Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Functional response of Euseius concordis to densities of different developmental stages of the cassava green mite

Both prey density and developmental stage of pests and natural enemies are known to influence the effectiveness of biological control. However, little is known about the interaction between prey density and population structure on predation and fecundity of generalist predatory mites. Here, we evaluated the functional response (number of prey eaten by predator in relation to prey density) of adult females and nymphs of the generalist predatory mite Euseius concordis to densities of different developmental stages of the cassava green mite Mononychellus tanajoa, as well as the fecundity of adult females of the predator. We further assessed the instantaneous rate of increase, based on fecundity and mortality, of E. concordis fed on eggs, immatures and adults of M. tanajoa. Overall, nymphs and adults of E. concordis feeding on eggs, immatures and females of M. tanajoa had a type III functional response curve suggesting that the predator increased prey consumption rate as prey density increased. Both nymphs and adult females of the predator consumed more eggs than immatures of M. tanajoa from the density of 20 items per leaf disc onwards, revealing an interaction between prey density and developmental stage in the predatory activity of E. concordis. In addition, population growth rate was higher when the predator fed on eggs and immatures in comparison with females. Altogether our results suggest that E. concordis may be a good candidate for the biological control of M. tanajoa populations. However, the efficiency of E. concordis as a biological control agent of M. tanajoa is contingent on prey density and population structure.     

Effect of disease-selective predation on prey infected by contact and external sources

We propose and analyze a simple mathematical model for susceptible prey (S)–infected prey (I)–predator (P) interaction, where the susceptible prey population (S) is infected directly from external sources as well as through contact with infected class (I) and the predator completely avoids consuming the infected prey. The model is analyzed to obtain different thresholds of the key parameters under which the system exhibits stability around the biologically feasible equilibria. Through numerical simulations we display the effects of external infection and the infection through contact on the system dynamics in the absence as well as in the presence of the predator. We compare the system dynamics when infection occurs only through contact, with that when it occurs through contact and external sources. Our analysis demonstrates that under a disease-selective predation, stability and oscillations of the system is determined by two key parameters: the external infection rate and the force of infection through contact. Due to the introduction of external infection, the predator and the prey population show limit-cycle oscillations over a range parametric values. We suggest that while predicting the dynamics of such an eco-epidemiological system, the modes of infection and the infection rates might be carefully investigated.     

Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator-prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator-prey system, Science 290 (2000), pp. 1358-1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Implications of floral resources for predation by an omnivorous lacewing

Understanding the relationship between a predator and its prey requires consideration of the other food resources used by the predator. In the case of true omnivores, these include plant-provided foods such as leaf tissue and nectar. The presence of plant resources can increase or decrease predation depending on the degree to which they are complementary to, or substitutable for, the prey. This has implications for the role of omnivores in biological control and some groups, notably heteropteran bugs and phytoseiid mites, have been studied in this context. However, few experiments have considered the effects of plant resources both on prey consumption by individual omnivores (which have an immediate effect on the prey population) and on attributes such as longevity and fecundity which act over the longer term to affect predation at the population level. In this study, a model system comprising an omnivorous adult lacewing (Micromus tasmaniae), buckwheat flowers and aphid prey was used to investigate how floral resources affected per capita predation rate, longevity and fecundity of the lacewing. Flowers reduced prey consumption. In the absence of prey, longevity was higher for lacewings with flowers than those without. In an experiment where aphids were provided in abundance, lacewing fecundity was unaffected by flowers. However, when aphids were less abundant, providing flowers decreased the pre-oviposition period and increased the daily oviposition rate. The results demonstrate that floral resources can mediate omnivore–prey relationships and that, in the context of biological control, their effects may be either positive or negative.     

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.     

Reciprocity in predator–prey interactions: exposure to defended prey and predation risk affects intermediate predator life history and morphology

A vast body of literature exists documenting the morphological, behavioural and life history changes that predators induce in prey. However, little attention has been paid to how these induced changes feed back and affect the predators’ life history and morphology. Larvae of the phantom midge Chaoborus flavicans are intermediate predators in a food web with Daphnia pulex as the basal resource and planktivorous fish as the top predator. C. flavicans prey on D. pulex and are themselves prey for fish; as D. pulex induce morphological defences in the presence of C. flavicans this is an ideal system in which to evaluate the effects of defended prey and top predators on an intermediate consumer. We assessed the impact on C. flavicans life history and morphology of foraging on defended prey while also being exposed to the non-lethal presence of a top fish predator. We tested the basic hypothesis that the effects of defended prey will depend on the presence or absence of top predator predation risk. Feeding rate was significantly reduced and time to pupation was significantly increased by defended morph prey. Gut size, development time, fecundity, egg size and reproductive effort respond to fish chemical cues directly or significantly alter the relationship between a trait and body size. We found no significant interactions between prey morph and the non-lethal presence of a top predator, suggesting that the effects of these two biological factors were additive or singularly independent. Overall it appears that C. flavicans is able to substantially modify several aspects of its biology, and while some changes appear mere consequences of resource limitation others appear facultative in nature.     

