Evolution Our inspiration Model Results Summary Evolution of Complex Density-Dependent Dispersal Strategies Kalle Parvinen, Anne Seppänen and John Nagy University of Turku, Finland Department of Mathematics [email protected] Applied Mathematics seminar 2.3.2012 Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Outline 1 Evolution 2 Our inspiration 3 Model Local model Metapopulation model Adaptive dynamics Example 4 Results 5 Summary Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Example Theory Questions Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Example Theory Questions Not examples of evolution butterfly metamorphosis cell differentiation: osteoclast [Boyle et al, Nature 2003] human development Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Example Theory Questions Evolution The theory of evolution by natural selection assumes: individuals differ by their trait, different survival probability traits are heritable too many offspring, not all can survive struggle for existence Charles Darwin Individuals that survive and reproduce before dying (1809-1882) are on average, better suited to their local environment Therefore... As time passes by the distribution of traits change. More fitted become more general: Adaptation Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Example Theory Questions Evolution Some questions: Why nature is green? Why multicellular organism evolved? Why helping behavior evolved? Why we have two genders? Why senescence? Why dispersing evolved? Can we prevent evolution of drug resistance? Can we control cancer? And so many more! Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary American pika Question American pika - Ochotona princeps distribution map order: Lagomorpha small, diurnal herbivore native to cold climates Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary American pika Question American pika - Ochotona princeps talus - rocky slopes Kalle Parvinen, Anne Seppänen and John Nagy hay pile for winter Evolution of dispersal Evolution Our inspiration Model Results Summary American pika Question Bodie metapopulation ore dumps - patches Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary American pika Question Question Dispersal is selected for, because of... Resource competition Fluctuating environment Kin competition Density dependent dispersal Large patches: threshold strategy [Gyllenberg and Metz:2001] Larger patch models “dilute” the kin benefits Our question: How density-dependent dispersal evolves in metapopulation with small local populations? Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary American pika Question Strategy types Examples of density dependent dispersal strategies a) threshold for large patch-sizes b) threshold emigr. 1 emigr. 1 0 0 x̃ density local population 1 c) monotone 2 3 4 5 6 7 8 9 10 7 8 9 10 local population size d) non-monotone emigr. 1 emigr. 1 0 0 1 2 3 4 5 6 7 8 local population size 9 10 Kalle Parvinen, Anne Seppänen and John Nagy 1 2 3 4 5 6 local population size Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Strategies Emigration: Natal dispersal: probability that an individual born in a patch with n inhabitants will emigrate immediately after birth Adult dispersal: rate to emigrate from a patch with n inhabitants e = (e1 , e2 , ..., eK ) Natal dispersal: 0 6 ei 6 1 Adult dispersal: ei > 0 Immigration: Probability to stay in a patch when encountered m = (m0 , m1 , ..., mK −1 ) 0 6 mi 6 1 Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Local model a) Adult dispersal, local dynamics Birth or immigration Immigration 0 1 Birth or immigration Death or emigration Death or emigration 2 Death or emigration Death or emigration Birth or immigration Birth or immigration Birth or immigration Death or emigration Death or emigration k K Local extinction b) Natal dispersal, local dynamics Immigration or birth without emigration Immigration 0 1 Death Immigration or birth without emigration Immigration or birth without emigration 2 Death Immigration or birth without emigration Immigration or birth without emigration k Death Death K Death Local extinction Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Death Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Metapopulation model Infinite number of patches Patches connected via global dispersal Forward Kolmogorov equations for natal dispersal d dt p0 d dt pn = −αm0 Dp0 + d1 p1 + µ(1 − p0 ), = [αmn−1 D + (n − 1)bn−1 (1 − en−1 )]pn−1 −[n(bn (1 − en ) + dn ) + αmn D + µ]pn +(n + 1)dn+1 pn+1 , d p = [αmK −1 D + (K − 1)bK −1 (1 − eK −1 )]pK −1 dt K −[KdK + µ]pK PK −1 d p m dt D = −αD PK n=0 n n + n=1 nbn en pn + KbK (1 − eK )(1 − η)σpK − νD. Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Assumptions Assume Separation of time scales: fast ecological dynamics slow evolutionary dynamics The resident is in its population dynamical attractor A mutant appears rare slightly different phenotype What happens? Can the mutant invade? Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Evolution Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Fitness Invasion fitness r Exponential growth when rare. If r > 0 then invasion is possible. [Metz et al., 1992, 1996; Geritz et al., 1997, 1998] Rmetapop - metapopulation reproduction ratio Operates on dispersal generations lnRmetapop and r sign equivalent Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Local model Metapopulation model Adaptive dynamics Example Adaptive dynamics cont. Singular strategy Singular strategies Fitness gradient equals to 0. Attracting vs. repelling Unbeatable vs. beatable Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Strategy types Examples of density dependent dispersal strategies a) threshold for large patch-sizes b) threshold emigr. 