Rebecca C. Tyson

Modelling the Canada lynx and snowshoe hare population cycle: the role of specialist predators

Mathematical models of the snowshoe hare (Lepus americanus) and Canada lynx (Lynx canadensis) population cycles in the boreal forest have largely focused on the interaction between a single specialist predator and its prey. Here, we consider the role that other hare predators play in shaping the cycles, using a predator–prey model for up to three separate specialist predators. We consider the Canada lynx, coyote (Canis latrans) and great horned owl (Bubo virginianus). Our model improves on past modelling efforts in two ways: (1) our model solutions more closely represent the boreal hare and predator cycles with respect to the cycle period, maximum and minimum hare densities and maximum and minimum predator densities for each predator, and (2) our model sheds light on the role each specialist plays in regulation of the hare cycle, in particular, the dynamics of the raptor appear to be crucial for characterising the low hare densities correctly.

The Effect of Habitat Fragmentation on Cyclic Population Dynamics: A Numerical Study

Through four spatially explicit models, we investigate how habitat fragmentation affects cyclic predator–prey population dynamics. We use a Partial Differential Equation (PDE) framework to describe the dispersal of predators and prey in a heterogeneous landscape made of high quality and low quality habitat patches, subject to increasing fragmentation through habitat separation and/or habitat loss. Our results show that habitat fragmentation decreases the amplitude of the predator–prey population cycles while average population density is not as strongly affected in general. Beyond these simple trends however, the four models show differing responses to fragmentation, indicating that when making predictions about population survival and persistence in the face of habitat fragmentation, the choice of model is important. Our results may inform conservation efforts in fragmented habitats for cyclic species such as the snowshoe hare and Canada lynx.

Pattern Formation in a Model for Mountain Pine Beetle Dispersal: Linking Model Predictions to Data

Pattern formation occurs in a wide range of biological systems. This pattern formation can occur in mathematical models because of diffusion-driven instability or due to the interaction between reaction, diffusion, and chemotaxis. In this paper, we investigate the spatial pattern formation of attack clusters in a system for Mountain Pine Beetle. The pattern formation (aggregation) of the Mountain Pine Beetle in order to attack susceptible trees is crucial for their survival and reproduction. We use a reaction-diffusion equation with chemotaxis to model the interaction between Mountain Pine Beetle, Mountain Pine Beetle pheromones, and susceptible trees. Mathematical analysis is utilized to discover the spacing in-between beetle attacks on the susceptible landscape. The model predictions are verified by analysing aerial detection survey data of Mountain Pine Beetle Attack from the Sawtooth National Recreation Area. We find that the distance between Mountain Pine Beetle attack clusters predicted by our model closely corresponds to the observed attack data in the Sawtooth National Recreation Area. These results clarify the spatial mechanisms controlling the transition from incipient to epidemic populations and may lead to control measures which protect forests from Mountain Pine Beetle outbreak.

The effect of habitat fragmentation on cyclic population dynamics: a reduction to ordinary differential equations

Habitat fragmentation is known to be a key factor affecting population dynamics. In a previous study by Strohm and Tyson (Bull Math Biol 71:1323–1348, 2009), the effect of habitat fragmentation on cyclic population dynamics was studied using spatially explicit predator–prey models with four different sets of reaction terms. The difficulty with spatially explicit models is that often analytical tractability is lost and the mechanisms behind the behaviour of the models are difficult to analyse. In this study, we employ a simplification procedure based on a Fourier series first-term truncation of the spatially explicit models Strohm and Tyson (Bull Math Biol 71:1323–1348, 2009) to obtain spatially implicit models. These simpler models capture the main features of the spatially explicit models and can be used to explain the dynamics observed by Strohm and Tyson. We find that the spatially implicit models and the spatially explicit models produce similar responses to habitat fragmentation for larger high-quality patch sizes. Additionally, we find that the critical patch size of the spatially implicit models provides an upper bound on the critical patch size of the spatially explicit models. Finally, we derive an approximation of the multi-patch habitat by a single-patch habitat with partial flux boundary conditions which allows for a lower bound on the critical patch size to be calculated.

Mating rates between sterile and wild codling moths (Cydia pomonella) in springtime: A simulation study

The sterile insect technique (SIT) can be a powerful method for pest control without the negative environmental effects of conventional pesticides. The goal is to induce pest population collapse by arranging conditions where wild females mate only with sterile males and thus do not produce offspring. In applying the SIT, it can be important to understand both how subtle alterations of sterile and wild insect behaviour alter the effectiveness of the SIT in different applications, and how this is reflected in the data gathered through associated monitoring devices, often pheromone traps. Our work in this paper is motivated by the use of SIT against orchard pests, particularly the codling moth (Cydia pomonella). We investigate how individual behaviours affect the mating rate between wild females and sterile males, and the corresponding sterile to wild trap catch ratio, through a preliminary individual-based model. Our analysis suggests that the sterile males may not be effective at interfering with mating between wild moths during springtime releases, while at the same time monitoring information gathered from trap catches may give no indication of reduced effectiveness of the SIT.

