Changes over time in the spatiotemporal dynamics of cyclic populations of field voles (Microtus agrestis L.)

Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction-diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction-diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

Cowpox virus infection in natural field vole Microtus agrestis populations: significant negative impacts on survival

1. Cowpox virus is an endemic virus circulating in populations of wild rodents. It has been implicated as a potential cause of population cycles in field voles Microtus agrestis L., in Britain, owing to a delayed density-dependent pattern in prevalence, but its impact on field vole demographic parameters is unknown. This study tests the hypothesis that wild field voles infected with cowpox virus have a lower probability of survival than uninfected individuals. 2. The effect of cowpox virus infection on the probability of an individual surviving to the next month was investigated using longitudinal data collected over 2 years from four grassland sites in Kielder Forest, UK. This effect was also investigated at the population level, by examining whether infection prevalence explained temporal variation in survival rates, once other factors influencing survival had been controlled for. 3. Individuals with a probability of infection, P(I), of 1 at a time when base survival rate was at median levels had a 22.4% lower estimated probability of survival than uninfected individuals, whereas those with a P(I) of 0.5 had a 10.4% lower survival. 4. At the population level, survival rates also decreased with increasing cowpox prevalence, with lower survival rates in months of higher cowpox prevalence. 5. Simple matrix projection models with 28 day time steps and two stages, with 71% of voles experiencing cowpox infection in their second month of life (the average observed seroprevalence at the end of the breeding season) predict a reduction in 28-day population growth rate during the breeding season from lambda = 1.62 to 1.53 for populations with no cowpox infection compared with infected populations. 6. This negative correlation between cowpox virus infection and field vole survival, with its potentially significant effect on population growth rate, is the first for an endemic pathogen in a cyclic population of wild rodents.

Tuberculosis (Mycobacterium microti) in wild field vole populations

Vole tuberculosis (TB; Mycobacterium microti) is an understudied endemic infection. Despite progressing slowly, it causes severe clinical pathology and overt symptoms in its rodent host. TB was monitored for 2 years in wild field voles in Kielder Forest, UK. The prevalence of characteristic cutaneous TB lesions was monitored longitudinally at 4 sites, with individuals live-trapped and repeatedly monitored. A prevalence of 5.2% of individuals with lesions was recorded (n=2791). In a cross-sectional study, 27 sites were monitored bi-annually, with TB assessed by post-mortem examination for macroscopic lesions, and by culture and histopathology. Seventy-nine voles (10.78%; n=733) were positive for mycobacteria, with the highest prevalence in spring (13.15%; n=327). TB prevalence varied, with between 0% and 50% of voles infected per site. Prevalence increased with age (mass), and apparent seasonality was due to a higher proportion of older animals in spring. Survival analysis supported this result, with cutaneous lesions only manifesting in the advanced stages of infection, and therefore only being found on older voles. The body condition of individuals with lesions declined at the time when the lesion was first recorded, when compared to individuals without lesions, suggesting there may be an acute phase of infection during its advanced stage. Although predicted survival following the appearance of a cutaneous lesion was lower than for uninfected individuals, this was not significant.

The effects of obstacle size on periodic travelling waves in oscillatory reaction–diffusion equations

Many natural populations undergo multi-year cycles, and field studies have shown that these can be organized into periodic travelling waves (PTWs). Mathematical studies have shown that large-scale landscape obstacles represent a natural mechanism for wave generation. Here, we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size. We firstly consider a large circular obstacle in an infinite domain for a reaction–diffusion system of ‘ λ – ω ’ type. We use perturbation theory to derive a leading order approximation to the wave generated by the obstacle. This shows the dependence of the wave properties on both parameter values and obstacle size. We find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge, rather than exponentially in the case of a flat boundary. We use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator–prey system, via a reduction of the predator–prey model to normal form close to Hopf bifurcation. Our predictions compare well with numerical simulations. We also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries.

Disease effects on reproduction can cause population cycles in seasonal environments

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.

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.

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.

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.

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.