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The stabilizing and destabilizing effects of eco-evolutionary feedbacks ECOLOGY EVOLUTION 1 2 1 Swati Patel , Michael Cortez , Sebastian Schreiber ESA 2016 1: University of California Davis 2: Utah State University Eco-‐evolutionary feedbacks affect communities Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles Becks et al. 2010 Ecology Letters Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles prey evolve to form clumps for defense Becks et al. 2010 Ecology Letters Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles genetic variation low high measure of relative rate of evolution Becks et al. 2010 Ecology Letters (c) (d) (c) Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles genetic variation low (e) high (f) Figure 5 Dynamics prey density predator density clump size STABLE Figure 4 Dynamics of population abundances and mean algal clump size in six replicate predator–prey chemostat experiments with ungrazed Chlamydomonas as the prey (panels a–f). Predator Becks et al. 2010 Ecology )1 Letters clump size in four with grazed Chlamy a function of p availability no lo when two prey ty present, resulting cyclical relationsh by undefended c negative to positiv (c) (d) (c) (c) (d) Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles genetic variation al. low (e) high (b) (e) (a) (f) (f) ( Figure 5 Dynamics prey density predator density clump size clump size in four with grazed Chlamy a function of p availability no lo when two prey ty (d) (c) present, resulting( STABLE Figure 4 Dynamics of population and meanrelationsh algal Figure 4 Dynamics of population abundances and meanabundances algal UNSTABLE cyclical clump size in six chemostat replicate predator–prey experiments clump size in six replicate predator–prey experimentschemostat by undefended c with ungrazed Chlamydomonas as the prey (panels a–f). Predator with ungrazed Chlamydomonas as the prey (panels a–f). Predator negative to positiv Becks et al. 2010 Ecology )1 Letters density (rotifers mL)1) – red curve and solid circles; prey density (c) (d) (c) (c) (d) Eco-‐evolutionary feedbacks affect communities Example: Rotifer-‐algae experiment alters predator-‐prey cycles genetic variation al. low (e) high (b) (e) (a) (f) (f) ( Figure 5 Dynamics prey density predator density clump size clump size in four with grazed Chlamy a function of p availability no lo when two prey ty (d) (c) present, resulting( STABLE Figure 4 Dynamics of population and meanrelationsh algal Figure 4 Dynamics of population abundances and meanabundances algal UNSTABLE cyclical clump size in six chemostat replicate predator–prey experiments clump size in six replicate predator–prey experimentschemostat by undefended c ungrazed Chlamydomonas as the Predator suggests that rwith elative o(panels f evolution cprey an a(panels lter sa–f). tability with ungrazed Chlamydomonas as therate prey a–f). Predator negative to positiv )1 density (rotifers mL)1) – red curve and solid circles; prey density Eco-‐evolutionary feedbacks affect communities predator-‐prey competition P N2 N1 N Abrams and Matsuda 1997 Cortez 2016 apparent competition intraguild predation P P N1 Vasseur et al. 2011 N N2 R Schreiber et al. 2011 Patel and Schreiber 2015 2 Density or Density or Density or 3 2 Relative rate of evolution can stabilize and destabilize 0 0.8 1.4 0 2 0 Genetic Variation 2 D min 0.5 0 0.5 Genetic Variation 0 1 2 1 0 0 0.5 1 Genetic Variation 50 100 Time Density or Trait 1 0 0 3 E max Density or Trait N Density or Trait P 2 Genetic Variation predator-‐prey 1.5 1 1.5 F 1 0 0 50 100 Time Figure 4: With increased additive genetic variation, stabilizing selection can stabilize ecological oscillations (A–C) or destabilize a stable ecological system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and for intermediate genetic variation values when F a 2 Ga is small in magnitude (B, E). A, B, D, E, Maximum and minimum long-term predator