Effects of a disease affecting a predator on the dynamics of a predator-prey system

We study the effects of a disease affecting a predator on the dynamics of a predator-prey system. We couple an SIRS model applied to the predator population, to a Lotka-Volterra model. The SIRS model describes the spread of the disease in a predator population subdivided into susceptible, infected and removed individuals. The Lotka-Volterra model describes the predator-prey interactions. We consider two time scales, a fast one for the disease and a comparatively slow one for predator-prey interactions and for predator mortality. We use the classical “aggregation method” in order to obtain a reduced equivalent model. We show that there are two possible asymptotic behaviors: either the predator population dies out and the prey tends to its carrying capacity, or the predator and prey coexist. In this latter case, the predator population tends either to a “disease-free” or to a “disease-endemic” state. Moreover, the total predator density in the disease-endemic state is greater than the predator density in the “disease-free” equilibrium (DFE).     

Individual base model of predator-prey system shows predator dominant dynamics

Individual base model of predator-prey system is constructed. Both predator and prey species have age structure and cohorts of early reproductive age have competitive advantage. The model has linear functional response in predation behavior and includes the effect of interference among predators and delay of population growth from resource intake, not by functional response but by calculation procedure. Each foraging action is calculated successively and surplus or scarce of acquired resources is interpreted into population size through individual birth and death. This model shows that biomass of prey killed by predator is dependent on demand of predator and that heterogeneity in predator population is essential in persistency and stability of predator-prey system. Heterogeneity of predator makes predator individuals of less competing ability die rapidly. Rapid death of weak individuals causes rapid decrease of total demand of predator and that makes enough room for survived predators. Therefore, the biomass of killed prey is dependent on predator's demand. As young or infant population of predator are the more vulnerable to shortage of prey, and when many of them cannot survive to reproductive age, they can stabilize the system by wasting excessive prey with only temporal numerical increase of predator population.     

Analysis of a predator-prey system with predator switching

In this paper, we consider an interaction of prey and predator species where prey species have the ability of group defence. Thresholds, equilibria and stabilities are determined for the system of ordinary differential equations. Taking carrying capacity as a bifurcation parameter, it is shown that a Hopf bifurcation can occur implying that if the carrying capacity is made sufficiently large by enrichment of the environment, the model predicts the eventual extinction of the predator providing strong support for the so-called ‘paradox of enrichment’.     

Persistence in predator-prey systems with ratio-dependent predator influence

Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes. Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A4823. Research carried out while visiting the University of Alberta.     

Unique coevolutionary dynamics in a predator-prey system

In this paper, we study the predator-prey coevolutionary dynamics when a prey's defense and a predator's offense change in an adaptive manner, either by genetic evolution or phenotypic plasticity, or by behavioral choice. Results are: (1) The coevolutionary dynamics are more likely to be stable if the predator adapts faster than the prey. (2) The prey population size can be nearly constant but the predator population can show very large amplitude fluctuations. (3) Both populations may oscillate in antiphase. All of these are not observed when the handling time is short and the prey's density dependence is weak. (4) The population dynamics and the trait dynamics show resonance: the amplitude of the population fluctuation is the largest when the speed of adaptation is intermediate. These results may explain experimental studies with microorganisms.     

Determining alternative models for vegetation response analysis: a non-parametric approach

Abstract. Vegetation models based on multiple logistic regression are of growing interest in environmental studies and decision making. The relatively simple sigmoid Gaussian optimum curves are most common in current vegetation models, although several different other response shapes are known. However, improvements in the technical means for handling statistical data now facilitate fast and interactive calculation of alternative complex, more data-related, non-parametric models. The aim in this study was to determine whether, and if so how often, a complex response shape could be more adequate than a linear or quadratic one. Using the framework of Generalized Additive Models, both parametric (linear and quadratic) and non-parametric (smoothed) stepwise multiple logistic regression techniques were applied to a large data set on wetlands and water plants and to six environmental variables: pH, chloride, orthophosphate, inorganic nitrogen, thickness of the sapropelium layer and depth of the water-body. All models were tested for their goodness-of-fit and significance. Of all 156 generalized additive models calculated, 77 % were found to contain at least one smoothed predictor variable, i.e. an environmental variable with a response better fitted by a complex, non-parametric, than by a linear or quadratic parametric curve. Chloride was the variable with the highest incidence of smoothed responses (48 %). Generally, a smoothed curve was preferable in 23 % of all species-variable correlations calculated, compared to 25 % and 18 % for sigmoid and Gaussian shaped curves, respectively. Regression models of two plant species are presented in detail to illustrate the potential of smoothers to produce good fitting and biologically sound response models in comparison to linear and polynomial regression models. We found Generalized Additive Modelling a useful and practical technique for improving current regression-based vegetation models by allowing for alternative, complex response shapes.     

