trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points

Trade-off & invasion plots, Trade-off & invasion plots, accelerating/decelerating costs and accelerating/decelerating costs and

evolutionary branching points.evolutionary branching points.

By Andy Hoyle & Roger Bowers.By Andy Hoyle & Roger Bowers.

(In collaboration with Andy White & Mike Boots.)(In collaboration with Andy White & Mike Boots.)

Outline of Talk.Outline of Talk.

Adaptive dynamics & TIPs:Adaptive dynamics & TIPs:– Evolution in the adaptive dynamics world,Evolution in the adaptive dynamics world,– Possible evolutionary outcomes,Possible evolutionary outcomes,– Trade-off and invasion plots,Trade-off and invasion plots,– Accelerating/decelerating costs.Accelerating/decelerating costs.

Examples of interactions:Examples of interactions:– Single species,Single species,– Competition,Competition,– Predator-prey,Predator-prey,– Host-parasite.Host-parasite.

The evolutionary cycle in adaptive The evolutionary cycle in adaptive dynamics.dynamics.

Resident Population (Resident Population (xx) existing at equilibrium.) existing at equilibrium.


Resident Population (Resident Population (xx) existing at equilibrium.) existing at equilibrium. Mutation in a few individuals ( Mutation in a few individuals ( y=xy=x±±εε ).).


Resident Population (Resident Population (xx) existing at equilibrium.) existing at equilibrium. Mutation in a few individuals ( Mutation in a few individuals ( y=xy=x±±εε ).). Fitness of Fitness of yy given by given by ssxx(y)(y),,

if if ssxx(y)<0 y(y)<0 y will die out. will die out.


Resident Population (Resident Population (xx) existing at equilibrium.) existing at equilibrium. Mutation in a few individuals ( Mutation in a few individuals ( y=xy=x±±εε ).). Fitness of Fitness of yy given by given by ssxx(y)(y),,

if if ssxx(y)<0 y(y)<0 y will die out. will die out.

if if ssxx(y)>0(y)>0 yy may invade may invade x.x. yy spreads becoming the new resident. spreads becoming the new resident.

Co-existence.Co-existence.

When When ssxx(y)>0(y)>0 AND AND ssyy(x)>0(x)>0……

Evolutionary outcomes.Evolutionary outcomes.

Attractor


Attractor Repellor


Attractor Repellor

Branching point

Where a TIP exists.Where a TIP exists.

Trade-off Trade-off f,f, yy11 vs. vs. yy22

(defines feasible (defines feasible strains).strains).


Trade-off Trade-off f,f, yy11 vs. vs. yy22 (defines feasible (defines feasible strains).strains).

Fixed strain Fixed strain xx on on ff..


Trade-off Trade-off f,f, yy11 vs. vs. yy22 (defines feasible (defines feasible strains)strains)

Fixed strain Fixed strain xx on on ff..

Axes of the TIP (strain Axes of the TIP (strain yy varies). varies).

The invasion boundaries.The invasion boundaries.

yy2 2 = f= f11(x,y(x,y11) ) ssxx(y)=0.(y)=0.


yy2 2 = f= f22(x,y(x,y11) ) ssyy(x)=0.(x)=0.


yy2 2 = f= f11(x,y(x,y11) ) ssxx(y)=0.(y)=0.

yy2 2 = f= f22(x,y(x,y11) ) ssyy(x)=0.(x)=0.

The singular TIP.The singular TIP.


Attractor – curvature of f is less than that of f1.


Repellor – curvature of f is greater than the mean curvature.


If sx(y)>0 and sy(x)>0, then branching points occur if curvature of f is between that of f1 and the mean curvature.

Accelerating/decelerating costs.Accelerating/decelerating costs.

Each improvement comes at an Each improvement comes at an ever…ever…



increasing cost – increasing cost – acceleratingly acceleratingly costly trade-offcostly trade-off..



decreasing cost – decreasing cost – deceleratingly deceleratingly costly trade-offcostly trade-off..



increasing cost – increasing cost – acceleratingly acceleratingly costly trade-offcostly trade-off..

decreasing cost – decreasing cost – deceleratingly deceleratingly costly trade-offcostly trade-off..

Applications of TIPs.Applications of TIPs.

Study a range of biological models.Study a range of biological models.

Primarily to investigate potential branching points.Primarily to investigate potential branching points.

Type, and magnitude, of costs necessary.Type, and magnitude, of costs necessary.

Single species – single stage.Single species – single stage.

Single species – single stage.Single species – single stage.

Fitness: sx(y)= -Asy(x) f1 = f2.

No possibility of branching points.

Single species - Maturation.Single species - Maturation.

Single species - Maturation.Single species - Maturation.

Carrying capacity tied to births q’= q’’=0

sx(y)= -Asy(x) f1 = f2 No branching points.

Carrying capacity tied to births q’= q’’=0

sx(y)= -Asy(x) f1 = f2 No branching points.

Carrying capacity tied to deaths q=0

No branching points.

Single Species - Maturation.Single Species - Maturation.

Competition.Competition.


Competition relation: czx=g(cxz).

Trade-off: r vs. c.


Competition relation: czx=g(cxz).

Trade-off: r vs. c.

Branching points iff g’(cxz)<0, with (gentle) deceleratingly costly trade-offs.

eg. red/grey squirrels czx=1/cxz

Predator-prey.Predator-prey.

Predator-prey.Predator-prey.

Branching points with (gentle) deceleratingly costly trade-offs.

Host-parasite – without recovery.Host-parasite – without recovery.

Trade-off – r vs. β

Host-parasite – with recovery.Host-parasite – with recovery.

Trade-offs

1) r vs. β

2) r vs. γ

3) r vs. α


1) r vs. β



2) r vs. γ

Branching points with (moderately) deceleratingly costly trade-offs.

Attractors with (gentle) deceleratingly costly trade-offs.


3) r vs. α

No possibility of branching points.

Conclusion.Conclusion.

Single Species – Single Species – – No branching points.No branching points.

Two Species + Single Class – Two Species + Single Class – – Branching points with (gentle) deceleratingly costly Branching points with (gentle) deceleratingly costly

trade-offs.trade-offs.

Two Species + Two Classes –Two Species + Two Classes –– Branching points and attractors with deceleratingly costly trade-Branching points and attractors with deceleratingly costly trade-

offs. offs.

trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points

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