Table 1.
Glossary.
Fig 1.
Mean time to extinction, T, is plotted against the initial frequency of the focal species x.
Black circles are the outcomes of a Monte-Carlo simulation, and the colored lines are the theoretical predictions for the corresponding case (see legend) as obtained from numerical solutions of Eq (8) with the μ(x) and σ2(x) from Eqs (9)–(12). Details of the solution are presented in the Methods section. The hierarchy TGP > TGS > TLP > TLS is evident. In panel (a), both species have the same mean fitness (s0 = 0) and therefore all the lines are symmetric around x = 1/2. Other parameters in panel (a) are N = 600, γ = 0.25 and δ = 0.55. In panel (b) the focal species is slightly advantageous, with s0 = 0.01 (all other parameters are the same), so the maximum time to extinction appears at x < 1/2.
Fig 2.
The relationships between the mean time to extinction T and the size of the community, N, in different scenarios.
Filled red circles were obtained from Monte-Carlo simulations, Blue circles are theoretical predictions based on numerical integration of Eq (8) with the μ(x) and σ2(x) from Eqs (9)–(12), and the dashed black line is a linear fit to these blue circles, presented to guide the eye. As expected, in the global-periodic case the mean time to extinction grows exponentially with N, in the global-periodic the grows satisfies a power-law, the local-periodic case behaves like the neutral model (T is linear in N), whereas the local-stochastic dynamics yields log2 N growth. In all cases s0 = 0 (so the T shown here is the mean time to extinction starting from x = 1/2), δ = 0.2 and γ = 0.4.
Fig 3.
The outcome of a typical simulation.
Shown are the species richness (panel A) and Shannon’s entropy (panel B) of a specific run of our simulation. The parameters of this run are δ = 0.2, N = 10000, γ = 0.4 and ν = 1/N. Competition is global, variations are periodic and the number of niches is Q = 3. At the initial state all individuals belong to a single species, hence SR = 1 and the entropy is zero. The system equilibrates after less than 1000 generations, and from this point species richness fluctuates around 8.8 and the entropy fluctuates around 1.2.
Fig 4.
Species richness (SR, upper panels) and Shannon’s entropy (SE, lower panels) vs. the rate in which new species are trying to invade the community, νN.
In all cases, when νN is small the diversity of a community and its evenness obey the same relationships as T in the previous section: global competition is better (for coexistence) than local due to the storage effect, and stochastic variations are worse than periodic variations. When the number of temporal niches is large, as in the Q = 30 case in panels (c) and (f), an increase of νN leads to an increase in the number of species, their different response buffers the effect of environmental variations and the results converge to the predictions of the neutral model (cyan dashed line). However, as the number of temporal niches decreases global competition puts a hurdle against invasion, as every invader must compete with niche-specialists. Therefore SR, and in particular SE, are much smaller for global competition if νN is large. A comparison between the case of Q = 10 [panels (b) and (e)] and Q = 3 [panels (a) and (d)] suggest that the crossover from the storage-dominated regime, where global competition is better, to the regime in which global competition acts to decrease biodiversity, occurs when the number of species is equal to the number of temporal niches. For all panels, parameters are: N = 10, 000, δ = 0.2 and γ = 0.4. SR and SE were monitored every generation and the results presented here reflect an average over the period between 6, 000 and 100, 000 generations. For the neutral model γ = 0 and δ is an irrelevant parameter.
Fig 5.
The dynamic of a Q = 3 community.
The abundance of all species is plotted against time (generations) for global-periodic (left panel) and local-periodic (right panel) competition. Global competition gives an advantage to specialists, therefore the community is dominated by three specialist species, and all other invaders either stay small or (if the invader is a better specialist) take over a given temporal niche and replace the existing specialist. Under local competition, many species coexist, with no gap between the abundances of the dominant species and the abundance of all other species. For both panels, parameters are: Q = 3, N = 10, 000, ν = 0.005, γ = 0.4, and δ = 0.2.