Effects of demographic stochasticity on evolving biological communities Yohsuke Murase Collaborators: Takashi Shimada, Nobuyasu Ito, Per Arne Rikvold

Reference Y. Murase et al., Phys. Rev. E, 81, 041908 (2010)

noise on population dynamics • Since the birth-death process of individuals is stochastic, the population of each species always fluctuates due to the finiteness of the number of individuals, known as demographic stochasticity. • Population dynamics with demographic stochasticity often show nontrivial dynamics which cannot be predicted by the deterministic equations. ˙ xi = bixi + aijxixj species i species j ˙ xj = bjxj + ajixjxi

previous studies McKane et al., PRL (2005) predator-prey models Reichenbach et al., PRE (2006) rock-paper-scissors game The effect of the demographic stochasticity may be much larger than the one estimated by naive O(1/√N) estimates.

A major goal of this work • to investigate the effects of demographic stochasticity in models of biological community assembly on evolutionary time scales ˙ ni = f({nj } , . . . ) new species extinction

• a mutant species has a tiny initial population • a species which is close to extinction has a small population new species extinction • In real ecosystems, species often shows wide range of population sizes. Volkov et al. Nature(2005) population fluctuations, which inevitably exist in any finite system, may drastically alter the predictions of deterministic models

Model: Tangled-Nature model - Christensen et al., JTB, 216, 73 (2002) - Hall et al., PRE, 66, 011904 (2002) - P.A. Rikvold and R.K.P. Zia, PRE 68, 031913 (2003) - P.A. Rikvold, PRE, 75, 051920 (2007) • Each individual of species I gives rise to F offsprings with a reproduction probability PI which depends on its birth cost and couplings. • An offspring "mutate" into another species with probability μ. generation t generation t+1

controlling the stochasticity Individual-based model: # of successful individuals follows a binomial distribution A stochastic difference equation is used instead to control the noise effect. Gaussian noise a parameter to control the noise effect • κ=1 : a good approximation to the individual-based model • κ=0 : deterministic population update pI(k) = ✓ nI k ◆ Pk I (1 PI)nI k

reproduction probability Model B (predator-prey model) Model A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -2 0 2 4 PI Delta MIJ is limited to anti-symmetric. No restriction on the form of MIJ . a measure of species fitness

introduction of new species 001001010 001001010 001001010 101110010 101110010 001001010 101110010 101110010 001001010 001001010 001001010 001001010 001001110 001001010 101110010 101110010 101110010 101110010 .... mutation! .... Each species has a "genome" bit string of length L. A mutation (bit flip) happens to the genomes of the existing offspring at the moment with probability mu. Interaction coefficients are predetermined at the beginning of the simulation and fixed during the simulation. t t+1 - Christensen et al., JTB, 216, 73 (2002) - Hall et al., PRE, 66, 011904 (2002) - P.A. Rikvold and R.K.P. Zia, PRE 68, 031913 (2003) - P.A. Rikvold, PRE, 75, 051920 (2007)

Results: Model A (mutualism) with noise without noise no significant difference between deterministic and stochastic population updates

Results: Model B (predator-prey) diversity total population size As the noise level increases, both the diversity and the total population size decrease remarkably.

linear-stability analysis from simulation snapshots, we obtained "core fixed-point communities". • updated the population dynamics without noise and mutation until we reach the fixed-point populations. fixed-point populations linear-stability matrix

distribution of n and λs population sizes eigenvalues of the LS matrix more stable communities are selected under the noise species with larger population sizes are selected under the noise

birth cost, interaction matrix less selection pressure for b under the noise stronger couplings are selected under the noise

Contradiction to Stability-Complexity Relation? The off-diagonal parts of the linear stability matrix is closer to an antisymmetric form. ~ -1 Eigenvalues originating from antisymmetric off-diagonal matrix elements are all pure imaginary. A community tends to be less stable when interactions are dense and strong. (Gardner et al., Nature (1970), R. May Nature (1972) )

survival condition for species To maintain a large population size, species should have either low birth cost large interaction constants => neutrally stable systems => stable systems Although the former is allowed in a deterministic population dynamics, the latter is selected in a stochastic population dynamics. Demographic noise may make the consequence significantly different! or

long-term evolutionary dynamics Model A does not show significant dependence on the demographic noise. power spectrum density intermittent time series approximate 1/f fluctuations

results for Model B The dynamics in evolutionary time-scale may be altered by the noise for predator-prey models. power spectrum density 1/f fluctuations for stochastic population dynamics 1/f2 fluctuations for noiseless dynamics

Conclusion • The effects of demographic stochasticity are explored for two types of biological macroevolution models. • The predator-prey model shows a remarkable decline in diversity at higher noise levels, while it does not for the mutualistic communities. • The communities that emerge under influence of the noise consist of species strongly coupled with each other and have stronger linear stability around the fixed-point populations than the corresponding noiseless model. • The dynamics on evolutionary time scales for the predator-prey model are also altered by the noise.