An impulsively controlled pest management model with n predator species and a common prey

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.     

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.     

Cannibalism in an age-structured predator-prey system

Recently, Kohlmeier and Ebenhöh showed that cannibalism can stabilize population cycles in a Lotka-Volterra type predator-prey model. Population cycles in their model are due to the interaction between logistic population growth of the prey and a hyperbolic functional response. In this paper, we consider a predator-prey system where cyclic population fluctuations are due to the age structure in the predator species. It is shown that cannibalism is also a stabilizing mechanism when population oscillations are due to this age structure. We conclude that in predator-prey systems, cannibalism by predators can stabilize both externally generated (consumer-resource) as well as internally generated (agestructure) fluctuations.     

Predator overcomes the Allee effect due to indirect prey–taxis

A mathematical model for spatiotemporal dynamics of prey–predator system was studied by means of linear analysis and numerical simulations. The model is a system of PDEs of taxis–diffusion–reaction type, accounting for the ability of predators to detect the locations of higher prey density, which is formalized as indirect prey–taxis, according to hypothesis that the taxis stimulus is a substance being continuously emitted by the prey, diffusing in space and decaying with constant rate in time (e.g. odour, pheromone, exometabolit). The local interactions of the prey and predators are described by the classical Rosenzweig – MacArthur system, which is modified in order to take into account the Allee effect in the predator population. The boundary conditions determine the absence of fluxes of population densities and stimulus concentration through the habitat boundaries. The obtained results suggest that the prey–taxis activity of the predator can destabilize both the stationary and periodic spatially-homogeneous regimes of the species coexistence, causing emergence of various heterogeneous patterns. In particular, it is demonstrated that formation of local dense aggregations induced by prey–taxis allows the predators to overcome the Allee effect in its population growth, avoiding the extinction that occurs in the model in the absence of spatial effects.     

Complexity in a prey-predator model with prey refuge and diffusion

Prey-predator interaction is one of the most commonly observed relationships in ecosystem. In the study of prey-predator models, it is frequently assumed that the changes in population densities are only time-dependent and the dynamics is generally represented by coupled nonlinear ordinary differential equations. In natural system, however, either prey or predator or both move from one place to another for various reasons. In such a case, their dynamic interaction depends both on time and space and requires coupled nonlinear partial differential equations for its dynamic representation. It is also well documented that prey refuges affect the interaction between prey and predator significantly. In this paper, we studied the dynamics of a diffusive prey-predator interaction with prey refuge and type III response function. We have considered both one and two dimensional diffusivity in the model system and presented different stability results under the assumptions that one or both species may be mobile or sedentary. Our results showed that the system may exhibit different spatiotemporal (non-Turing) patterns, like spiral waves, patchy structures, spot pattern, or even spatiotemporal chaos depending on the refuge availability and diffusion rate of species. Another interesting finding was that the dynamic complexity in a prey-predator model increases in case of mobile predator and sedentary prey compare to mobile prey and sedentary predator while refuge availability is varied.     

Influence of density dependence on predator-prey seabird interactions at large spatio-temporal scales

Theoretical investigations of competitive dynamics have noted that numbers of predator and prey influence each other. However, few empirical studies have demonstrated how a life-history trait of the prey (such as fecundity) can be affected simultaneously by its own density and the density of predators. For instance, density dependence can reduce fecundity with increasing number of prey, while inverse density dependence or Allee effects may occur especially when the prey is a social organism. Here we analysed an intraguild predator-prey system of two seabird species at a large spatio-temporal scale. As expected, we found that fecundity of prey was negatively affected by predator density. Nevertheless, fecundity of prey also increased nonlinearly with its own density and strikingly with the prey-predator ratio. Small groups of prey were probably not able to defend their nests especially against large number of predators. At the highest prey densities (i.e. when anti-predator strategies should be most efficient), prey fecundity also lowered, suggesting the appearance of density dependence mediated by food competition. Allee effects and density dependence occurred across a broad range of population sizes of both the prey and the predator at several local populations facing different ecological environments.     

Interaction strength between a predator and dangerous prey: Sinistrofulgur predation on Mercenaria

The lack of direct empirical evidence of predator evolution in response to prey adaptation is a fundamental weakness of the arms race analogy of predator-prey coevolution. I examined the interaction between the predatory busyconine whelk Sinistrofulgur sinistrum and its bivalve prey Mercenaria mercenaria to evaluate whether reciprocal adaptation was likely in this predator-prey system. Thick-lipped whelks use their shell lip to chip open the shell of their prey, often resulting in breakage to their own shell. Thus, hard-shelled prey, such as Mercenaria, may be considered dangerous because they are able to inflict damage to the predator as a consequence of the interaction. The strength of interaction between whelks and their bivalve prey was viewed by regressing predator performance (the incidence of shell breakage in encounters with prey) on prey phenotype (a function of size). Interaction with Mercenaria of varying sizes has strong and predictable consequences (r²=0.946; p=0.028) for Sinistrofulgur. Predators that select large, thick bivalve prey increase the likelihood that their shell lip will be broken in the process of attempting to open their prey. Ecological consequences of feeding-induced breakage may include reduced growth rate, reproductive success, and survivorship. These results suggest that natural selection should favor predator phenotypes that reduce feeding-induced breakage when interactions with damage-inducing prey occur.     

