The Ideal Free Distribution in a Predator–Prey Model with Multifactor Taxis

Biophysics - The concept of an ideal free distribution (IFD) is analyzed for the predator–prey system in an inhomogeneous ring-shaped habitat. Diffusion–reaction–advection...     

Permanence and Extinction of a Diffusive Predator–Prey Model with Robin Boundary Conditions

The main concern of this paper is to study the dynamic of a predator–prey system with diffusion. It incorporates the Holling-type-II and a modified Leslie–Gower functional responses under Robin boundary conditions. More concretely, we study the dissipativeness of the system by using the comparison principle, and we derive a criteria for permanence and for predator extinction.     

Modeling of Multifactor Taxis in a Predator–Prey System

Biophysics - A model of predator–prey dynamics in a spatially heterogeneous range is considered using a system of two nonlinear equations of the diffusion–advection reaction. The...     

Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey

Biophysics - This paper mainly deals with the prey?predator dynamics where both the prey and predator exhibit herd behavior. Positivity, boundedness, some extinction criteria, stability of...     

Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

This work aims to examine the global behavior of a Gause type predator–prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.     

How Predator Food Preference can Change the Destiny of Native Prey in Predator–Prey Systems

The aim of this work is to develop and analyse a mathematical model for a predator-2 preys system arising in insular environments. We are interested in the evolution of a native prey population without behavioural traits to cope with predation or competition, after the introduction of alien species. Here, we consider a long living bird population with low fertility rate. We point out the effects of the preference of the predator for either juvenile or adult stages. In addition, we study the impact of alien prey introduction in such a model. We use a reaction-diffusion system with a singular logistic right hand side. The aim of this work is to bring interesting dynamics to the fore. As a first example, oscillatory behaviour takes place in the model without alien preys and when predators have an average preference coefficient. Introduction of alien preys can lead to species extinction.     

Bayesian Inference for Functional Response in a Stochastic Predator–Prey System

We present a Bayesian method for functional response parameter estimation starting from time series of field data on predator–prey dynamics. Population dynamics is described by a system of stochastic differential equations in which behavioral stochasticities are represented by noise terms affecting each population as well as their interaction. We focus on the estimation of a behavioral parameter appearing in the functional response of predator to prey abundance when a small number of observations is available. To deal with small sample sizes, latent data are introduced between each pair of field observations and are considered as missing data. The method is applied to both simulated and observational data. The results obtained using different numbers of latent data are compared with those achieved following a frequentist approach. As a case study, we consider an acarine predator–prey system relevant to biological control problems.     

Along Came a Spider: Using Live Arthropods in a Predator–Prey Activity

We developed a predator–prey activity with eighth-grade students in which they used wolf spiders (Lycosa carolinensis), house crickets (Acheta domestica), and abiotic factors to address how (1) adaptations in predators and prey shape their interaction and (2) abiotic factors modify the interaction between predators and prey. We tested student understanding with pre- and postquizzes, written observations, and interpretations of graphical results.     

Impact of Fear in a Delayed Predator–Prey System with Prey Refuge in Presence of Additional Food

Biophysics - It is observed from field experiments on terrestrial vertebrates that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect...     

A Jump-Growth Model for Predator–Prey Dynamics: Derivation and Application to Marine Ecosystems

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker–Planck equation for the stochastic fluctuations. The McKendrick–von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation, and the McKendrick–von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.     

A New Stochastic Individual-Based Model for Pattern Formation and its Application to Predator–Prey Systems

Reaction–diffusion theory has played a very important role in the study of pattern formation in biology. However, a group of individuals is described by a single state variable representing population density in reaction–diffusion models, and interaction between individuals can be included only phenomenologically. In this paper, we propose a new scheme that seamlessly combines individual-based models with elements of reaction–diffusion theory and apply it to predator–prey systems as a test of our scheme. In the model, starvation periods and the time to reproductive maturity are modeled for individual predators. Similarly, the life cycle and time to reproductive maturity of an individual prey are modeled. Furthermore, both predators and prey migrate through a two-dimensional space. To include animal migration in the model, we use a relationship between the diffusion and the random numbers generated according to a two-dimensional bivariate normal distribution. Despite the simplicity of this model, our scheme successfully produces logistic patterns and oscillations in the population size of both predator and prey. The peak for the predator population oscillation lags slightly behind the prey peak. The simplicity of this scheme will aid additional study of spatially distributed negative-feedback systems.     

The Effects of Predator Evolution and Genetic Variation on Predator–Prey Population-Level Dynamics

This paper explores how predator evolution and the magnitude of predator genetic variation alter the population-level dynamics of predator–prey systems. We do this by analyzing a general eco-evolutionary predator–prey model using four methods: Method 1 identifies how eco-evolutionary feedbacks alter system stability in the fast and slow evolution limits; Method 2 identifies how the amount of standing predator genetic variation alters system stability; Method 3 identifies how the phase lags in predator–prey cycles depend on the amount of genetic variation; and Method 4 determines conditions for different cycle shapes in the fast and slow evolution limits using geometric singular perturbation theory. With these four methods, we identify the conditions under which predator evolution alters system stability and shapes of predator–prey cycles, and how those effect depend on the amount of genetic variation in the predator population. We discuss the advantages and disadvantages of each method and the relations between the four methods. This work shows how the four methods can be used in tandem to make general predictions about eco-evolutionary dynamics and feedbacks.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

Adaptive Control Based Harvesting Strategy for a Predator–Prey Dynamical System

This paper deals with designing a harvesting control strategy for a predator–prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy.     

Effects of Temperature and Turbulence on the Predator–Prey Interactions between a Heterotrophic Flagellate and a Marine Bacterium

Biotic and abiotic factors can influence interactions between microbial grazers and their prey, thus impacting both the cycling of biogenic carbon within the surface layer of the ocean and the export of carbon to the deep ocean and higher trophic levels. In this study, microcosm experiments were used to evaluate the combined effect of temperature and turbulence on the growth rate of a marine bacterium (Vibrio splendidus), a protistan predator (Paraphysomonas sp.), and the community grazing impact of Paraphysomonas sp. on V. splendidus. It was found that the artificial turbulence generated (1.35 × 10⁻¹ cm² s⁻³) significantly increased the rates of growth of Paraphysomonas sp. at high (>10°C), but not low (<5°C) temperatures, and that turbulence had no effect on the growth of V. splendidus. Both flagellate and bacterial growth were temperature dependent and decreased 4- to 6-fold as temperatures decreased from 15 to 0°C. Bacterial grazing mortality by Paraphysomonas sp. was 1.3- to 2.5-fold greater in the turbulent than static treatments among all four temperatures, and the rates of cell-specific ingestion of bacteria by Paraphysomonas sp. was 2-fold greater at 15 and 10°C in the turbulent than in the static treatment. Hence, this study shows that turbulence can influence nanoflagellate grazing at temperatures >5°C and suggests that at low temperatures, increased viscosity may limit the size of organisms that can be affected by small-scale turbulence.Present address (M.P. Delaney): Goddard Earth Sciences & Technology Center, NASA Goddard Space Flight Center, Mail Code 900.1, Greenbelt, MD 20771     

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.