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Trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points. By Andy Hoyle & Roger Bowers. (In collaboration with Andy White & Mike Boots.)
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Outline of Talk. Adaptive dynamics & TIPs: –Evolution in the adaptive dynamics world, –Possible evolutionary outcomes, –Trade-off and invasion plots, –Accelerating/decelerating costs. Examples of interactions: –Single species, –Competition, –Predator-prey, –Host-parasite.
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The evolutionary cycle in adaptive dynamics. Resident Population ( x ) existing at equilibrium.
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The evolutionary cycle in adaptive dynamics. Resident Population ( x ) existing at equilibrium. Mutation in a few individuals ( y=x±ε ).
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The evolutionary cycle in adaptive dynamics. Resident Population ( x ) existing at equilibrium. Mutation in a few individuals ( y=x±ε ). Fitness of y given by s x (y), if s x (y)<0 y will die out.
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The evolutionary cycle in adaptive dynamics. Resident Population ( x ) existing at equilibrium. Mutation in a few individuals ( y=x±ε ). Fitness of y given by s x (y), if s x (y)<0 y will die out. if s x (y)>0 y may invade x. y spreads becoming the new resident.
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Co-existence. When s x (y)>0 AND s y (x)>0 …
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Evolutionary outcomes. Attractor
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Evolutionary outcomes. AttractorRepellor
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Evolutionary outcomes. AttractorRepellor Branching point
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Where a TIP exists. Trade-off f, y 1 vs. y 2 (defines feasible strains).
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Where a TIP exists. Trade-off f, y 1 vs. y 2 (defines feasible strains). Fixed strain x on f.
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Where a TIP exists. Trade-off f, y 1 vs. y 2 (defines feasible strains) Fixed strain x on f. Axes of the TIP (strain y varies).
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The invasion boundaries. y 2 = f 1 (x,y 1 ) s x (y)=0.
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The invasion boundaries. y 2 = f 2 (x,y 1 ) s y (x)=0.
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The invasion boundaries. y 2 = f 1 (x,y 1 ) s x (y)=0. y 2 = f 2 (x,y 1 ) s y (x)=0.
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The singular TIP.
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Attractor – curvature of f is less than that of f 1.
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The singular TIP. Repellor – curvature of f is greater than the mean curvature.
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The singular TIP. If s x (y)>0 and s y (x)>0, then branching points occur if curvature of f is between that of f 1 and the mean curvature.
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Accelerating/decelerating costs. Each improvement comes at an ever…
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Accelerating/decelerating costs. Each improvement comes at an ever… increasing cost – acceleratingly costly trade-off.
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Accelerating/decelerating costs. Each improvement comes at an ever… decreasing cost – deceleratingly costly trade-off.
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Accelerating/decelerating costs. Each improvement comes at an ever… increasing cost – acceleratingly costly trade-off. decreasing cost – deceleratingly costly trade-off.
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Applications of TIPs. Study a range of biological models. Primarily to investigate potential branching points. Type, and magnitude, of costs necessary.
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Single species – single stage.
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Fitness: s x (y)= -As y (x) f 1 = f 2. No possibility of branching points.
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Single species - Maturation.
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Carrying capacity tied to births q’= q’’=0 s x (y)= -As y (x) f 1 = f 2 No branching points.
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Carrying capacity tied to births q’= q’’=0 s x (y)= -As y (x) f 1 = f 2 No branching points. Carrying capacity tied to deaths q=0 No branching points. Single Species - Maturation.
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Competition.
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Competition. Competition relation: c zx =g(c xz ). Trade-off: r vs. c.
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Competition. Competition relation: c zx =g(c xz ). Trade-off: r vs. c. Branching points iff g’(c xz )<0, with (gentle) deceleratingly costly trade-offs. eg. red/grey squirrels c zx =1/c xz
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Predator-prey.
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Predator-prey. Branching points with (gentle) deceleratingly costly trade-offs.
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Host-parasite – without recovery. Trade-off – r vs. β
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Host-parasite – without recovery. Trade-off – r vs. β Branching points with (gentle) deceleratingly costly trade-offs.
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Host-parasite – with recovery. Trade-offs 1) r vs. β 2) r vs. γ 3) r vs. α
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Host-parasite – with recovery. 1) r vs. β Branching points with (gentle) deceleratingly costly trade-offs.
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Host-parasite – with recovery. 2) r vs. γ Branching points with (moderately) deceleratingly costly trade-offs. Attractors with (gentle) deceleratingly costly trade-offs.
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Host-parasite – with recovery. 3) r vs. α No possibility of branching points.
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Conclusion. Single Species – –No branching points. Two Species + Single Class – –Branching points with (gentle) deceleratingly costly trade-offs. Two Species + Two Classes – – Branching points and attractors with deceleratingly costly trade- offs.
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