A predator-prey model with infected prey

A predator-prey model with logistic growth in the prey is modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds are identified which determine when the predator population survives and when the disease remains endemic. For some parameter values the greater vulnerability of the infected prey allows the predator population to persist, when it would otherwise become extinct. Also the predation on the more vulnerable prey can cause the disease to die out, when it would remain endemic without the predators.     

Stability regions in predator-prey systems with constant-rate prey harvesting

Summary We analyze the global behavior of a predator-prey system, modelled by a pair of non-linear ordinary differential equations, under constant-rate prey harvesting. By methods analogous to those used to study predator harvesting, we characterize the theoretically possible structures and transitions. With the aid of a computer simulation we construct examples to show which of these transitions can be realized in a biologically plausible model.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada, Grant No. 67-3138.     

Availability of prey resources drives evolution of predator-prey interaction

Productivity is predicted to drive the ecological and evolutionary dynamics of predator-prey interaction through changes in resource allocation between different traits. Here we report results of an evolutionary experiment where prey bacteria Serratia marcescens was exposed to predatory protozoa Tetrahymena thermophila in low- and high-resource environments for approximately 2400 prey generations. Predation generally increased prey allocation to defence and caused prey selection lines to become more diverse. On average, prey became most defensive in the high-resource environment and suffered from reduced resource use ability more in the low-resource environment. As a result, the evolution of stronger prey defence in the high-resource environment led to a strong decrease in predator-to-prey ratio. Predation increased temporal variability of populations and traits of prey. However, this destabilizing effect was less pronounced in the high-resource environment. Our results demonstrate that prey resource availability can shape the trade-off allocation of prey traits, which in turn affects multiple properties of the evolving predator-prey system.     

Individual variation in prey choice in a predator-prey community

One predator-two prey community models are studied with an emphasis on individual variation in predator behavior. The predator behaves according to a well-known prey choice model. The behavioral model predicts that predators should always attack the primary prey (more profitable prey of the two), but only attack the alternative prey (less profitable prey of the two) when the density of the primary prey is below a threshold density. The predator that accepts the alternative prey does not discriminate between the primary and alternative prey (all-or-nothing preference for the alternative prey). However, empirical studies do not result in clear all-or-nothing responses. Previous models examined the relaxation of the all-or-nothing response by assuming partial preference (e.g., predators preferentially forage on the primary prey even when they also attack the alternative prey). In this study, I consider individual variation in two predator traits (prey density perception and handling time) as the sources of the variation in the threshold density, which can make empirical data appear deviated from the expectation. I examine how community models with partial preference and individual variation differ in their dynamics and show that the differences can be substantial. For example, the dynamics of a model based on individual variation can be more stable (e.g., stable in a wider parameter region) than that of a model based on partial preference. As the general statistical property (Jensen’s inequality) is a main factor that causes the differences, the results of the study have general implications to the interpretation of models based on average per-capita rates.     

The limits of adaptation: humans and the predator-prey arms race

In the history of life, species have adapted to their consumers by evolving a wide variety of defenses. By contrast, animal species harvested in the wild by humans have not adapted structurally. Nonhuman predators have high failure rates at one or more stages of an attack, indicating that victim species have spatial refuges or phenotypic defenses that permit further functional improvement. A new compilation confirms that species in the wild cannot achieve immunity from human predation with structural defenses. The only remaining options are to become undesirable or to live in or escape to places where harvesting by people is curtailed. Escalation between prey defenses and predators' weapons may be restricted under human dominance to interactions involving those low-level predators that have benefited from human overexploitation of top consumers.     

Analysis of a predator-prey model with herd behavior and disease in prey incorporating prey refuge

In this work,we have introduced an eco-epidemiological model of an infected predator prey system.Incorporation of prey refuge gives that a fraction of the infected prey is available to the predator for consumption.Moreover,to make the model more realistic to the environment,we have introduced strong Allee effect in the susceptible population.Boundedness and positivity of the solution have been established.Local stability conditions of the equilibrium points have been found with the help of Routh-Hurwitz criterion and it has been observed that if a prey population is infected with a lethal disease,then both the prey(susceptible and infected)and predator cannot survive simultaneously in the system for any parametric values.The disease transmission rate and the attack rate on the susceptible have an important role to control the system dynamics.For different values of these two key parameters,we have got only healthy or disease-free or predation-free or a fluctuating disease-free or even a fluctuating predator-free system with some certain parametric conditions.     

Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline

We consider a special form of the Gause model of interactions between predator and prey populations. Using the ideas of Cheng, we prove the uniqueness of the limit cycle for more general systems, satisfying some additional conditions. These include also a condition due to Kuang and Freedman. Moreover, in this paper it is shown that the similar generalization of Cheng's uniqueness proof by Conway and Smoller is not correct.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Slight phenotypic variation in predators and prey causes complex predator-prey oscillations

Predator-prey oscillations are expected to show a 1/4-phase lag between predator and prey. However, observed dynamics of natural or experimental predator-prey systems are often more complex. A striking but hardly studied example are sudden interruptions of classic 1/4-lag cycles with periods of antiphase oscillations, or periods without any regular predator-prey oscillations. These interruptions occur for a limited time before the system reverts to regular 1/4-lag oscillations, thus yielding intermittent cycles. Reasons for this behaviour are often difficult to reveal in experimental systems. Here we test the hypothesis that such complex dynamical behaviour may result from minor trait variation and trait adaptation in both the prey and predator, causing recurrent small changes in attack rates that may be hard to capture by empirical measurements. Using a model structure where the degree of trait variation in the predator can be explicitly controlled, we show that a very limited amount of adaptation resulting in 10–15% temporal variation in attack rates is already sufficient to generate these intermittent dynamics. Such minor variation may be present in experimental predator-prey systems, and may explain disruptions in regular 1/4-lag oscillations.     

Social geographies of race: connecting race and space

This article examines what it means to study race from a spatial perspective. In order to make explicit connections between spatial and racial processes, we present an interdisciplinary snapshot of the work being done under the banners of space and race, and offer a framework – a theory of racial space – for linking these concepts. We provide four key characteristics of space – contested, fluid and historical, interactional and relational, and defined by inequality and difference – that overlap with prevalent theorizing on race and racialization. We suggest that a framework of racial space provides a language for explaining the persistence of racial inequality in its varied, and often subtle, forms today, and reveals a useful analytical and practical pathway for challenging and changing the existing racial order.     

Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment

The Lotka-Volterra predator-prey model with prey density dependence shows the final prey density to be independent of its vital rates. This result assumes the community to be well mixed so that encounters between predators and prey occur as a product of the landscape densities, yet empirical evidence suggests that over small spatial scales this may not be the normal pattern. Starting from an individual-based model with neighborhood interactions and movements, a deterministic approximation is derived, and the effect of local spatial structure on equilibrium densities is investigated. Incorporating local movements and local interactions has important consequences for the community dynamics. Now the final prey density is very much dependent on its birth, death, and movement rates and in ways that seem counterintuitive. Increasing prey fecundity or mobility and decreasing the coefficient of competition can all lead to decreases in the final density of prey if the predator is also relatively immobile. However, analysis of the deterministic approximation makes the mechanism for these results clear; each of these changes subtly alters the emergent spatial structure, leading to an increase in the predator-prey spatial covariance at short distances and hence to a higher predation pressure on the prey.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation

Interplay between predator and prey is a complex process in ecosystems due to its nature. The population dynamics can be affected by many extrinsic and intrinsic factors. In this paper, we make an attempt to uncover the effects from environmental disturbances when populations are subject to habitat complexity and aggregation effect. We firstly propose a stochastic predator-prey model with habitat complexity and aggregation efficiency for prey. We then mathematically analyze the model, to demonstrate the existence, uniqueness and the stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the existence of ergodic stationary distribution of the solution. We also establish sufficient conditions under which either only predator population dies out or the entire predator-prey model becomes extinct. Our theoretical and numerical results indicate that: (1) the environmental noises are disadvantage for the survival of biological populations; (2) when the density of prey is greater than one, prey aggregation can heighten the capability of predator species to capture prey and reduce the effect of environmental fluctuations, while when the density of prey is less than one, the results are opposite; (3) habitat complexity is propitious to the survival of prey population and may seriously threaten the persistence of the predator population.     

Consequences of size structure in the prey for predator-prey dynamics: the composite functional response

