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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS. 10. Branching scenarios in prey -predator communities (P. Landi)
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ANALYSIS OF MULTI-SPECIES ECOLOGICAL AND EVOLUTIONARY DYNAMICS 10. Branching scenarios in prey-predator communities (P. Landi) Overview of branching scenarios in coevolvingcommunities with differenttypes of ecologicalinteractions. Classification of branching scenarios. Full catalog of branching scenarios in prey-predator coevolvingcommunitiesthrough the numericalcontinuationof branching bifurcations. Furtherreading SIAM J. Appl. Math (2013) 73:1634-1658 Ecole Normale Supérieure, Paris December 9-13, 2013
BRANCHING SCENARIOS No Branching 1 Single 2 1 Branching Unilateral 2 Alternate Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate
BRANCHING SCENARIOS No Branching (B) 1 Single 2 1 Branching Unilateral 2 Alternate Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate
BRANCHING SCENARIOS No Branching (B) 1 (A) Single 2 1 Branching Unilateral 2 Alternate Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate
BRANCHING SCENARIOS No Branching (B) 1 (A) Single 2 1 (B) Branching Unilateral 2 (B) Alternate Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate
Alternate Branching: Doebeli, Dieckmann (2000) (prey-predator) 1 2
Alternate Branching: Ferrièreet al. (2002) (mutualism) 11 9 7 5 3 1 12 10 8 6 4 2
Alternate Branching: Best et al. (2010) (host-parasite) 7 6 5 4 3 2 1
BRANCHING SCENARIOS No Branching (B) 1 (A) Single 2 1 (B) Branching Unilateral 2 (B) Alternate (A,B,D) Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate
BRANCHING SCENARIOS No Branching (B) 1 (A) Single 2 1 (B) Branching Unilateral 2 (B) Alternate (A,B,D) Multiple A Doebeli, Dieckmann, AmNat 2000 B Ferrièreet al., PRSB 2002 C Ito, Ikegami, JTB 2006 D Best et al.,AmNat 2010 Bilateral Non Alternate (C)
No BR BR p2 p1
No BR S BR 1 S BR 2 Multiple BR p2 p1
No BR S BR 1 S BR 2 U BR 1 U BR 2 Bilateral BR p2 p1
No BR S BR 1 S BR 2 U BR 1 U BR 2 Alt BR Alt BR Non Alternate BR p2 p1
B A B B A,B,D C A Doebeli, Dieckmann (2000) B Ferrièreet al. (2002) C Ito, Ikegami (2006) D Best et al. (2010)
How can weobtain branching scenarios? General principles (i.e. competitive exclusion, environmentalfeedbacks) Simulation Continuation
Continuation p2 p1
Continuation: first iteration No BR p2 p1
Continuation On the boundary: 2 solutions 2 1 5 equations and 6 unknowns
Continuation p2 0 1 4 2 3 p1
Case study: prey-predator coevolution Resident model
Case study: prey-predator coevolution (Prey) Resident-mutant model Competitionmortality Competitionfunction Symmetriccompetition (Gaussianbell)
No BR S BR 1 U BR 1 Alt BR Non Alternate BR p2 p1
Conclusions An iterative methodbased on continuationhasbeenproposed for obtaining the full catalog of branching scenarios. The full cataloghasbeenderived for a prey-predator system. Continuation can be used to discussprobability vs richness of branching scenarios. Influence of otherparameters can be easilyobtained. Itwould be interesting to apply the method to varioustypicaltwo-speciessystems.