density (solid black), prey density (dashed black), and mean prey trait (dashed-dotted gray). For each genetic variation value, a single curve for each variable denotes the stable equilibrium value, and two curves denote the maximum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05 in A; the predator-prey phase lag is less than a quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a period. See appendix E for models and parameter values. cost of reduced sharing of resources among conspecifics are destabilizing. Cortez 2016 (De)stabilization Occurs When the Amount of Genetic stability of the ecological subsystem (measured by the leading eigenvalue of the Jacobian of the ecological subsystem; for stable subsystems this is the subsystem resilience sensu Pimm and Lawton [1977]). Rewriting this equation as V y p 2S reveals that at the critical value of V, the 2 Density or Density or Density or 3 2 Relative rate of evolution can stabilize and destabilize 0 0.8 1.4 0 2 0 Genetic Variation D 50 0.5 E 1 0 1 Genetic Variation 0 0.5 Genetic Variation A F 1 0 1.5Effects of0Prey Genetic Variation 50 Time 1 4 100 2 Density or Trait Density or Trait 0.5 0 0 Time max min 0 0 3 2 1 4 B 000100 C 1.4 2 0 Genetic Variation 1 Cortez 2016 3 0 Genetic Variation cost of reduced sharing of resources among conspecifics are destabilizing. 1.5 2 2 D it (De)stabilization Occurs When the Amount of Genetic 50 Time 100 stability of the ecological subsystem (measured by the leading eigenvalue of the Jacobian of the ecological subsystem; 2 for stable E subsystems this is the subsystem resilience F sensu Pimm and Lawton [1977]). Rewriting this equation as V y p 2S reveals that at the critical value of V, the it 0.8 Density or Trait Density or Trait Density or Trait 6 With increased additive genetic variation, stabilizing selection can stabilize ecological oscillations (A–C) or destabilize a stable ecoFigure 4: logical system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and 2 for intermediate genetic variation values2when F a 2 Ga is small in magnitude (B, E). A, B, D, 3E, Maximum and minimum long-term predator density (solid black), prey density (dashed black), and mean prey trait (dashed-dotted gray). For each genetic variation value, a single curve for each variable denotes the stable equilibrium value, and two curves denote the maximum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05 in A; the predator-prey phase lag is less than a quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a period. See 0 appendix E for models and parameter values. 0 0 it N Density or Trait P 2 Genetic Variation predator-‐prey 1.5 1 σG .3 and evolutionary .7 =.1 coupling of 2theσ ecological dyna change the stability of the long-term community d When there is oscillatory phenotype coexisten heritability, increased heritability leads to conver 0 notype coexistence by stabilizing the unstable co 3 0 50 100 Density or Density or Density or vergent phenotype coexistence often occurs and culminatesN 2 3 in the IGP module resembling a food chain. Alternatively,R when the predator evolves to specialize on the resource (!x ≈ vR ), the IGP module resembles exploitative competition,00.8 and either the1.4 predator or the prey 0is0excluded1 (figs. 4A,2 2 Relative rate of evolution can stabilize and destabilize 0 0 0 0.5 θR R x0 0 1 Genetic Variation θN B F C 6 .1 .7 .3 predator 6 4 N 1 Density or Trait 0.5 .7 2 1 .3 θN 2 E 0 σG = .1 σ P population densities D N R A 2 Density or Trait Density or Trait P 1 prey resource θN θR 0 x 50 1.5Effects of0Prey Genetic Variation 1.0 Genetic Variation Time 0.5 1 000100 1.0 x 8 4 6 population densities 4 C D A variation, stabilizing selection B Ca stable ecopredator 6 With increased additive genetic Figure 4: can stabilize ecological oscillations (A–C) or destabilize predator 0.8 1.4 2 0 Genetic Variation 1 2 Density or Trait 6 2 4 Density or Trait 4 2 0 Density or Trait logical system (D–F). Destabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga 1 0), and stabilization occurs when individual fitness decreases with higher mean defense (F a 2 Ga ! 