Path-integral calculation for the emergence of rapid evolution from demographic stochasticity

Genetic variation in a population can sometimes arise so fast as to modify ecosystem dynamics. Such phenomena have been observed in natural predator-prey systems and characterized in the laboratory as showing unusual phase relationships in population dynamics, including a $\pi$ phase shift between predator and prey (evolutionary cycles) and even undetectable prey oscillations compared to those of the predator (cryptic cycles). Here we present a generic individual-level stochastic model of interacting populations that includes a subpopulation of low nutritional value to the predator. Using a master equation formalism and by mapping to a coherent state path integral solved by a system-size expansion, we show that evolutionary and cryptic quasicycles can emerge generically from the combination of intrinsic demographic fluctuations and clonal mutations alone, without additional biological mechanisms.

Figure 1

Stochastic simulations for (a) evolutionary cycles emerging from normal cycles due to random mutation and (b) cryptic cycles. Phase portraits of the steady states of (c) normal cycles and (d) evolutionary cycles from the stochastic simulations show that the phase differences between predator and the total prey population are roughly $\pi /2$ and $\pi$, respectively, while for (e) cryptic cycles there is no obvious phase relationship. (f)–(h) Power spectrum of the wild-type prey (thick curve) and phase difference spectrum (thin curve) from analytic calculations based on ILMs. The estimated phase differences are $-0.55\pi$ and $0.905\pi$ for (f) normal cycles and (g) evolutionary cycles and for (h) cryptic cycles the predicted phase difference between the wild-type prey and the defended prey is approximately $0.874\pi$. The parameter values are (a) $V=1000,c_W=0.3,p_W=0.6,c_D/c_W=0.8,p_D/p_W=0.01,d_D/d_W=1,{\varphi }_{N,\text{max}}=1$, and $b=0.1$ and (b) $V=380,c_W=60,p_W=0.92,c_D/c_W=0.95,p_D/p_W=0.001,d_D/d_W=7.5,{\varphi }_{N,\text{max}}=16$, and $b=0.1$.

Figure 2

Phase diagrams for evolutionary cycles (EC) and cryptic cycles (CC) calculated from ILMs with respect to the ratio of the prey reproduction rate $c_D/c_W$, the ratio of the predation rate $p_D/p_W$, the maximum nutrient concentration ${\varphi }_N^{\text{max}}$, and the dilution rate $b$. The gradient-colorful region corresponds to the coexistence of all species and in the other regions the rapid evolution is not stable, with corresponding letters indicating the coexistence of only certain species. The coexistence states are decided by the mean-field densities and their ratio to the fluctuations; when fluctuations are larger than mean-field solutions, the dynamics is under high risk of extinction. The color legend represents the predicted phase difference between the wild-type prey and the defended prey ${\theta }_{WD}$ for rapid evolution, in units of $\pi$. The contours are the estimated amplitude ratios of wild-type prey to predator, indicating the tendency to be cryptic cycles. In the gray region near transition, the two types of prey start to decouple, leading to degenerate peaks in power spectra, and thus the phase is not well defined. Except for the axis specified in each diagram, the parameters in the calculations are $V=300,c_W=1,p_W=1,c_D/c_W=0.8,p_D/p_W=0.01,d_D/d_W=3.5,{\varphi }_N^{\text{max}}=16$, and $b=0.6$. The predicted phase diagram is consistent with stochastic simulation.