1 emigr. 1 0 0 x̃ density local population 1 c) monotone 2 3 4 5 6 7 8 9 10 7 8 9 10 local population size d) non-monotone emigr. 1 emigr. 1 0 0 1 2 3 4 5 6 7 8 local population size 9 10 Kalle Parvinen, Anne Seppänen and John Nagy 1 2 3 4 5 6 local population size Evolution of dispersal Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Density-dependence vs. density-independence K = 10, k = 7 Natal dispersal Adult dispersal 10 ESS uniform TSS dens.dep. e TSS dens.dep. e,m 8 pop.size pop.size 10 6 4 2 0 ESS uniform TSS dens.dep. e TSS dens.dep. e,m 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 µ 0.6 0.8 1 µ 1 av.emigr. 1 av.emigr. 0.4 0.8 0.6 0.4 ESS uniform TSS dens.dep. e TSS dens.dep. e,m 0.2 0 0.8 0.6 0.4 ESS uniform TSS dens.dep. e TSS dens.dep. e,m 0.2 0 0 0.2 0.4 0.6 0.8 1 µ Kalle Parvinen, Anne Seppänen and John Nagy 0 0.2 0.4 µ Evolution of dispersal 0.6 0.8 1 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=0 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=1 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=2 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=3 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=4 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=5 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=6 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=7 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=8 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t=9 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 10 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 20 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 30 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 40 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 50 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 100 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 150 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 160 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 170 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 180 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 190 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 200 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 210 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 220 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 230 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 240 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 245 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 250 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 255 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 258 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal A Trait Substitution Sequence K = 10, k = 7, ν = 0.08, µ = 0.06 t = 260 emigration 1 0 1 2 3 4 5 6 pop.size Kalle Parvinen, Anne Seppänen and John Nagy 7 8 Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Just an exception? Attracting vertices of the strategy space ñ single-step multi-step 0 0.2 0.4 µ 0.6 0.8 1 8 c) K = 8 emigration and immigration 1st step up 10 9 8 7 6 5 4 3 2 1 b) K = 8 emigration only 1st step up 1st step up a) K = 10 emigration only ñ 7 single-step multi-step 6 5 4 3 2 1 0 0.2 0.4 µ 0.6 0.8 8 single-step multi-step 6 5 4 3 2 1 1 Expected threshold ñ = k 1 − µb < k Kalle Parvinen, Anne Seppänen and John Nagy ñ 7 Evolution of dispersal 0 0.2 0.4 µ 0.6 0.8 1 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Phase plane plot a) b) µ = 0.048 ) µ = 0.055 d) µ = 0.058 0.6 0.6 0.6 0.6 0.4 0.4 e6 1 0.8 e6 1 0.8 e6 1 0.8 e6 1 0.8 0.4 0.4 0.2 0.2 0.2 laements PSfrag replaements PSfrag replaements PSfrag replaements 0 0 0 0.2 0.4 0.6 0.8 1 0.2 e5 e) 0.4 0.6 0.8 1 0 0 0.2 e5 f) µ = 0.070 0.2 0 0 0.4 0.6 0.8 1 0 g) µ = 0.075 h) µ = 0.080 0.8 0.8 0.6 0.6 0.6 0.4 0 0.2 0.4 0.6 0.8 1 0 0 e5 0.2 0.4 0.6 0.8 1 e5 1 µ = 0.085 0.2 0 0 0.2 0.4 0.6 0.8 1 e5 e1 = e2 = e3 = e4 = 0 e7 = e8 = e9 = e10 = 1 Kalle Parvinen, Anne Seppänen and John Nagy 0.8 0.4 0.2 0.2 0.2 laements PSfrag replaements PSfrag replaements PSfrag replaements 0 0.6 e6 0.8 0.6 e6 0.8 e6 1 e6 1 0.4 0.4 e5 1 0 0.2 e5 1 0.4 µ = 0.065 Evolution of dispersal 0 0.2 0.4 e5 0.6 0.8 1 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Strategy Complements? Emigration and immigration are not always complements mi 6= 1 − ei Immigrant does not know his neighbors Newborn knows his mother is there K = 10, k = 7, µ = 0.07 1 strategy strategy K = 10, k = 7, µ = 0.1 e mnn 0 0 2 4 6 8 local population size, n 10 Kalle Parvinen, Anne Seppänen and John Nagy 1 e mnn 0 0 2 4 6 8 local population size, n Evolution of dispersal 10 Evolution Our inspiration Model Results Summary Density-dependence Natal dispersal Adult dispersal Adult dispersal Non-monotone density dependence? a) t ∈ [0, 10], µ = 0.25 c) t ∈ [80, 750], µ = 0.25 6 2 emigr. emigr. 5 1 4 3 2 1 0 0 1 2 3 4 5 6 7 8 local population size, n 9 10 b) t ∈ [10, 80], µ = 0.25 1 2 3 4 5 6 7 8 local population size, n 9 10 d) t ∈ [750, 10000], µ = 0.25 8 3 emigr. emigr. 7 2 1 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 local population size, n 9 10 Kalle Parvinen, Anne Seppänen and John Nagy 1 2 3 4 5 6 7 8 local population size, n Evolution of dispersal 9 10 Evolution Our inspiration Model Results Summary Take-home messages Nature is full of exciting questions. Intuition is not always right. Complex density dependent dispersal may evolve. Adult dispersers may benefit more from conditional dispersal strategies. Hypothesis: bang-bang type dispersal at Bodie. Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal Evolution Our inspiration Model Results Summary Thank you! Kalle Parvinen, Anne Seppänen and John Nagy Evolution of dispersal
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