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.     

Remarks on antipredator behavior and food chain dynamics

When consumers feeding on a resource spend time in avoiding high risks of predation, the predator functional response declines with predator density. While this is well established, less attention has been paid to the dependence of the consumer functional response on predator density. Here we show how the separation of behavioral and ecological timescales allows one to determine both responses starting from an explicit behavioral model. Within the general set-up considered in this paper, the two functional responses can tend toward Holling type II responses when consumers react only weakly to predation. Thus, the main characteristics of the standard Rosenzweig-MacArthur tritrophic food chain (logistic resource and Holling type II consumer and predator) remain valid also when consumers have weak antipredator behavior. Moreover, through numerical analysis, we show that in a particular but interesting case pronounced antipredator behaviors stabilize the system.     

Handling time promotes the coevolution of aggregation in predator-prey systems

Predators often have type II functional responses and live in environments where their life history traits as well as those of their prey vary from patch to patch. To understand how spatial heterogeneity and predator handling times influence the coevolution of patch preferences and ecological stability, we perform an ecological and evolutionary analysis of a Nicholson-Bailey type model. We prove that coevolutionarily stable prey and searching predators prefer patches that in isolation support higher prey and searching predator densities, respectively. Using this fact, we determine how environmental variation and predator handling times influence the spatial patterns of patch preferences, population abundances and per-capita predation rates. In particular, long predator handling times are shown to result in the coevolution of predator and prey aggregation. An analytic expression characterizing ecological stability of the coevolved populations is derived. This expression implies that contrary to traditional theoretical expectations, predator handling time can stabilize predator-prey interactions through its coevolutionary influence on patch preferences. These results are shown to have important implications for classical biological control.     

Habitat choice in predator-prey systems: spatial instability due to interacting adaptive movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey "chasing" each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.     

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.     

Impacts of Foraging Facilitation Among Predators on Predator-prey Dynamics

Whereas impacts of predator interference on predator-prey dynamics have received considerable attention, the “inverse” process—foraging facilitation among predators—have not been explored yet. Here we show, via mathematical models, that impacts of foraging facilitation on predator-prey dynamics depend on the way this process is modeled. In particular, foraging facilitation destabilizes predator-prey dynamics when it affects the encounter rate between predators and prey. By contrast, it might have a stabilizing effect if the predator handling time of prey is affected. Foraging facilitation is an Allee effect mechanism among predators and we show that for many parameters, it gives rise to a demographic Allee effect or a critical predator density in need to be crossed for predators to persist. We explore also the effects of predator interference, to make the picture “symmetric” and complete. Predator interference is shown to stabilize predator-prey dynamics once its strength is not too high, and thus corroborates results of others. On the other hand, there is a wide range of model parameters for which predator interference gives rise to three co-occurring co-existence equilibria. Such a multi-equilibrial regime is rather robust as we observe it for all the functional response types we explore. This is a previously unreported phenomenon which we show cannot occur for the Beddington–DeAngelis functional response. An interesting topic for future research thus might be to seek for general conditions on predator functional responses that would produce multiple co-existence equilibria in a predator-prey model.     

Body masses,functional responses and predator–prey stability

The stability of ecological communities depends strongly on quantitative characteristics of population interactions (type‐II vs. type‐III functional responses) and the distribution of body masses across species. Until now, these two aspects have almost exclusively been treated separately leaving a substantial gap in our general understanding of food webs. We analysed a large data set of arthropod feeding rates and found that all functional‐response parameters depend on the body masses of predator and prey. Thus, we propose generalised functional responses which predict gradual shifts from type‐II predation of small predators on equally sized prey to type‐III functional‐responses of large predators on small prey. Models including these generalised functional responses predict population dynamics and persistence only depending on predator and prey body masses, and we show that these predictions are strongly supported by empirical data on forest soil food webs. These results help unravelling systematic relationships between quantitative population interactions and large‐scale community patterns.     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

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.     