Oscillations in a size-structured prey-predator model

This article introduces a predator–prey model with the prey structured by body size, based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator–prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka–Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey’s population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining with prey population size. We find that predation, even size-non-specific linear predation can destabilize a stable prey-only equilibrium, if reproduction is size specific and limited to individuals of large enough size. Furthermore, we show that size-specific predation can also destabilize the predator–prey equilibrium in the PDE model. We surmise that size-specific predation allows for temporary prey escape which is responsible for destabilization in the predator–prey dynamics.     

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.     

Differential effects of native vs. invasive predators on a common Caribbean reef fish

Predators may have consumptive (lethal) and non-consumptive (sub-lethal) effects on prey. Non-consumptive effects include altered behavior and reduced growth and fecundity. Native prey may not recognize non-native predators as a threat, and therefore may suffer pronounced effects. Additionally, non-native predators may elicit different behavioral responses from prey compared to native predators. Theory predicts that consumptive effects should be greater for non-native predators (due to prey naiveté), and non-consumptive effects should be greater for native predators (due to predator recognition). To test these hypotheses, I monitored bicolor damselfish (Stegastes partitus) in the presence of invasive predatory Pacific lionfish (Pterois spp.), a native predator (graysby, Cephalopholis cruentata), and an egg predator (bluehead wrasse, Thalassoma bifasciatum). Body size and location of lionfish and graysby were monitored on reefs in the Bahamas. Bicolor fecundity was measured as the number and size of egg-masses that individual fish laid. Bicolor fecundity was negatively correlated with lionfish density but not graysby or bluehead density. Neither predator had a detectable effect on bicolor body size, but lionfish density was negatively correlated with the size of mature adult damselfish. I observed behavioral responses of bicolors to the two piscivores, to bluehead wrasse, and to two herbivorous fishes (Acanthurus coeruleus, Scarus spp.) as non-aggressive controls. Bicolors changed behavior (feeding and aggression) in the presence of all native fishes, but not in the presence of lionfish. Thus, differential effects exist between native and non-native predators, and invasive lionfish pose a non-consumptive threat to bicolor damselfish via reduced growth and fecundity.     

Decomposing Predation: Testing for Parameters that Correlate with Predatory Performance by a Social Bacterium

Predator–prey interactions presumably play major roles in shaping the composition and dynamics of microbial communities. However, little is understood about the population biology of such interactions or how predation-related parameters vary or correlate across prey environments. Myxococcus xanthus is a motile soil bacterium that feeds on a broad range of other soil microbes that vary greatly in the degree to which they support M. xanthus growth. In order to decompose predator–prey interactions at the population level, we quantified five predation-related parameters during M. xanthus growth on nine phylogenetically diverse bacterial prey species. The horizontal expansion rate of swarming predator colonies fueled by prey lawns served as our measure of overall predatory performance, as it incorporates both the searching (motility) and handling (killing and consumption of prey) components of predation. Four other parameters—predator population growth rate, maximum predator yield, maximum prey kill, and overall rate of prey death—were measured from homogeneously mixed predator–prey lawns from which predator populations were not allowed to expand horizontally by swarming motility. All prey species fueled predator population growth. For some prey, predator-specific prey death was detected contemporaneously with predator population growth, whereas killing of other prey species was detected only after cessation of predator growth. All four of the alternative parameters were found to correlate significantly with predator swarm expansion rate to varying degrees, suggesting causal interrelationships among these diverse predation measures. More broadly, our results highlight the importance of examining multiple parameters for thoroughly understanding the population biology of microbial predation.     

Life-history trade-off in two predator species sharing the same prey: a study on cassava-inhabiting mites

In cassava fields, two species of predatory mites, Typhlodromalus aripo and T. manihoti, co-occur at the plant level and feed on Mononychellus tanajoa , a herbivorous mite. The two predator species are spatially segregated within the plant: T. manihoti dwells on the middle leaves, while T. aripo occurs in the apices of the plant during the day and moves to the first leaves below the apex at night.
To monitor the prey densities experienced by the two predator species in their micro-environment, we assessed prey and predator populations in apices and on the leaves of cassava plants in the field. Prey densities peaked from November to January and reached the lowest levels in July. They were higher on leaves than in the apices. To test whether the life histories of the two predator species are tuned to the prey density they experience, we measured age-specific fecundity and survival of the two predators under three prey density regimes (1 prey female/72 h, 1 prey female/24 h and above the predators level of satiation). T. manihoti had a higher growth rate than T. aripo at high prey densities, mainly due to its higher fecundity. T. aripo had a higher growth rate at low prey density regimes, due to its late fecundity and survival. Thus, each of the two species perform better under the prey density that characterizes their micro-habitat within the plant.