1. Current formulations of functional responses assume that the prey is homogeneous and independent of intraspecific processes. Most prey populations consist of different coexisting size classes that often engage in asymmetrical intraspecific interactions, including cannibalism, which can lead to nonlinear interaction effects. This may be important as the size structure with the prey could alter the overall density-dependent predation rates. 2. In a field experiment with damselfly and dragonfly larvae, 16 treatments manipulated the density of a small prey stage, the presence of large conspecific prey and the presence of heterospecific predators. 3. Size structure in the prey (i.e. when both prey stages were present) decreased the impact of the predator on overall prey mortality by 25-48% at mid and high prey densities, possibly due to density-dependent size-structured cannibalism in the prey. The predation rates on small prey stages were determined by the interaction of large prey and predators. Predation rates increased with prey density in the absence of large prey, but predation rates were constant across densities when large conspecifics were present. 4. The functional response for unstructured prey followed a Holling type III model, but the predation rate for size-structured prey was completely different and followed a complex pattern that could not be explained with any standard functional response. 5. Using additional laboratory experiments, a mortality model was developed and parameterized. It showed that the overall prey mortality of size-structured prey can be adequately predicted with a composite functional response model that modelled the individual functional responses of each prey stage separately and accounted for their cannibalistic interaction. 6. Thus, treating a prey population as a homogeneous entity will lead to erroneous predictions in most real-world food webs. However, if we account for the effects of size structure and the intraspecific interactions on functional responses by treating size classes as different functional groups, it is possible to reliably predict the dynamics of size-structured predator-prey systems.     

Plugging space into predator-prey models: an empirical approach

Extrapolating ecological processes from small-scale experimental systems to scales of natural populations usually entails a considerable increase in spatial heterogeneity, which may affect process rates and, ultimately, population dynamics. We demonstrate how information on the heterogeneity of natural populations can be taken into account when scaling up laboratory-derived process functions, using the technique of moment approximation. We apply moment approximation to a benthic crustacean predator-prey system, where a laboratory-derived functional response is made spatial by including correction terms for the variance in prey density and the covariance between prey and predator densities observed in the field. We also show how moment approximation may be used to incorporate spatial information into a dynamic model of the system. While the nonspatial model predicts stable dynamics, its spatial equivalent also produces bounded fluctuations, in agreement with observed dynamics. A detailed analysis shows that predator-prey covariance, but not prey variance, destabilizes the dynamics. We conclude that second-order moment approximation may provide a useful technique for including spatial information in population models. The main advantage of the method is its conceptual value: by providing explicit estimates of variance and covariance effects, it offers the possibility of understanding how heterogeneity affects ecological processes.     

Observations of predator-prey interaction on dispersal of an oligochaete prey,Limnodrilus hoffmeisteri

Clumped patches of the oligochaete, Limnodrilus hoffmeisteri, were subjected to predation by the leech, Erpobdella punctata to determine how the dispersion pattern might change. Two separate experiments showed that worms left the patch in 39% and 48% greater numbers when the leech predator was present. This suggests that L. hoffmeisteri uses a antipredatory strategy of dispersal rather than clumping.     

State-dependent invasion windows for prey in size-structured predator-prey systems: whole lake experiments

1. In size-structured communities where individuals grow in size over their life cycle, interactions between species will shift between competitive and predatory interactions depending on size relationships. The outcome of interactions will subsequently depend on the strength of competitive and predatory interactions, respectively. 2. In a whole lake experiment including four experimental lakes, it was tested under which conditions the competing prey, roach Rutilus rutilus, could successfully recruit into systems previously occupied by the predator, perch Perca fluviatilis. Two replicated introduction experiments were carried out 3 years apart. 3. Roach were able to successfully recruit into three of the four experimental lakes of which two were also inhabited by the top predator pike Esox lucius. Resource levels were unrelated to whether roach could successfully recruit into the systems as recruiting roach in all years were feeding close to their maximum rate. 4. High population fecundity of roach and low predation pressure by perch combined were necessary ingredients for successful recruitment and the presence of only one of these conditions did not result in successful recruitment. 5. It is hypothesized that, although roach were able to successfully recruit into one lake with only perch present in addition to the two lakes that also inhabited pike, long-term coexistence of roach and perch depends on the presence of another top predator (e.g. pike) selectively preying on perch. This hypothesis was supported by data on co-occurrence of perch and roach in different lakes. 6. Overall, the results are in accordance with expectation of size-structured life-history omnivory theory suggesting that coexistence between top predator and intermediate consumer is fragile.     

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.     

Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge

The non-linear behavior of a differential equations-based predator-prey model, incorporating a spatial refuge protecting a consant proportion of prey and with temperature-dependent parameters chosen appropriately for a mite interaction on fruit trees, is examined using the numerical bifurcation code AUTO 86. The most significant result of this analysis is the existence of a temperature interval in which increasing the amount of refuge dynamically destabilizes the system; and on part of this interval the interaction is less likely to persist in that predator and prey minimum population densities are lower than when no refuge is available. It is also shown that increasing the amount of refuge can lead to population outbreaks due to the presence of multiple stable states. The ecological implications of a refuge are discussed with respect to the biological control of mite pests.