Disease effects on reproduction can cause population cycles in seasonal environments

Abstract

1. Recent studies of rodent populations have demonstrated that certain parasites can cause juveniles to delay maturation until the next reproductive season. Furthermore, a variety of parasites may share the same host, and evidence is beginning to accumulate showing nonindependent effects of different infections. 2. We investigated the consequences for host population dynamics of a disease-induced period of no reproduction, and a chronic reduction in fecundity following recovery from infection (such as may be induced by secondary infections) using a modified SIR (susceptible, infected, recovered) model. We also included a seasonally varying birth rate as recent studies have demonstrated that seasonally varying parameters can have important effects on long-term host-parasite dynamics. We investigated the model predictions using parameters derived from five different cyclic rodent populations. 3. Delayed and reduced fecundity following recovery from infection have no effect on the ability of the disease to regulate the host population in the model as they have no effect on the basic reproductive rate. However, these factors can influence the long-term dynamics including whether or not they exhibit multiyear cycles. 4. The model predicts disease-induced multiyear cycles for a wide range of realistic parameter values. Host populations that recover relatively slowly following a disease-induced population crash are more likely to show multiyear cycles. Diseases for which the period of infection is brief, but full recovery of reproductive function is relatively slow, could generate large amplitude multiyear cycles of several years in length. Chronically reduced fecundity following recovery can also induce multiyear cycles, in support of previous theoretical studies. 5. When parameterized for cowpox virus in the cyclic field vole populations (Microtus agrestis) of Kielder Forest (northern England), the model predicts that the disease must chronically reduce host fecundity by more than 70%, following recovery from infection, for it to induce multiyear cycles. When the model predicts quasi-periodic multiyear cycles it also predicts that seroprevalence and the effective date of onset of the reproductive season are delayed density-dependent, two phenomena that have been recorded in the field.

Fig. 1

Predicted long-term dynamics for total population density (N) and individual component densities (susceptible, S, infected, I, recovered but not reproductive, Y, recovered and potentially reproductive, Z) for different values of τ. The thin dotted vertical lines denote the annual transition from the reproductive season to the non-reproductive season. The top row plots N through time, when it is sampled once a year only (at time T). The second row plots N as a function for continuous time t. Parameter values are the Kielder Forest field vole parameters (Table 1) with α = 4·3, β = 0·9, f = 0·225, 1/γ =14 days and 1/τ as detailed at the top of each row. S_C is the critical density of S required for I to increase (as detailed in the main text). All simulations started with S = 49 voles ha⁻¹, I = 1 voles ha⁻¹ and Y = Z = 0 voles ha⁻¹.

Fig. 2

(a) Date in the year at which the infection threshold (S_C) is first exceeded, as a function of 1/τ. Parameter values are as in Fig. 1. Dotted lines denote 1/τ values used for the examples in Fig. 1. The critical τ at which the multiyear cycles occur (τ_C) is at 1/τ = 27 days in this example. (b) τ_C as a function of the length of the reproductive season, L. Regular annual cycles occur in the regions denoted ‘NO M.Y. CYCLES’. Other parameter values and initial conditions are as in Fig. 1m–r. We wrote a numerical code to run the model simulations for the full range of L with these parameter values, and calculate τ_C to three decimal places when it could be found (dots), or give details of the predicted dynamics when τ_C could not be found.

Fig. 3

Effects of variation in disease parameters on the period (left column) and amplitude (right column) of the multiyear cycles predicted by model (1), for fixed values of β (labelled on plot) and different rodent population parameters (Table 1). Parameter values are: (a, b) Kielder forest field voles; (c, d) Manor Wood bank voles; (e, f) French common voles; (g, h) northern Fennoscandian field voles. Results for all values of β, for all rodent population parameters, are given in Appendix S4. In the colour bar ‘n.d.’ denotes simulations in which the disease prevalence decays to zero during the course of the simulation. In these cases the susceptible population density exhibits regular annual cycles. Similarly, ‘n.c.’ denotes simulations in which disease remains endemic in the population and all four population components exhibit regular annual cycles. The dominant period of the multiyear dynamics was measured by spectral analysis (using fast Fourier transform) of 256 years of equilibrium population data, measured annually (see Turchin 2003 for a review). The amplitude was the difference between the maximum and the minimum total population density in this data set. For brevity we do not distinguish between regularly repeated multiyear cycles and irregular (pseudo-periodic) multiyear cycles.

Fig. 4

Dominant period (a) and amplitude (b) predicted by the model as 1/τ and f are varied for the Kielder Forest field vole parameters (Table 1), with α = 4·3, β = 0·9 and 1/γ = 28 days. Colour bar conventions are as in Fig. 3. Dominant period and amplitude were measured as detailed in the legend of Fig. 3.

Fig. 5

(a) Correlation between ‘effective start date’ of the reproductive season, as defined in the main text, and the population density at varying times in the past. The dashed lines denote the correlation coefficient at which the probability of no significant relationship is P = 0.05. (c) The same analysis as in (a), but correlating seroprevalence at the start of the reproductive season with past population densities; (b, d) ‘effective season start date’ and seroprevalence plotted against past population density for the corresponding highest significant positive correlation in (a) and (c), respectively. Zero on the vertical axis in (b) corresponds to no effective delay in the onset of the reproductive season. Parameter values are the same as for Fig. 1m–r.