A mathematical model with young predation

We generalise the model of [21] in which the author considered a predator-prey system with predators eating only the young ones (or eggs) of the prey species. The prime assumption of the present paper is that the birth rate (per unit individual per unit time) of predators depends not only on the current prey egg-level but also on all previous prey egg-levels. It is seen that under this assumption an otherwise stable system may be stable as well as unstable leading to the conclusion that young predation with time delay is less stable than without it. Finally for the model of [21] we prove a result which shows that large predation rates help in the co-existence of both predator and prey species.     

Effect of prey size on attack components of the functional response by Notonecta undulata

The number of encounters per prey, the proportion of encounters resulting in attacks, and the proportion of attacks that were successful were observed while fourth-instar Notonecta undulata nymphs preyed on smaller N. undulata nymphs. While encounters per prey and proportion of encounters resulting in attacks increased with prey size, the proportion of attacks that were successful decreased. The increase in encounter rate per prey was due in part to an increase in the predator's reactive distance to prey as prey size increased. While none of the attack parameters varied significantly with prey density, logarithmic regression of the number of encounters per unit search time on prey density suggested that prey density tends to have a positive effect on encounters per first-instar prey but a negative effect on encounters per second-instar prey. A functional response model is presented that incorporates components of the predator's attack rate as exponential functions of prey density and allows for effects of the time the predator may spend evaluating prey encountered but not attacked and time spent attacking prey not captured. Estimates of the attack parameters derived from the experimental data are used in the model to generate functional response curves for fourth-instar N. undulata preying on first- or second-instar conspecifics. The predicted curve for second-instar prey is typical type II but the curve for firstinstar prey is slightly positively density dependent at low prey densities, i.e., type III.     

Diurnal and nocturnal functional response of juvenile Notonecta maculata considered as a consequence of shifting predation behaviour

Presence or absence of light is considered to significantly affect predation within freshwater communities. In relation to light regime, the scope of the present study is to identify proximate factors accounting for different rates of predation in juvenile Notonecta maculata. It has been shown that foraging of juvenile N. maculata on Daphnia magna is reduced under dark conditions. These differences are accounted for by the presence/absence of light and are not regulated by any endogenous circadian rhythm. Direct observation of predation components revealed that in the dark, daphnid prey is detected at shorter distances, thus lowering the rate of encounter which finally results in a decreasing number of prey items eaten. Handling time was found to significantly increase during dark periods. Since the amount of food ingested per prey item increases to a certain extent with increasing handling time, it is suggested that the decreasing number of daphnids consumed is compensated by gaining a higher amount of food per item during a lengthier, more thorough, handling of the prey. A mechanistic model that describes the foraging process along a general predation cycle was parameterized based on the results of direct observations, instead of using classic functional response curves. This allows the comparison of model output to independent functional response data. A good correlation between observed and predicted data confirms the assumption that the reduction of the nocturnal predation rate is caused by shifting predation behaviour and indicates that the crucial light/dark differences in the foraging of N. maculata are considered in the approach.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

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

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis

Summary Some of the processes that influence the emigration of prey and predatory mites from bean plants were investigated experimentally. The emigration of the prey depends on the damage they cause to the plants and on predator density. The predator's emigration rate is a decreasing function of prey density, and does not change (or it slightly decreases) when prey and predator numbers are increased maintaining the same prey/predator ratio. The probability of emigration of the predators is independent of their own density when prey are absent and density dependent when prey density is kep constant. Forty three per cent of the variability in the predator's instantaneous rate of emigration in the different experiments is accounted for by a two parameter negative exponential function of capture rate (number of prey eaten per predator and per unit of time).     

ONTOGENETIC SHIFT IN PREY SIZE SELECTION AND ITS RELATION TO PROFITABILITY IN NUCELLA LAPILLUS

According to the energy maximization premise an animal shouldchoose its diet to maximize its net energy gain per unit handlingtime (E/T_b). Previous studies on adult marine gastropods haveshown this index of profitability to be a monotonicaUy increasingfunction of prey size, yet they select smaller, apparently sub-optimalprey. Laboratory experiments were used to investigate the profitabilityof different sized mussels, as prey, to juvenile Nucella lapillususing two separate criteria. First the ability to promote growthand second the more usual energy gain per unit handling time. The major difference between the two measures of prey valuewas that E/T_h predicted an optimum prey size larger than thegrowth rate model. Selection experiments were used to comparepreferences exhibited by developing Nucella with the spectraof profitability defined from the two models. In general thewhelks' choice of prey size was best predicted by the growthrate model of prey value. Field studies illustrated a functionalrelationship between size of mussel eaten and size of whelk,the mean size of mussel eaten again conformed more closely tomaximization of growth rate. The energetic return in the E/T_h model is generally taken asthe amount of food ingested, however gross growth efficiencywas found to be a decreasing function of ingestion rate. Sinceingestion rate was an increasing function of prey size, theE/T_h model overestimated the value of large prey by ignoringthe efficiency with which food is used by the predator.     

