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
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.
effects of demographic stochasticity in models of biological community assembly on evolutionary time scales ˙ ni = f({nj } , . . . ) new species extinction
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
(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
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
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
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)
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) )
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
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
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.