are all equivalent in this model! And yet…

A special case of „balls-and-bins”

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

• Brian Arthur, Michael Mitzenmacher: „law of increasing returns”

• Theory of monopoly

• 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

• 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”

• 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?