Global dynamics of a ratio-dependent predator-prey system

Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey interactions where predation involves serious searching processes. However, such models have set up a challenging issue regarding their dynamics near the origin since these models are not well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey system at the origin in the interior of the first quadrant is studied. It is shown that the origin is indeed a critical point of higher order. There can exist numerous kinds of topological structures in a neighborhood of the origin including the parabolic orbits, the elliptic orbits, the hyperbolic orbits, and any combination of them. These structures have important implications for the global behavior of the model. Global qualitative analysis of the model depending on all parameters is carried out, and conditions of existence and non-existence of limit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.     

The anatomy of predator-prey dynamics in a changing climate

Humans are increasingly influencing global climate and regional predator assemblages, yet a mechanistic understanding of how climate and predation interact to affect fluctuations in prey populations is currently lacking. Here we develop a modelling framework to explore the effects of different predation strategies on the response of age-structured prey populations to a changing climate. We show that predation acts in opposition to temporal correlation in climatic conditions to suppress prey population fluctuations. Ambush predators such as lions are shown to be more effective at suppressing fluctuations in their prey than cursorial predators such as wolves, which chase down prey over long distances, because they are more effective predators on prime-aged adults. We model climate as a Markov process and explore the consequences of future changes in climatic autocorrelation for population dynamics. We show that the presence of healthy predator populations will be particularly important in dampening prey population fluctuations if temporal correlation in climatic conditions increases in the future.     

Population dynamics of a South American rodent: seasonal structure interacting with climate,density dependence and predator effects

Understanding the role of interactions between intrinsic feedback loops and external climatic forces is one of the central challenges within the field of population ecology. For rodent dynamics, the seasonal structure of the environment necessitates changes between two stages: reproductive and non-reproductive. Nevertheless, the interactions between seasonality, climate, density dependence and predators have been generally ignored. We demonstrate that direct climate effects, the nonlinear effect of predators and the nonlinear first-order feedback embedded in a seasonal structure are key elements underlying the large and irregular fluctuations in population numbers exhibited by a small rodent in a semi-arid region of central Chile. We found that factors influencing population growth rates clearly differ between breeding and non-breeding seasons. In addition, we detected nonlinear density dependencies as well as nonlinear and differential effects of generalist and specialist predators. Recent climatic changes may account for dramatic perturbations of the rodent's population dynamics. Changes in the predator guild induced by climate are likely to result, through the food web, in a large impact on small rodent demography and population dynamics. Assuming such interactions to be typical of ecological systems, we conclude that appropriate predictions of the ecological consequences of climate change will depend on having an in-depth understanding of the community-weather system.     

The subcritical collapse of predator populations in discrete-time predator-prey models.

Many discrete-time predator-prey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator's population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations and provide examples of predator-prey systems with a subcritical flip bifurcation.     

Prey: predator ratio dependence in the functional response of a freshwater amphipod

1. First known for their shredding activity, freshwater amphipods also behave as active predators with consequences for prey population regulation and amphipod coexistence in the context of biological invasions. 2. A way to quantify predation is to determine the average consumption rate per predator, also known as its functional response (FR). 3. Although amphipods are gregarious and can display social interactions that can alter per capita consumption rates, previous studies using the FR approach to investigate amphipod predation ignored such potential mutual interference because they did not consider variations in predator density. 4. We investigated the FR of Echinogammarus berilloni feeding on dipteran larvae with joint variations in prey and predator densities. This bivariate experimental design allowed us to estimate interference and to compare the fits of the three main classes of theoretical FR models, in which the predation rate is a function of prey density alone (prey‐dependent models), of both prey and predator densities (predator‐dependent models) or of the prey‐to‐predator ratio (ratio‐dependent models). 5. The Arditi–Ginzburg ratio‐dependent FR model provided the best representation of the FR of E. berilloni, whose predation rate showed a decelerating rise to a horizontal asymptote as prey abundance increased. 6. Ratio dependence means that mutual interference between amphipods leads to prey sharing. Mutual interference is likely to vary between amphipod species, depending on their level of aggressiveness.     

Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.     