Trait‐mediated indirect interactions in a marine intertidal system as quantified by functional responses

Studies of trait‐mediated indirect interactions (TMIIs) typically focus on effects higher predators have on per capita consumption by intermediate consumers of a third, basal prey resource. TMIIs are usually evidenced by changes in feeding rates of intermediate consumers and/or differences in densities of this third species. However, understanding and predicting effects of TMIIs on population stability of such basal species requires examination of the type and magnitude of the functional responses exhibited towards them. Here, in a marine intertidal system consisting of a higher‐order fish predator, the shanny Lipophrys pholis, an intermediate predator, the amphipod Echinogammarus marinus, and a basal prey resource, the isopod Jaera nordmanni, we detected TMIIs, demonstrating the importance of habitat complexity in such interactions, by deriving functional responses and exploring consequences for prey population stability. Echinogammarus marinus reacted to fish predator diet cues by reducing activity, a typical anti‐predator response, but did not alter habitat use. Basal prey, Jaera nordmanni, did not respond to fish diet cues with respect to activity, distribution or aggregation behaviour. Echinogammarus marinus exhibited type II functional responses towards J. nordmanni in simple habitat, but type III functional responses in complex habitat. However, while predator cue decreased the magnitude of the type II functional response in simple habitat, it increased the magnitude of the type III functional response in complex habitat. These findings indicate that, in simple habitats, TMIIs may drive down consumption rates within type II responses, however, this interaction may remain de‐stabilising for prey populations. Conversely, in complex habitats, TMIIs may strengthen regulatory influences of intermediate consumers on prey populations, whilst potentially maintaining prey population stability. We thus highlight that TMIIs can have unexpected and complex ramifications throughout communities, but can be unravelled by considering effects on intermediate predator functional response types and magnitudes. Synthesis Higher‐order predators and habitat complexity can influence behaviour of intermediate species, affecting their consumption of prey through trait‐mediated indirect interactions (TMIIs). However, it is not clear how these factors interact to determine prey population stability. Using functional responses (FRs), relating predator consumption to prey density, we detected TMIIs in a marine system. In simple habitats, TMIIs reduced consumption rates, but FRs remained de‐stabilising for prey populations. In complex habitats, TMIIs strengthened prey regulation with population stabilizing FRs. We thus demonstrate that FRs can assess interactions of environmental and biological cues that result in complex and unexpected outcomes for prey populations.     

Functional responses of contrasting seed predator guilds to masting in two Mediterranean oak species

The predator satiation hypothesis poses that synchronous and variable seed production during masting events increases seed escape through seed predator satiation. The success of this strategy depends upon the type of consumer functional response, in this case defined as the change in seed consumption rate by a predator as a function of change in seed density. Type II (where the proportion of seed consumed is highest at low levels of seed availability) and type III (where the proportion of seed consumed is highest at some intermediate level of seed availability and then declines towards zero) functional responses describe negative density‐dependence and indicate predator satiation. The type of function response should be contingent upon herbivore traits: type II responses are predicted for dietary specialist predators with low mobility, and type III responses are predicted for highly mobile, dietary generalist predators. Surprisingly, most studies have not evaluated whether functional responses vary among seed predator guilds. Here we describe the functional responses at population and individual tree level of highly mobile generalist (birds and rodents) and less mobile specialist (insects) pre‐dispersal seed predators attacking acorns of two sympatric oaks (Quercus suber and Q. canariensis) over a 10‐year period. Our results showed that in most cases specialist seed predators exhibited the predicted type II functional response at both the individual tree and population level for both oak species. However, generalist seed predators did not exhibit the predicted type III response; instead, they also exhibited a type II response at the individual tree and population level for both oak species. By independently assessing the effects of multiple seed predators associated with the same host tree species, our work highlights the influence of herbivore traits on the outcome of plant–seed predator interactions in masting species, and thus furthers our understanding of the ecological and evolutionary mechanisms underlying masting behaviour.     

A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment

In this paper, a mathematical model of a prey-predator system is proposed to resolve the paradox of enrichment in ecosystems. The model is based on the natural strategy that a predator takes, i.e, it produces resting eggs in harsh environment. Our result gives a criterion for a functional response, which ensures that entering dormancy stabilizes the population dynamics. It is also shown that the hatching of resting eggs can stabilize the population dynamics when the switching between non-resting and resting eggs is sharp. Furthermore, the bifurcation structure of our model suggests the simultaneous existence of a stable equilibrium and a large amplitude cycle in natural enriched environments.      

