Trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points. By Andy Hoyle & Roger Bowers. (In collaboration with Andy White & Mike Boots.)

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.

The evolutionary cycle in adaptive dynamics.  Resident Population ( x ) existing at equilibrium.

The evolutionary cycle in adaptive dynamics.  Resident Population ( x ) existing at equilibrium.  Mutation in a few individuals ( y=x±ε ).

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.

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.

Co-existence.  When s x (y)>0 AND s y (x)>0 …

Evolutionary outcomes. Attractor

Evolutionary outcomes. AttractorRepellor

Evolutionary outcomes. AttractorRepellor Branching point

Where a TIP exists.  Trade-off f, y 1 vs. y 2 (defines feasible strains).

Where a TIP exists.  Trade-off f, y 1 vs. y 2 (defines feasible strains).  Fixed strain x on f.

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).

The invasion boundaries.  y 2 = f 1 (x,y 1 )  s x (y)=0.

The invasion boundaries.  y 2 = f 2 (x,y 1 )  s y (x)=0.

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.

The singular TIP.

Attractor – curvature of f is less than that of f 1.

The singular TIP. Repellor – curvature of f is greater than the mean curvature.

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.

Accelerating/decelerating costs.  Each improvement comes at an ever…

Accelerating/decelerating costs.  Each improvement comes at an ever…  increasing cost – acceleratingly costly trade-off.

Accelerating/decelerating costs.  Each improvement comes at an ever…  decreasing cost – deceleratingly costly trade-off.

Accelerating/decelerating costs.  Each improvement comes at an ever…  increasing cost – acceleratingly costly trade-off.  decreasing cost – deceleratingly costly trade-off.

Applications of TIPs.  Study a range of biological models.  Primarily to investigate potential branching points.  Type, and magnitude, of costs necessary.

Single species – single stage.

Fitness: s x (y)= -As y (x)  f 1 = f 2. No possibility of branching points.

Single species - Maturation.

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 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.

Competition.

Competition. Competition relation: c zx =g(c xz ). Trade-off: r vs. c.

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

Predator-prey.

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

Host-parasite – without recovery. Trade-off – r vs. β

Host-parasite – without recovery. Trade-off – r vs. β Branching points with (gentle) deceleratingly costly trade-offs.

Host-parasite – with recovery. Trade-offs 1) r vs. β 2) r vs. γ 3) r vs. α

Host-parasite – with recovery. 1) r vs. β Branching points with (gentle) deceleratingly costly trade-offs.

Host-parasite – with recovery. 2) r vs. γ Branching points with (moderately) deceleratingly costly trade-offs. Attractors with (gentle) deceleratingly costly trade-offs.

Host-parasite – with recovery. 3) r vs. α No possibility of branching points.

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.