Modelling the dynamics of invasion and control of competing green crab genotypes

Establishment of invasive species is a worldwide problem. In many jurisdictions, management strategies are being developed in an attempt to reduce the environmental and economic harm these species may cause in the receiving ecosystem. Scientific studies to improve understanding of the mechanisms behind invasive species population growth and spread are key components in the development of control methods. The work presented herein is motivated by the case of the European green crab (Carcinus maenas L.), a remarkably adaptable organism that has invaded marine coastal waters around the globe. Two genotypes of European green crab have independently invaded the Atlantic coast of Canada. One genotype invaded the mid-Atlantic coast of the USA by 1817, subsequently spreading northward through New England and reaching Atlantic Canada by 1951. A second genotype, originating from the northern limit of the green crabs European range, invaded the Atlantic coast of Nova Scotia in the 1980s and is spreading southward from the Canadian Maritime provinces. We developed an integrodifference equation model for green crab population growth, competition and spread, and demonstrate that it yields appropriate spread rates for the two genotypes, based on historical data. Analysis of our model indicates that while harvesting efforts have the benefit of reducing green crab density and slowing the spread rate of the two genotypes, elimination of the green crab is virtually impossible with harvesting alone. Accordingly, a green crab fishery would be sustainable. We also demonstrate that with harvesting and restocking, the competitive imbalance between the Northern and Southern green crab genotypes can be reversed. That is, a competitively inferior species can be used to control a competitively superior one.

Moving forward in circles: challenges and opportunities in modelling population cycles

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer-resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research.

Seasonally Varying Predation Behavior and Climate Shifts Are Predicted to Affect Predator-Prey Cycles

The functional response of some predator species changes from a pattern characteristic for a generalist to that for a specialist according to seasonally varying prey availability. Current theory does not address the dynamic consequences of this phenomenon. Since season length correlates strongly with altitude and latitude and is predicted to change under future climate scenarios, including this phenomenon in theoretical models seems essential for correct prediction of future ecosystem dynamics. We develop and analyze a two-season model for the great horned owl (Bubo virginialis) and snowshoe hare (Lepus americanus). These species form a predator-prey system in which the generalist to specialist shift in predation pattern has been documented empirically. We study the qualitative behavior of this predator-prey model community as summer season length changes. We find that relatively small changes in summer season length can have a profound impact on the system. In particular, when the predator has sufficient alternative resources available during the summer season, it can drive the prey to extinction, there can be coexisting stable states, and there can be stable large-amplitude limit cycles coexisting with a stable steady state. Our results illustrate that the impacts of global change on local ecosystems can be driven by internal system dynamics and can potentially have catastrophic consequences.

A Lévy-flight diffusion model to predict transgenic pollen dispersal

The containment of genetically modified (GM) pollen is an issue of significant concern for many countries. For crops that are bee-pollinated, model predictions of outcrossing rates depend on the movement hypothesis used for the pollinators. Previous work studying pollen spread by honeybees, the most important pollinator worldwide, was based on the assumption that honeybee movement can be well approximated by Brownian motion. A number of recent studies, however, suggest that pollinating insects such as bees perform Lévy flights in their search for food. Such flight patterns yield much larger rates of spread, and so the Brownian motion assumption might significantly underestimate the risk associated with GM pollen outcrossing in conventional crops. In this work, we propose a mechanistic model for pollen dispersal in which the bees perform truncated Lévy flights. This assumption leads to a fractional-order diffusion model for pollen that can be tuned to model motion ranging from pure Brownian to pure Lévy. We parametrize our new model by taking the same pollen dispersal dataset used in Brownian motion modelling studies. By numerically solving the model equations, we show that the isolation distances required to keep outcrossing levels below a certain threshold are substantially increased by comparison with the original predictions, suggesting that isolation distances may need to be much larger than originally thought.

Opinion strength influences the spatial dynamics of opinion formation

ABSTRACT Opinions are rarely binary; they can be held with different degrees of conviction, and this expanded attitude spectrum can affect the influence one opinion has on others. Our goal is to understand how different aspects of influence lead to recognizable spatio-temporal patterns of opinions and their strengths. To do this, we introduce a stochastic spatial agent-based model of opinion dynamics that includes a spectrum of opinion strengths and various possible rules for how the opinion strength of one individual affects the influence that this individual has on others. Through simulations, we find that even a small amount of amplification of opinion strength through interaction with like-minded neighbors can tip the scales in favor of polarization and deadlock.