Functional responses modified by predator density

Realistic functional responses are required for accurate model predictions at the community level. However, controversy remains regarding which types of dependencies need to be included in functional response models. Several studies have shown an effect of very high predator densities on per capita predation rates, but it is unclear whether this predator dependence is also important at low predator densities. We fit integrated functional response models to predation data from 4-h experiments where we had varied both predator and prey densities. Using an information theoretic approach we show that the best-fit model includes moderate predator dependence, which was equally strong even at low predator densities. The best fits of Beddington–DeAngelis and Arditi–Akçakaya functional responses were closely followed by the fit of the Arditi–Ginzburg model. A Holling type III functional response did not describe the data well. In addition, independent behavioral observations revealed high encounter rates between predators. We quantified the number of encounters between predators and the time the focal predator spent interacting with other individuals per encounter. This time “wasted” on conspecifics reduced the total time available for foraging and may therefore account for lower predation rates at higher predator densities. Our findings imply that ecological theory needs to take realistic levels of predator dependence into account.     

PREY ADAPTATION AS A CAUSE OF PREDATOR-PREY CYCLES

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.     

Modeling changes in predator functional response to prey across spatial scales

Extrapolation of predator functional responses from laboratory observations to the field is often necessary to predict predation rates and predator-prey dynamics at spatial and temporal scales that are difficult to observe directly. We use a spatially explicit individual-based model to explore mechanisms behind changes in functional responses when the scale of observation is increased. Model parameters were estimated from a predator-prey system consisting of the predator Delphastus catalinae (Coleoptera: Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) on tomato plants. The model explicitly incorporates prey and predator distributions within single plants, the search behavior of predators within plants, and the functional response to prey at the smallest scale of interaction (within leaflets) observed in the laboratory. Validation revealed that the model is useful in scaling up from laboratory observations to predation in whole tomato plants of varying sizes. Comparing predicted predation at the leaflet scale, as observed in laboratory experiments, with predicted predation on whole plants revealed that the predator functional response switches from type II within leaflets to type III within whole plants. We found that the magnitude of predation rates and the type of functional response at the whole plant scale are modulated by (1) the degree of alignment between predator and prey distributions and (2) predator foraging behavior, particularly the effect of area-concentrated search within plants when prey population density is relatively low. The experimental and modeling techniques we present could be applied to other systems in which active predators prey upon sessile or slow-moving species.     

Foraging facilitation among predators and its impact on the stability of predator–prey dynamics

Predator foraging facilitation may strongly influence the dynamics of a predator–prey system. This behavioral pattern is well-observed in real life interactions, but less is known about its possible impacts on the predator–prey dynamics. In this paper we analyze a modified Rosenzweig–MacArthur model, where a predator-dependent family of functions describing predator foraging facilitation is introduced into the Holling type II functional response. As the general assumption of foraging facilitation is that higher predator densities give rise to an increased foraging efficiency, we model predator facilitation with an increasing encounter rate function. Using the tools of bifurcation analysis we describe all the nonlinear phenomena that occur in the system provoked by foraging facilitation, these include the fold, Hopf, transcritial, homoclinic and Bogdanov–Takens bifurcation. We show that foraging facilitation can stabilize the coexistence in the predator–prey system for specific rates, but in most of the cases it can have fatal consequences for the predators themselves.     

Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential-integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey-unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick-von Foerster model with exponential growth rates determined by the Euler-Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).     

Experimental evidence for innate predator recognition in the Seychelles warbler.

Nest predation is a major determinant of fitness in birds and costly nest defence behaviours have evolved in order to reduce nest predation. Some avian studies have suggested that predator recognition is innate whereas others have stressed the importance of learning. However, none of these studies controlled for the genetic origin of the populations investigated and the effect of unfamiliarity with the predator. Here we determined whether experience with a nest predator is a prerequisite for nest defence by comparing predator recognition responses between two isolated but genetically similar Seychelles warbler (Acrocephalus sechellensis) populations, only one of which had experience of the egg predating Seychelles fody (Foudia sechellarum). Individuals in the predator-free population significantly reduced nest guarding compared to individuals in the population with the predator, which indicates that this behaviour was adjusted to the presence of nest predators. However, recognition responses (measured as both alarm call and attack rates) towards a mounted model of the fody were equally strong in both populations and significantly higher than the responses towards either a mounted familiar non-predator and a mounted, novel, non-predator bird species. Responses did not differ with a warbler's age and experience with the egg predator, indicating that predator recognition is innate.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.