0). (De)stabilization occurs for all sufficiently large genetic variation when F a 2 Ga is large in magnitude (A, D) and −1.0 values2when F a 2 Ga is small in magnitude (B, E). −1.0 2 for intermediate genetic variation preyprey density (dashed A, B, D, 3E, Maximum and minimum long-term predator density (solid black), prey black), and mean prey trait (dashed-dotted gray). For each genetic variation value, a single curve for each variable denotes the stable equilibrium value, and two curves denote the maxresource imum and minimum values during the cycle. C, Numerical example of the cycle at V p 0:05 in A; the predator-prey is less than 0.2 0.4 0.6phase lag0.8 1.0a resource quarter period. F, Numerical example of the cycle at V p 0:25 in D; the predator-prey phase lag is between one-quarter and one-half of a heritability period. See appendix E for models and parameter values. Genetic Variation 1.0 1.0 0 0 0 3 0 Genetic Variation 50 Time 100 it it x the ecological (measured the leadcost of reduced sharing of resources among conspecifics are 6: stability Figure Rate of ofevolution affectssubsystem community stability.byThe dynamics ingphase eigenvalue of various the Jacobian of the ecological destabilizing. In A and C, slow evolutionary plots for heritabilities h p jG =j.subsystem; 1.5 2 2 stable this the subsystem resilience sensu lution, the for dynamics are a stabilized to aisconvergent In Cortez 2016 D Patel nd Schreiber 2015phenotype Esubsystems F coexistence. and Rewritinggenerates this equation as at a fitnessPimm minimum, whereas[1977]). faster evolution oscillations. T −1.0 Lawton (De)stabilization −1.0 Occurs When the Amount of Genetic V yfrom p 2S reveals (generated that at the critical value of V, the and 6 mum values the dynamics numerically) between 40,000 it N max 1.5 x Time 8 intraguild predation predator-‐prey θR 4 Genetic Variation 2 Genetic Variation 0 Main Objective: General theory on how eco-‐ evolutionary feedbacks affect community stability ECOLOGY EVOLUTION Main Objective: General theory on how eco-‐ evolutionary feedbacks affect community stability ECOLOGY population densities EVOLUTION population traits Main Objective: General theory on how eco-‐ evolutionary feedbacks affect community stability ECOLOGY population densities EVOLUTION population traits Stability: response to perturbations Main Objective: General theory on how eco-‐ evolutionary feedbacks affect community stability Outline 1. general model 2. simple mathematical stability conditions for slow and fast evolution and intuition 3. implications of conditions through example Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach per-‐capita Vitness time scale separation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach per-‐capita Vitness time scale separation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach per-‐capita Vitness time scale separation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach per-‐capita Vitness time scale separation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach per-‐capita Vitness time scale separation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: ECOLOGY -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) dNi = Ni fi (N, x) dt per-‐capita Vitness Example: classic Lotka Volterra equations modiVied with trait-‐ dependent parameters Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt Example: quantitative genetics framework; Lande’s approach selection equation Model • • many species interacting in a community: -‐ where N i is population density of species i -‐ N = (N1 , . . . , Nk ) some or all evolving in one or more traits: -‐ where x j is the trait value of trait j -‐ x = (x1 , . . . , x ) ECOLOGY dNi = Ni fi (N, x) dt EVOLUTION dxj = εgj (N, x) dt relative rate of evolution Lande’s Approach single trait in species i (freq. independent selection) dxj fi = Vi dt xj Vitness • trait Lande’s Approach single trait in species i (freq. independent selection) dxj fi = Vi dt xj Vitness • trait Traits change in direction of Vitness gradient Lande’s Approach single trait in species i (freq. independent selection) dxj fi = Vi dt xj Vitness • Evolutionary equilibria occur at Vitness peaks or valleys Vitness trait trait Lande’s Approach single trait in species i (freq. independent selection) dxj fi = Vi dt xj Vitness • Genetic variance inVluences relative rate of evolution to ecology Vitness trait trait Model ECOLOGY EVOLUTION dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt equilibrium (N , x ) Model ECOLOGY EVOLUTION dNi = Ni fi (N, x) dt equilibrium dxj = εgj (N, x) dt (N , x ) STABILITY-‐ three different perspectives: 1. Mathematically 2. Eco, evo, and eco-‐evo feedbacks 3. Trajectory after perturbation dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Mathematically Reminder: determined from the eigenvalue of the Jacobian with the largest real part Ṅi Nj J= ... .. . x˙k Nj Ṅj x .. . ... x˙k x dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Mathematically Reminder: determined from the eigenvalue of the Jacobian with the largest real part Ṅi Nj J= ... .. . x˙k Nj Ṅj x .. . ... x˙k x s(J) = stability modulus dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Mathematically Reminder: determined from the eigenvalue of the Jacobian with the largest real part Ṅi Nj J= ... .. . x˙k Nj Ṅj x .. . ... x˙k x s(J) = stability modulus s(J) < 0 STABLE dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Mathematically Reminder: determined from the eigenvalue of the Jacobian with the largest real part Ṅi Nj J= ... .. . x˙k Nj Ṅj x .. . ... x˙k x s(J) = stability modulus s(J) < 0 s(J) > 0 STABLE UNSTABLE dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability MATHEMATICALLY A J= εC B εD FEEDBACKS trait pop. dens. dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability MATHEMATICALLY A J= εC B εD FEEDBACKS trait pop. dens. A DIRECT ECOLOGICAL INTERACTIONS: how population densities affect population Vitness dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability MATHEMATICALLY A J= εC B εD FEEDBACKS D trait pop. dens. A DIRECT EVOLUTIONARY INTERACTIONS: how traits affect selection dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability MATHEMATICALLY A J= εC FEEDBACKS B εD D trait B pop. dens. A ECO-‐EVOLUTIONARY INTERACTIONS: how traits affect population Vitness dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability MATHEMATICALLY A J= εC B εD FEEDBACKS D trait C pop. dens. A ECO-‐EVOLUTIONARY INTERACTIONS: how population densities affect selection B dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Eco and Evo Uncoupled MATHEMATICALLY A J= 0 0 εD 1.) ecologically stable: s(A) < 0 2.) evolutionarily stable: s(D) < 0 FEEDBACKS D trait pop. dens. A dNi = Ni fi (N, x) dt dxj = εgj (N, x) dt Stability: Eco and Evo Coupled FEEDBACKS two feedback pathways: D trait B 1.) evo-eco-evo pathway 2.) eco-evo-eco pathway C pop. dens. A EVO-ECO-EVO PATHWAY EVO-ECO-EVO PATHWAY traits change EVO-ECO-EVO PATHWAY traits change alters Vitness EVO-ECO-EVO PATHWAY traits change population densities change alters Vitness EVO-ECO-EVO PATHWAY traits change population densities change alters Vitness alters selection EVO-ECO-EVO PATHWAY traits change population densities change alters Vitness traits change alters selection EVO-ECO-EVO PATHWAY traits change population densities change alters Vitness traits change alters selection STABILIZING EVO-ECO-EVO PATHWAY traits change population densities change alters Vitness traits change alters selection DESTABILIZING EVO-ECO-EVO PATHWAY traits change population densities D changetrait alters Vitness B traits change C pop. dens. alters selection A feedback effects captured by matrix product: CA 1 ( B) DESTABILIZING ECO-EVO-ECO PATHWAY ECO-EVO-ECO PATHWAY population densities change