Age structure effects in predator-prey interactions

A mathematical model of predator-prey interactions is proposed which incorporates both age structure in the predators and density dependence in the prey. The properties of the model are investigated by a linearized analysis, which enables the conditions for stability to be formulated. The analysis indicates that for a substantial domain of parameter space, a stable equilibrium is possible with the prey well below its carrying capacity. The effect of violating the stability conditions on the behaviour of the model was investigated by computer simulation. Two further types of behaviour are noted in which coexistence is possible. The first is a two point limit cycle in which young and old predators occur in alternate time periods. The second involves a limit cycle in which the component population trajectories lie on closed curves in phase space.     

Scale dependent effects of predatory fish on stream benthos

In open predation experiments the effects of predators on prey densities can be influenced by predator consumption and by prey movements in to and out of experimental arenas. A published model predicts that the predator effects observed in such experiments are scale dependent over the scale range where there is a transition from movement control (of prey densities) to consumption control. The scale dependence follows from the assumption that per capita rate of emigration out of an experimental arena decreases with increasing arena size.
To test this model the effects of a small benthic fish ( Cottus gobio ) on densities of stream invertebrates was investigated in instream channels of different length (0.5, 2 and 8 m). The effect of fish predation was scale dependent for four prey taxa. For three of these taxa predator effects increased with experimental scale, which is in agreement with model predictions. However, this proved to be a case of "making the right prediction for the wrong reason" as the basic assumption of scale dependent emigration rate was not upheld. By analyzing the behaviour of the model, parameterized with emigration and consumption rates observed in the experimental channels, it was found that observed scale effects occurred because prey emigration in response to the predator treatment was modified by the experimental scale. Further analysis of the parameterized model suggested that the densities of most prey taxa were controlled by prey movements and not by consumption by the sculpins.     

Global analysis of a predator–prey model with variable predator search rate

We consider a modified Holling-type II predator–prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Holling-type II model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Holling-type II model. The main differences are that: 1. The paradox of enrichment which always occurs in the Holling-type II model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system.

     

The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Surface tension prey transport in shorebirds: how widespread is it?

Surface tension prey transport is a feeding mechanism employing the surface tension of water surrounding prey to transport prey from bill tip to mouth. Previously, it has been demonstrated only in the Red-necked Phalarope Phalaropus lobatus. On the basis of a model of the bill morphology necessary for this method of prey transport, I suggest that many species of shorebird should be capable of surface tension feeding. Laboratory investigations of the feeding mechanics of Wilson's Phalarope Phalaropus tricolor , Western Sandpiper Calidris mauri and Least Sandpiper Calidris minutilla demonstrated that all three use surface tension transport of prey when feeding in water. I examined interspecific variation in the performance of this feeding mechanism with a high-speed video system and a customized motion analysis system. Exploratory analyses indicated significant interspecific variation in distance the prey is transported per cycle of mandibular spreading, gape increase per unit transport, speed of transport, total number of cycles necessary to complete transport and total time to complete transport. The calidrid sandpipers also occasionally used other feeding mechanisms in conjunction with surface tension transport of prey. The discovery that these sandpipers, which normally obtain prey by probing, are capable of surface tension transport of prey implies that the capacity to employ this feeding mechanism may be widespread in the Scolopacidae and may have been a significant factor in the evolutionary radiation of phalaropes into aquatic environments.     

Numerical exploration of the parameter plane in a discrete predator–prey model

We propose a variant of the discrete Lotka–Volterra model for predator–prey interactions. A detailed stability and numerical analysis of the model are presented to explore the long time behaviour as each of the control parameter is varied independently. We show how the condition for survival of the predator depends on the natural death rate of predator and the efficiency of predation. The model is found to support different dynamical regimes asymptotically including predator extinction, stable fixed point and limit cycle attractors for co-existence of predator and prey and more complex dynamics involving chaotic attractors. We are able to locate exactly the domain of chaos in the parameter plane using a dimensional analysis.     

A powerful truncated tail strength method for testing multiple null hypotheses in one dataset

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.     

On a predator–prey system of Gause type

In this paper a Gause type model of interactions between predator and prey population is considered. We deal with the sufficient condition due to Kuang and Freedman in the generalized form including a kind of weight function. In a previous paper we proved that the existence of such weight function implies the uniqueness of limit cycle. In the present paper we give a new condition equivalent to the existence of a weight function (Theorem 4.4). As a consequence of our result, it is shown that some simple qualitative properties of the trophic function and the prey isocline ensure the uniqueness of limit cycle.     