Testing for density dependence

Summary The robustness and sensitivity of a test for density dependence in an animal population against departures from the assumed null and alternative model is assessed via simulation. The test is shown to be nonrobust and insensitive to departures from the assumed models.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

Effect of resource subsidies on predator-prey population dynamics: a mathematical model

The influence of a resource subsidy on predator-prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator-prey-subsidy interactions also increases.     

Ionic strength dependence of F-actin and glycolytic enzyme associations: a Brownian dynamics simulations approach

The association of glycolytic enzymes with F-actin is proposed to be one mechanism by which these enzymes are compartmentalized, and, as a result, may possibly play important roles for: regulation of the glycolytic pathway, potential substrate channeling, and increasing glycolytic flux. Historically, in vitro experiments have shown that many enzyme/actin interactions are dependent on ionic strength. Herein, Brownian dynamics (BD) examines how ionic strength impacts the energetics of the association of F-actin with the glycolytic enzymes: lactate dehydrogenase (LDH), glyceraldehyde-3-phosphate dehydrogenase (GAPDH), fructose-1,6-bisphosphate aldolase (aldolase), and triose phosphate isomerase (TPI). The BD simulations are steered by electrostatics calculated by Poisson-Boltzmann theory. The BD results confirm experimental observations that the degree of association diminishes as ionic strength increases but also suggest that these interactions are significant, at physiological ionic strengths. Furthermore, BD agrees with experiments that muscle LDH, aldolase, and GAPDH interact significantly with F-actin whereas TPI does not. BD indicates similarities in binding regions for aldolase and LDH among the different species investigated. Furthermore, the residues responsible for salt bridge formation in stable complexes persist as ionic strength increases. This suggests the importance of the residues determined for these binary complexes and specificity of the interactions. That these interactions are conserved across species, and there appears to be a general trend among the enzymes, support the importance of these enzyme-F-actin interactions in creating initial complexes critical for compartmentation.     

Effect of predator density dependent dispersal of prey on stability of a predator-prey system

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.     

Testing the threat-sensitive predator avoidance hypothesis: physiological responses and predator pressure in wild rabbits

Predation is a strong selective force with both direct and indirect effects on an animal’s fitness. In order to increase the chances of survival, animals have developed different antipredator strategies. However, these strategies have associated costs, so animals should assess their actual risk of predation and shape their antipredator effort accordingly. Under a stressful situation, such as the presence of predators, animals display a physiological stress response that might be proportional to the risk perceived. We tested this hypothesis in wild European rabbits (Oryctolagus cuniculus), subjected to different predator pressures, in Doñana National Park (Spain). We measured the concentrations of fecal corticosterone metabolites (FCM) in 20 rabbit populations. By means of track censuses we obtained indexes of mammalian predator presence for each rabbit population. Other factors that could modify the physiological stress response, such as breeding status, food availability and rabbit density, were also considered. Model selection based on information theory showed that predator pressure was the main factor triggering the glucocorticoid release and that the physiological stress response was positively correlated with the indexes of the presence of mammalian carnivore predators. Other factors, such as food availability and density of rabbits, were considerably less important. We conclude that rabbits are able to assess their actual risk of predation and show a threat-sensitive physiological response.     

Multiple dynamics in a single predator-prey system: experimental effects of food quality.

Recent work with the freshwater zooplankton Daphnia has suggested that the quality of its algal prey can have a significant effect on its demographic rates and life-history patterns. Predator-prey theory linking food quantity and food quality predicts that a single system should be able to display two distinct patterns of population dynamics. One pattern is predicted to have high herbivore and low algal biomass dynamics (high HBD), whereas the other is predicted to have low herbivore and high algal biomass dynamics (low HBD). Despite these predictions and the stoichiometric evidence that many phytoplankton communities may have poor access to food of quality, there have been few tests of whether a dynamic predator-prey system can display both of these distinct patterns. Here we report, to the authors' knowledge, the first evidence for two dynamical patterns, as predicted by theory, in a single predator-prey system. We show that the high HBD is a result of food quantity effects and that the low HBD is a result of food quality effects, which are maintained by phosphorus limitation in the predator. These results provide an important link between the known effects of nutrient limitation in herbivores and the significance of prey quality in predator-prey population dynamics in natural zooplankton communities.     

Evolutionary dynamics of predator-prey systems: an ecological perspective

 Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals – the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator–prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator–prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species. Received 10 November 1993; received in revised form 25 July 1994