traits change alters selection population densities change STABILIZING alters Vitness DESTABILIZING ECO-EVO-ECO PATHWAY population densities change traits changeD alters selection trait B C pop. dens. A alters population densities change Vitness feedback effects captured by matrix product: BD 1 ( C) STABILIZING DESTABILIZING Stability: Eco and Evo Coupled QUESTION: How do these eco-‐evolutionary feedbacks affect stability? Stability: Eco and Evo Coupled QUESTION: How do these eco-‐evolutionary feedbacks affect stability? relative rate of evolution slow (small ε ) fast (large ε ) Stability: Eco and Evo Coupled QUESTION: How do these eco-‐evolutionary feedbacks affect stability? relative rate of evolution slow (small ε ) evo-eco-evo feedback fast (large ε ) eco-evo-eco feedback Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A CA 1 ( B) Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A CA pop. dens. TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1 ( B) Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A CA pop. dens. TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1 ( B) Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A CA 1 ( B) pop. dens. TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1. FAST ECOLOGICAL RESPONSE Slow Evolution: Evo-‐Eco-‐Evo feedback important for stability MATHEMATICALLY 1. ecologically stable: s(A) < 0 2. sum of evolution with evo-‐eco-‐evo feedback stable: pop. dens. s(D + CA 1 ( B)) < 0 FEEDBACKS D trait pop. dens. A 1 CA ( B) TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1. FAST ECOLOGICAL RESPONSE 2. SLOW ECO-EVOLUTIONARY RESPONSE (with feedbacks) Fast Evolution: Eco-‐Evo-‐Eco feedback important for stability Fast Evolution: Eco-‐Evo-‐Eco feedback important for stability MATHEMATICALLY 1. evolution stable: s(D) < 0 2. sum of ecology with eco-‐evo-‐eco feedback stable: s(A + BD 1 ( C)) < 0 FEEDBACKS D trait pop. dens. Fast Evolution: Eco-‐Evo-‐Eco feedback important for stability 1 BD ( C) MATHEMATICALLY FEEDBACKS 1. evolution stable: s(D) < 0 2. sum of ecology with eco-‐evo-‐eco feedback stable: s(A + BD 1 ( C)) < 0 D trait pop. dens. A Fast Evolution: Eco-‐Evo-‐Eco feedback important for stability 1 BD ( C) MATHEMATICALLY FEEDBACKS 1. evolution stable: s(D) < 0 trait 2. sum of ecology with eco-‐evo-‐eco feedback stable: pop. dens. s(A + BD 1 D ( C)) < 0 pop. dens. A TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1. FAST EVOLUTION RESPONSE Fast Evolution: Eco-‐Evo-‐Eco feedback important for stability 1 BD ( C) MATHEMATICALLY FEEDBACKS 1. evolution stable: s(D) < 0 trait 2. sum of ecology with eco-‐evo-‐eco feedback stable: pop. dens. s(A + BD 1 D ( C)) < 0 pop. dens. A TWO FOLD RESPONSE TO PERTURBATION: eco eq uili bri a a i r b i l i u q e o ev trait 1. FAST EVOLUTION RESPONSE 2. SLOW ECO-EVOLUTIONARY RESPONSE (with feedbacks) Summary: Stability Uncoupled Coupled with slow evolution 1. ecologically 1. ecologically stable stable 2. sum of evolution with 2. evolutionarily evo-‐eco-‐evo feedback stable stable 1. evolutionarily stable 2. sum of ecology with eco-‐evo-‐eco feedback stable s(D) < 0 1 s(A + BD ( C)) < 0 pop. dens. s(D) < 0 s(A) < 0 1 s(D + CA ( B)) < 0 pop. dens. s(A) < 0 Coupled with fast evolution trait trait Applications to a competition model N1 N2 • two competing species • one species evolving in a trait that affects intra-‐ and inter-‐ speciVic competition • trade off between optimal trait Applications to a competition model N1 N2 • two competing species • one species evolving in a trait that affects intra-‐ and inter-‐ speciVic competition • trade off between optimal trait Pruitt et al. 2008 Animal Behavior Applications to a competition model N2 N1 • two competing species • one species evolving in a trait that affects intra-‐ and inter-‐ speciVic competition • trade off between optimal trait USE THIS INFORMATION TO DETERMINE SIGNS OF MATRICES A, B, C D trait B C pop. dens. A Pruitt et al. 2008 Animal Behavior Applications