Mechanistic modelling of gastric evacuation in predatory gadoids applying the square root model to describe surface‐dependent evacuation

A new model approach to gastric evacuation in predatory fishes was shown to give accurate and reliable estimates of evacuation. It is believed that such a model would prove particularly useful in understanding predator‐prey interactions in natural systems. The model is a simple, geometric abstraction of the square root model predicting that evacuation time is proportional to the square root of meal size in accordance with extensive empirical evidence. Digestive processes are assumed to be restricted to an outer surface of total stomach content that is represented by the curved side of a cylinder. This way, total stomach content is considered a cylinder of constant length the mass of which is gradually reduced by successive peeling off its side leaving the ends unaffected. The fundamental rate δ describes mean thickness of the layer that is peeled off the cylinder per time unit. In the model, δ is constant independent of cylinder thickness but inversely proportional to the square root of cylinder length. Anatomical and dynamic characteristics of the stomach render this formulation of δ plausible. Using mean evacuation rate over time during evacuation of a meal, the model disregards prey heterogeneities that influence instantaneous evacuation rate. Relationships between prey heterogeneities and evacuation patterns were indicated by application of a general power model to evacuation data, and it was suggested how the effects of prey characteristics might be incorporated into the square root model to produce a generic model of gastric evacuation.     

Optimal Bayesian foraging policies and prey population dynamics-some comments on Rodriguez-Girones and Vasquez

In this paper we show the density-dependent harvest rates of optimal Bayesian foragers exploiting prey occurring with clumped spatial distribution. Rodríguez-Gironés and Vásquez (1997) recently treated the issue, but they used a patch-leaving rule (current value assessment rule) that is not optimal for the case described here. An optimal Bayesian forager exploiting prey whose distribution follows the negative binomial distribution should leave a patch when the potential (and not instantaneous) gain rate in that patch equals the best long-term gain rate in the environment (potential value assessment rule). It follows that the instantaneous gain rate at which the patches are abandoned is an increasing function of the time spent searching in the patch. It also follows that the proportion of prey harvested in a patch is an increasing sigmoidal function of the number of prey initially present. In this paper we vary several parameters of the model to evaluate the effects on the forager's intake rate, the proportion of prey harvested per patch, and the prey's average mortality rate in the environment. In each case, we study an intake rate maximizing forager's optimal response to the parameter changes. For the potential value assessment rule we find that at a higher average prey density in the environment, a lower proportion of the prey is taken in a patch with a given initial prey density. The proportion of prey taken in a patch of a given prey density also decreases when the variance of the prey density distribution is increased and if the travel time between patches is reduced. We also evaluate the effect of using predation minimization, rather than rate maximization, as the currency. Then a higher proportion of the prey is taken for each given initial prey density. This is related to the assumption that traveling between patches is the most risky activity. Compared to the optimal potential value assessment rule, the current value assessment rule performs worse, in terms of long-term intake rate achieved. The difference in performance is amplified when prey density is high or highly aggregated. These results pertain to the foraging patch spatial scale and may have consequences for the spatial distribution of prey in the environment.     

The role of virus infection in a simple phytoplankton zooplankton system

Many planktonic species show spectacular bursts ("blooms") in population density. Though viral infections are known to cause behavioural and other changes in phytoplankton and other aquatic species, yet their role in regulating the phytoplankton population is still far from being understood. To study the role of viral diseases in the planktonic species, we model the phytoplankton-zooplankton system as a prey-predator system. Here the prey (phytoplankton) species is infected with a viral disease that divides the prey population into susceptible and infected classes, with the infected prey being more vulnerable to predation by the predator (zooplankton). The dynamical behaviour of the system is investigated from the point of view of stability and persistence both analytically and numerically. The model shows that infection can be sustained only above a threshold of force of infection, and, there exists a range in the infection rate where this system shows "bloom"-like stable limit cycle oscillations. The time series of natural "blooms" with different types of irregular oscillations can arise in this model simply from a biologically realistic feature, i.e., by the random variation of the epidemiological parameter (rate of infection) in the infected prey population. The difference in mean strength of infection alone can lead to the different types of patterns observed in natural planktonic blooms.     

The functional response of a predatory plant preying on swarming zooplankton

In a laboratory study, we determined the functional response of the carnivorous aquatic plant Utricularia vulgaris feeding on Polyphemus pediculus, a cladoceran zooplankton that forms swarms. The number of prey eaten increased linearly with prey density up to a density of 35 prey per 125 ml and decreased slightly above this density. Independent estimates of handling time showed that the number eaten was not limited by handling. Thus, we hypothesized that the functional response levelled off because attack rate decreased with increasing density. Direct observations of the predation act at high and low prey densities showed that prey per capita mortality rate was markedly lower at high densities. An analysis of the components of the predation cycle showed that encounter rate and attack probability but not capture success decreased with increasing prey density. We, then, studied the degree of aggregation and the movement behaviour of Polyphemus . The tendency to form swarms increased with density and this was associated with reduced swimming speed and swimming along a more tortuous path. Presence of Utricularia leaves did not influence the spatial distribution and swimming behaviour of Polyphemus . We concluded that the unusual shape of the functional response was due to density dependent prey mortality rates that resulted from a density dependent tendency to form swarms. We, therefore, suggested a modification of Holling's type II functional response model that included density dependent attack rate and this model fitted data significantly better than the original model.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.