Generating functionals and Gaussian approximations for interruptible delay reactions

We develop a generating functional description of the dynamics of non-Markovian individual-based systems in which delay reactions can be terminated before completion. This generalizes previous work in which a path-integral approach was applied to dynamics in which delay reactions complete with certainty. We construct a more widely applicable theory, and from it we derive Gaussian approximations of the dynamics, valid in the limit of large, but finite, population sizes. As an application of our theory we study predator-prey models with delay dynamics due to gestation or lag periods to reach the reproductive age. In particular, we focus on the effects of delay on noise-induced cycles.

Figure 1

Relationship between survival probability at the fixed point, $\chi$, and delay period, $\overline{\tau }$. Lines are parametric plots using Eq. (46). Blue lines and circles correspond to the gestation model; yellow lines and triangles correspond to the juvenile model. Parameters are $b=p=k=1$ and $d=0.2$. For the gestation model the survival probability approaches $0.5$, whereas for the juvenile model it approaches 0. Symbols are the fraction of delay reactions which completed in simulations using the MNRM, averaged over 100 realizations and with $\mathrm{\Omega }=1000$.

Figure 2

Relationship between fixed point location and delay period, $\overline{\tau }$. Lines are parametric plots using Eq. (46) and Eqs. (43, 44, 45). Blue lines and circles correspond to the gestation model; yellow lines and triangles correspond to the juvenile model. Parameters are $b=p=k=1$ and $d=0.2$. The total prey concentration, $x_{\mathrm{tot}}^{*}$, is not affected by the delay (black line), and is the same for both models. Symbols are from simulations using the MNRM averaged over 100 realizations with $\mathrm{\Omega }=1000$.

Figure 3

Sample trajectories of $x$ using the gestation model for $\overline{\tau }=0$ (top line) and $\overline{\tau }=5$ (bottom line). Parameters are $b=p=k=1$ and $d=0.2$. Simulations performed using the MNRM with $\mathrm{\Omega }=10000$. Both trajectories are observed to oscillate about their respective deterministic fixed points. The oscillations for $\overline{\tau }=5$ are much more defined, with a period $T\sim 50$.

Figure 4

Variation of the power spectrum of $x$, $S_x\left(\omega \right)$ with the delay period, $\overline{\tau }$. The top panel corresponds to the gestation model; the bottom panel corresponds to the juvenile model. Dashed lines are theoretical predictions using Eq. (46); solid lines are from simulations using the MNRM with $\mathrm{\Omega }=10000$. Parameters are $b=p=k=1$ and $d=0.2$.