to a competition model BIOLOGY • D trait B C pop. dens. A two competing species with inter-‐ and intra speciVic competition N1 N2 SIGNS OF MATRICES A= Applications to a competition model BIOLOGY • two competing species with inter-‐ and intra speciVic competition • at equilibrium, evolving species is at Vitness peak or valley aggressiveness has negative impact on species 2 D trait B C pop. dens. A • N2 N1 SIGNS OF MATRICES A= B= 0 Applications to a competition model BIOLOGY • two competing species with inter-‐ and intra speciVic competition • at equilibrium, evolving species is at Vitness peak or valley aggressiveness has negative impact on species 2 D trait B C pop. dens. A • • • N2 N1 SIGNS OF MATRICES high density of species 1 selects for less aggressiveness high density of species 2 selects for more aggressiveness A= B= C= 0 + Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow fast Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow fast Assuming ecological stability, evo-‐eco-‐evo feedback is stabilizing: CA 1 ( B) = Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow fast Assuming ecological stability, evo-‐eco-‐evo feedback is stabilizing: CA 1 ( B) = s(D + CA 1 ( B)) < 0 Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow fast Assuming ecological stability, evo-‐eco-‐evo feedback is stabilizing: CA 1 ( B) = s(D + CA 1 ( B)) < 0 Even if evolution is unstable, adding the evo-‐eco-‐evo feedback can lead equilibria to be stable Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow Assuming ecological stability, evo-‐eco-‐evo feedback is stabilizing: CA 1 ( B) = s(D + CA fast Assuming evolutionary stability, eco-‐evo-‐eco feedback is stabilizing: BD 1 ( B)) < 0 Even if evolution is unstable, adding the evo-‐eco-‐evo feedback can lead equilibria to be stable 1 0 ( C) = + 0 Use signs of matrices to determine effects of eco-‐evolutionary feedbacks relative evolutionary time scale slow Assuming ecological stability, evo-‐eco-‐evo feedback is stabilizing: CA 1 ( B) = s(D + CA fast Assuming evolutionary stability, eco-‐evo-‐eco feedback is stabilizing: BD 1 ( B)) < 0 Even if evolution is unstable, adding the evo-‐eco-‐evo feedback can lead equilibria to be stable 1 s(A + BD 0 ( C) = + 1 0 ( C)) < 0 Even if ecology is unstable, adding the eco-‐evo-‐eco feedback can lead equilibria to be stable Applications to a competition model simulated model from Vasseur et al. 2011 1.2 0.8 1.0 trade off 1.4 1.6 • 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.2 0.8 1.0 trade off 1.4 1.6 unstable stable eco 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.2 0.8 1.0 trade off 1.4 unstable stable evo 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.4 1.2 stable for slow and fast evolution stable for fast evolution 0.8 1.0 trade off stable for slow evolution unstable stable evo unstable for slow and fast evolution 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.4 1.2 stable for slow and fast evolution stable for fast evolution 0.8 1.0 trade off stable for slow evolution unstable stable evo unstable for slow and fast evolution For slow evolution, equilibrium stable despite being evolutionarily unstable 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.4 1.2 stable for slow and fast evolution stable for fast evolution 0.8 1.0 trade off stable for slow evolution unstable stable evo unstable for slow and fast evolution For slow evolution, equilibrium stable despite being evolutionarily unstable 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition For fast evolution, equilibrium stable despite being ecologically unstable Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.4 1.2 stable for slow and fast evolution stable for fast evolution 0.8 1.0 trade off stable for slow evolution 0.2 0.4 0.6 0.8 unstable stable evo unstable for slow and fast evolution For slow evolution, equilibrium stable despite being evolutionarily unstable For fast evolution, equilibrium stable despite being