Stochasticity in host-parasitoid models informs mechanisms regulating population dynamics

Abstract

Population dynamics of host-parasitoid interactions have been traditionally studied using a discrete-time formalism starting from the classical work of Nicholson and Bailey. It is well known that differences in parasitism risk among individual hosts can stabilize the otherwise unstable equilibrium of the Nicholson-Bailey model. Here, we consider a stochastic formulation of these discrete-time models, where the host reproduction is a random variable that varies from year to year and drives fluctuations in population densities. Interestingly, our analysis reveals that there exists an optimal level of heterogeneity in parasitism risk that minimizes the extent of fluctuations in the host population density. Intuitively, low variation in parasitism risk drives large fluctuations in the host population density as the system is on the edge of stability. In contrast, high variation in parasitism risk makes the host equilibrium sensitive to the host reproduction rate, also leading to large fluctuations in the population density. Further results show that the correlation between the adult host and parasitoid densities is high for the same year, and gradually decays to zero as one considers cross-species correlations across different years. We next consider an alternative mechanism of stabilizing host-parasitoid population dynamics based on a Type III functional response, where the parasitoid attack rate accelerates with increasing host density. Intriguingly, this nonlinear functional response makes qualitatively different correlation signatures than those seen with heterogeneity in parasitism risk. In particular, a Type III functional response leads to uncorrelated adult and parasitoid densities in the same year, but high cross-species correlation across successive years. In summary, these results argue that the cross-correlation function between population densities contains signatures for uncovering mechanisms that stabilize consumer-resource population dynamics.

Figure 1

Extent of fluctuations in the host population density are minimized at an intermediate level of heterogeneity in parasitism risk. The steady-state coefficient of variation squared of the host population density as predicted by (14) and (16) is plotted as a function of the heterogeneity in parasitism risk CV. Three examples of host density fluctuations are generated by performing stochastic simulations of model (3)-(4) for different values of CV assuming $\overline{c}=1$, $R=2$ and ${\sigma }_R^2=1$. All time series are normalized to have a mean value of one.

Figure 2

Randomness in host reproduction induces strong positive correlations between host-parasitoid densities for model (3)–(4) that incorporates heterogeneity in parasitism risk. Predicted Pearson correlation coefficient between the host and parasitoid densities for $R=2$ and $R=10$ as a function of $H_R$ as given in (14).

Figure 3

Different stabilizing mechanisms of host-parasitoid population dynamics can be discriminated from the cross-correlation function profile. Left: Scatter plot of $H_t$ and $P_t$ from a stochastic simulation of model (3)-(4) for $CV=2$, $\overline{c}=1$, $R=2$, ${\sigma }_R^2=1$ along with the time-series of host/parasitoid population densities. Heterogeneity in parasitism risk results in strong correlation between $H_t$ and $P_t$, that shows a modest increase followed by decay to zero with increasing generation lag $\mathrm{\Delta }t$ for correlations between $H_t$ and $P_{t+\mathrm{\Delta }t}$. Right: Scatter plot of $H_t$ and $P_t$ from a stochastic simulation of model (1) and (18) for $m=2$, $\overline{c}=1$, $R=2$, ${\sigma }_R^2=1$ reveals uncorrelated fluctuations in host/parasitoid densities for a Type III functional response. As can be seen in the simulated time-series and the cross-correlation function, $h_t$ and $p_{t+1}$ show a strong positive correlation that decays back to zero with increasing generation lag $\mathrm{\Delta }t$.