Gunog Seo

A comparison of two predator-prey models with Holling's type I functional response.

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.

A Predator–Prey Model with a Holling Type I Functional Response Including a Predator Mutual Interference

The most widely used functional response in describing predator–prey relationships is the Holling type II functional response, where per capita predation is a smooth, increasing, and saturating function of prey density. Beddington and DeAngelis modified the Holling type II response to include interference of predators that increases with predator density. Here we introduce a predator-interference term into a Holling type I functional response. We explain the ecological rationale for the response and note that the phase plane configuration of the predator and prey isoclines differs greatly from that of the Beddington–DeAngelis response; for example, in having three possible interior equilibria rather than one. In fact, this new functional response seems to be quite unique. We used analytical and numerical methods to show that the resulting system shows a much richer dynamical behavior than the Beddington–DeAngelis response, or other typically used functional responses. For example, cyclic-fold, saddle-fold, homoclinic saddle connection, and multiple crossing bifurcations can all occur. We then use a smooth approximation to the Holling type I functional response with predator mutual interference to show that these dynamical properties do not result from the lack of smoothness, but rather from subtle differences in the functional responses.

Mathematical model of anaerobic digestion in a chemostat: effects of syntrophy and inhibition

Three of the four main stages of anaerobic digestion: acidogenesis, acetogenesis, and methanogenesis are described by a system of differential equations modelling the interaction of microbial populations in a chemostat. The microbes consume and/or produce simple substrates, alcohols and fatty acids, acetic acid, and hydrogen. Acetogenic bacteria and hydrogenotrophic methanogens interact through syntrophy. The model also includes the inhibition of acetoclastic and hydrogenotrophic methanogens due to sensitivity to varying pH-levels. To examine the effects of these interactions and inhibitions, we first study an inhibition-free model and obtain results for global stability using differential inequalities together with conservation laws. For the model with inhibition, we derive conditions for existence, local stability, and bistability of equilibria and present a global stability result. A case study illustrates the effects of inhibition on the regions of stability. Inhibition introduces regions of bistability and stabilizes some equilibria.

The effect of temporal variability on persistence conditions in rivers.

There has been great interest in the invasion and persistence of algal and insect populations in rivers. Recent modeling approaches assume that the flow speed of the river is constant. In reality, however, flow speeds in rivers change significantly on various temporal scales due to seasonality, weather conditions, or many human activities such as hydroelectric dams. In this paper, we study persistence conditions by deriving the upstream invasion speed in simple reaction-advection-diffusion equations with coefficients chosen to be periodic step functions. The key methodological idea to determine the spreading speed is to use the exponential transform in order to obtain a moment generating function. In a temporally periodic environment, the averages of each coefficient function determine the minimal upstream and downstream propagation speeds for a single-compartment model. For a two-compartment model, the temporal variation can enhance population persistence.

SPREAD RATES UNDER TEMPORAL VARIABILITY: CALCULATION AND APPLICATION TO BIOLOGICAL INVASIONS

In this paper, we introduce a technique to study the minimal wave speed in reaction-diffusion equations with temporal variability and apply it to two particular models for biological invasions. We use the exponential transform to avoid solving partial differential equations explicitly or finding inverse transforms. In a single reaction-diffusion equation with time-periodic coefficients, the minimal wave speed depends only on time-averages of each coefficient function. In a two-compartment system with mobile and stationary individuals, the invasion speed depends on the precise form of the coefficient functions and their temporal correlations; in some cases, a lower bound can be obtained. Our technique can be extended to more complex life histories of invading organisms.

Spatial Dynamics of an Age-Structured Population Model of Asian Clams

Asian clam (Corbicula fluminea) is one of the most important nonnative aquatic invasive species in the freshwater ecosystem of North America, rapidly spreading in lakes, canals, streams, and rivers. This species has remarkably distinct mobility patterns in different phases of its life cycle. We formulate a novel mathematical model, in the form of nonlocal delayed partial differential equations, to calculate and characterize the invasion speed, and show that the invasion speed coincides with the minimal speed of traveling wave fronts.

Sensitivity of the dynamics of the general Rosenzweig–MacArthur model to the mathematical form of the functional response: a bifurcation theory approach

The equations in the Rosenzweig–MacArthur predator–prey model have been shown to be sensitive to the mathematical form used to model the predator response function even if the forms used have the same basic shape: zero at zero, monotone increasing, concave down, and saturating. Here, we revisit this model to help explain this sensitivity in the case of three response functions of Holling type II form: Monod, Ivlev, and Hyperbolic tangent. We consider both the local and global dynamics and determine the possible bifurcations with respect to variation of the carrying capacity of the prey, a measure of the enrichment of the environment. We give an analytic expression that determines the criticality of the Hopf bifurcation, and prove that although all three forms can give rise to supercritical Hopf bifurcations, only the Trigonometric form can also give rise to subcritical Hopf bifurcation and has a saddle node bifurcation of periodic orbits giving rise to two coexisting limit cycles, providing a counterexample to a conjecture of Kooji and Zegeling. We also revisit the ranking of the functional responses, according to their potential to destabilize the dynamics of the model and show that given data, not only the choice of the functional form, but the choice of the number and/or position of the data points can influence the dynamics predicted.

Bifurcations and global dynamics in a toxin-dependent aquatic population model

The study of effects of environmental toxins on ecosystems is of great interest from both environmental and conservation points of view. In this paper, we present a global stability and bifurcation analysis of a toxin-dependent aquatic population model. Our analytical and numerical results show that both the environmental toxin level and the depuration capability of the population significantly affect the population persistence. The model exhibits a multifarious array of dynamics. While low levels of external toxin allow population persistence and high levels of toxin lead to an extirpation, intermediate toxin concentrations can produce very rich dynamics, such as transient oscillations, hysteresis, heteroclinic orbits, and a codimension-two bifurcation. In particular, a regime of bistability exists where the population is doomed to extinction or survival, depending on initial state of the system. As a practical implication of our study, the toxic effects of methylmercury on rainbow trout are scrutinized. The theory developed here provides a sound theoretical foundation for understanding the population effects of toxicity.