ecologically unstable strengthstabilizing of competitore2ffects intraspecific competitionwere really important of feedbacks Applications to a competition model • simulated model from Vasseur et al. 2011 1.6 unstable stable eco 1.2 stable for slow evolution stable for slow and fast evolution stable for fast evolution 0.8 1.0 trade off 1.4 unstable for slow and fast evolution 0.2 0.4 0.6 0.8 strength of competitor 2 intraspecific competition Applications to a competition model • simulated model from Vasseur et al. 2011 1.5 1.0 0.5 stable for slow evolution trait species 2 0.0 1.4 unstable for slow and fast evolution density/density trait value 1.6 unstable stable eco 0.8 2.0 0.4 0.6 0.5 1.0 1.5 N2 density trait 0.4 0.6 genetic variance 0.4 0.6 0.8 strength of competitor 2 intraspecific competition 1.0 heritability 0.2 0.2 0.8 0.0 density density/ trait value 1.2 stable for slow and fast evolution stable for fast evolution 1.0 trade off 0.2 0.8 1.0 Applications to a competition model • simulated model from Vasseur et al. 2011 1.5 1.0 0.5 stable for slow evolution trait species 2 0.0 1.4 unstable for slow and fast evolution density/density trait value 1.6 unstable stable eco 0.8 2.0 0.4 0.6 0.8 heritability 0.5 1.0 1.5 N2 density trait 0.2 0.4 0.6 0.8 genetic variance If either ecology or evolution is unstable, then this can lead to difference in stability for different relative rates of evolution 0.2 0.4 0.6 0.8 1.0 0.0 density density/ trait value 1.2 stable for slow and fast evolution stable for fast evolution 1.0 trade off 0.2 1.0 Main Conclusions • there are three important components to stability: ecology, evolution and eco-‐evolutionary feedbacks • two types of feedback pathways • when evolution is slow, evo-‐eco-‐evo feedback is important • • can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-‐evo-‐eco feedback is important • can lead to stability even if equilibria are ecologically unstable Main Conclusions • there are three important components to stability: ecology, evolution and eco-‐evolutionary feedbacks • two types of feedback pathways • when evolution is slow, evo-‐eco-‐evo feedback is important • • 1 can are ( C) D lead to stability even if equilibria BD trait evolutionarily unstable D when evolution is fast, eco-‐evo-‐eco feedback is important trait • can lead to stability even if equilibria are pop. dens. ecologically unstable A CA evo-eco-evo feedback 1 ( B) pop. dens. A eco-evo-eco feedback Main Conclusions • there are three important components to stability: ecology, evolution and eco-‐evolutionary feedbacks • two types of feedback pathways • when evolution is slow, evo-‐eco-‐evo feedback is important • • can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-‐evo-‐eco feedback is important • can lead to stability even if equilibria are ecologically unstable Main Conclusions • there are three important components to stability: ecology, evolution and eco-‐evolutionary feedbacks • two types of feedback pathways • when evolution is slow, evo-‐eco-‐evo feedback is important • • can lead to stability even if equilibria are evolutionarily unstable when evolution is fast, eco-‐evo-‐eco feedback is important • can lead to stability even if equilibria are ecologically unstable Main Conclusions • there are three important components to stability: ecology, evolution and eco-‐evolutionary feedbacks • two types of feedback pathways • when evolution is slow, evo-‐eco-‐evo feedback is important • • when evolution is fast, eco-‐evo-‐eco feedback is important • • can lead to stability even if equilibria are evolutionarily unstable can lead to stability even if equilibria are ecologically unstable differences in stability for different rates of evolution may be due to whether equilibria are ecologically or evolutionarily unstable Thank you Organizers: Casey terHorst Peter Zee Collaborators: Sebastian Schreiber Michael Cortez Feedback: Jacob Moore Masato Yamamichi Axel Saenz Thomas Schoener Funding: