The evolution of host resistance to microparasites Roger G. Bowers 1, Andrew Hoyle 1 & Michael Boots 2 1 Department of Mathematical Sciences, The University of Liverpool 2 Department of Animal and Plant Sciences, The University of Sheffield Andrew White – Heriot Watt University,

Contents Fitness and invasion boundaries –Reciprocal invasion plots Trade-offs –Acceleratingly/deceleratingly costly Trade-off and invasion plots (TIPs) The singular TIP The geometric characterisation of the singularity TIPs and simulations compared Host-pathogen TIPs – what dynamics, trade-off types and trade-off ‘strengths’ lead to –attractors, repellors and branching points (speciation) Discussion/Summary

Fitness and invasion boundaries Two strains labelled x = (x 1, x 2 ) and y = (y 1,y 2 ) Example A host species for a microparasite x 1’ y 1 rate of transmission of microparasite x 2, y 2 intrinsic growth rate of host Trade-off region y 1 < x 1 & y 2 < x 2 A gain in resistance to a parasite is bought at a cost in intrinsic growth rate Fitness – per capita growth rate of strain s x (y) fitness of invader y with resident x s y (x) fitness of invader x with resident y

y 2 = f 1 (x,y 1 )  s x (y)=0. Fitness and invasion boundaries

y 2 = f 2 (x,y 1 )  s y (x)=0. Fitness and invasion boundaries

y 2 = f 1 (x,y 1 )  s x (y)=0. y 2 = f 2 (x,y 1 )  s y (x)=0. Reciprocal Invasion Plots

Trade-offs and cost types y 2 = f(y 1 ), x 2 = f(x 1 ) The trade-off represents the cost (of resistance) and defines the set of feasible strains f′ is of fixed sign

Each improvement comes at an ever… increasing cost – acceleratingly costly trade- off. Trade-offs: accelerating/deceleratingly costly

Each improvement comes at an ever… decreasing cost – deceleratingly costly trade- off. Trade-offs: accelerating/deceleratingly costly

Each improvement comes at an ever… increasing cost – acceleratingly costly trade- off. decreasing cost – deceleratingly costly trade- off. Trade-offs: accelerating/deceleratingly costly

Trade-off and invasion plots (TIPs)

The singular TIP

The singular behaviour depends on the relative curvatures of the trade-off and invasion boundaries

Attractor – curvature of f is less than that of f 1. The geometric characterisation of the singularity

Repellor – curvature of f is greater than the mean curvature. The geometric characterisation of the singularity

Branching points – curvature of f is between that of f 1 and the mean curvature (and s x (y)>0 and s y (x)>0 there). The geometric characterisation of the singularity

TIPs and simulations compared

An applications of TIPs Study a range of host-microparasite models What type and ‘strength’ of trade-off/cost is associated with –Attractors –Repellors –Branching points (these particularly since they may indicate speciation)

Host-parasite system – without recovery. Trade-off – r vs. β (avoidance) Attractors … acceleratingly costly trade-offs Branching points…weakly deceleratingly costly trade-offs Repellors…strongly deceleratingly costly trade-offs

Calculation of fitness Dynamics of rare mutant Fitness Growth rates Residence times So … where the equilibrium resident densities are known in terms of the resident parameters

Host-parasite system – with recovery Trade-off – 1) r vs. β (avoidance) Attractors … acceleratingly costly trade-offs Branching points…weakly deceleratingly costly trade-offs Repellors…strongly deceleratingly costly trade-offs

Host-parasite system – with recovery Trade-off – 2) r vs. γ (recovery) Attractors … acceleratingly and weakly deceleratingly costly trade-offs Branching points…moderately deceleratingly costly trade-offs Repellors…strongly deceleratingly costly trade-offs

Host-parasite system – with recovery Trade-off - 3) r vs. α (tolerance) Attractors … strongly acceleratingly costly trade-offs Garden of Eden repellors…moderately acceleratingly trade-offs Repellors…weakly acceleratingly and all deceleratingly costly trade-offs

Host-parasite system – with reproduction from infecteds PreviouslyNow Branching points can occur with weakly acceleratingly costly trade-offs.

Discussion/Summary Trade-off and invasion plots allow a convenient geometric characterisation of evolutionary behaviour Host-pathogen systems with a single reproductive class –Branching points with intermediately deceleratingly costly trade-offs Host-pathogen systems with a multiple reproductive classes – Branching points can occur with weakly acceleratingly costly trade-offs.