Permanence of a stage-structured predator-prey system with a class of functional responses

A stage-structured predator-prey system incorporating a class of functional responses is presented in this article. By analyzing the system and using the standard comparison theorem, the sufficient conditions are derived for permanence of the system and non-permanence of predators.     

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.     

Effects of behavioral adaptation on a predator-prey model

The Volterra-Lotka predator-prey equations are modified so that the predator's ability to utilize the prey varies in proportion to the average number of encounters between the two species in the past. The behavior of this adaptive system is then described in terms of three parameters — the carrying capacity of the prey, the relative death rate of the predator, and the predator's memoryspan. The most stable situation is shown to occur when the carrying capacity of the prey is large, the predator's death rate is close to zero, and the predator is able to adapt quickly to changing levels of prey density.     

Behavioral responses to prey density by three acarine predator species with different degrees of polyphagy

Behavioral responses by three acarine predators, Phytoseiulus persimilis, Typhlodromus occidentalis, and Amblyseius andersoni (Acari: Phytoseiidae), to different egg and webbing densities of the spider mite Tetranychus urticae (Acari: Tetranychidae) on rose leaflets were studied in the laboratory. Prey patches were delineated by T. urticae webbing and associated kairomones, which elicit turning back responses in predators near the patch edge. Only the presence of webbing affected predator behavior; increased webbing density did not increase patch time. Patch time increased with increased T. urticae egg density in the oligophagous P. persimilis, but was density independent in the polyphagous species T. occidentalis and A. andersoni. Patch time in all three species was more strongly correlated with the number of prey encounters and attacks than with the actual prey number present in the patch. Patch time was determined by (a) the turning back response near the patch edge; this response decayed through time and eventually led to the abandonment of the patch, and (b) encounters with, and attacks upon, prey eggs; these prolonged patch time by both an increment of time spent in handling or rejecting prey and an increment of time spent searching between two successive prey encounters or attacks. Although searching efficiency was independent of prey density in all three species, the predation rate by P. persimilis decreased with prey density because its searching activity (i.e. proportion of total patch time spent in searching) decreased with prey density. Predation rates by T. occidentalis and A. andersoni decreased with prey density because their searching activity and success ratio both decreased with prey density. The data were tested against models of predator foraging responses to prey density. The effects of the degree of polyphagy on predator foraging behavior were also discussed.     

Global stability of predator-prey interactions

Summary A Lyapunov function is given that extends functions used by Volterra, Goh, and Hsu to a wide class of predator-prey models, including Leslie type models, and a biological interpretation of this function is given. It yields a simple stability criterion, which is used to examine the effect on stability of intraspecific competition among both prey and predators, of a refuge for the prey, and of Holling type II and type III functional responses. Although local stability analysis of these specific models has been done previously, the Lyapunov function facilitates study of global stability and domains of attraction and provides a unified theory which depends on the general nature of the interactions and not on the specific functions used to model them.     

How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

In Rosenzweig-MacArthur models of predator-prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator's functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator-prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents.     

带有食饵保护的Gause型扩散捕食系统的稳定性分析

考虑了具有扩散项和食饵保护的Gause型捕食系统.该模型带有齐次Neumann边界条件.讨论了系统的全局吸引性以及系统非负常数平衡态的局部稳定性和全局稳定性.其条件依赖于食饵保护参数,表明了食饵保护对系统动力学行为的影响.     

The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model

 A predator–prey interaction model based on a system of differential equations with temperature-dependent parameters chosen appropriately for a mite interaction on apple trees is analyzed to determine how the type of functional response influences bifurcation and stability behavior. Instances of type I, II, III, and IV functional responses are considered, the last of which incorporates prey interference with predation. It is shown that the model systems with the type I, II, and III functional responses exhibit qualitatively similar bifurcation and stability behavior over the interval of definition of the temperature parameter. Similar behavior is found in the system with the type IV functional response at low levels of prey interference. Higher levels of interference are destabilizing, as illustrated by the prevalence of bistability and by the presence of three attractors for some values of the model parameters. All four systems are capable of modeling population oscillations and outbreaks. Received 12 March 1996; received in revised form 25 October 1996     

The effect of prey size and prey density on the functional response,survival, growth and development of a predatory pentatomid bug,Podisus maculiventris say

Two experiments on the nymphal predation of Podisus maculiventris were conducted using Spodoptera litura larvae as prey. First experiment: The predator nymphs divided into three groups were reared individually from second instar to adult in a small vessel. Each nymph in the groups 1, 2 and 3 was allowed to attack the serially growing larvae (these were supplied at the rate of one per day) from 3-, 5- and 7-day old after hatching, respectively. The first prey used for the group 1 was so small that it was not only insufficient to satiate the predator but also was difficult to be searched out. But these disadvantages were soon recuperated due to the rapid growth of the prey and all nymphs could survive to adults. The survival rate of third and fourth instar nymphs in the group 3 was severely affected by vigorous counterattack of older prey larvae. Second experiment: The predator nymphs were individually reared either in a small vessel or in a large one at various rates of food supply (the prey larvae of 7-day old were used). The functional response curves obtained for each instar of the predator took a saturation type within a certain range of the prey density. The saturation level specific to each instar was generally higher for the predator reared in the large vessel than in the small one. The functional response of fourth and fifth instar nymphs was accelerated at a high prey density, viz. 16 larvae per vessel. Even at the low rate of food supply, viz. one larva per day per predator, the predator nymphs could survive to adults, but the size of resultant adults were abnormally small.     

An eco-epidemiological system with infected prey and predator subject to the weak Allee effect

In this article, we propose a general prey–predator model with disease in prey and predator subject to the weak Allee effects. We make the following assumptions: (i) infected prey competes for resources but does not contribute to reproduction; and (ii) in comparison to the consumption of the susceptible prey, consumption of infected prey would contribute less or negatively to the growth of predator. Based on these assumptions, we provide basic dynamic properties for the full model and corresponding submodels with and without the Allee effects. By comparing the disease free submodels (susceptible prey–predator model) with and without the Allee effects, we conclude that the Allee effects can create or destroy the interior attractors. This enables us to obtain the complete dynamics of the full model and conclude that the model has only one attractor (only susceptible prey survives or susceptible-infected coexist), or two attractors (bi-stability with only susceptible prey and susceptible prey–predator coexist or susceptible prey-infected prey coexists and susceptible prey–predator coexist). This model does not support the coexistence of susceptible-infected-predator, which is caused by the assumption that infected population contributes less or are harmful to the growth of predator in comparison to the consumption of susceptible prey.     

Encounters in predator-prey systems: a simple discrete model

Predator-prey systems are often described by exploitation models. These models can mimic experimental data very accurately, but it is sometimes difficult to realize the relationships between the models and the behavior of individual predator and prey animals. A simple discrete model is proposed here that tries to elucidate the connections between: the animals' movements, the predator/prey encounters; and the dynamics in the system as globally represented by the exploitation models. In these models, the term "area of discovery" plays an essential role. This term is shown to be a predictable coefficient that is composed of measurable physical properties of the analyzed predator-prey system. The model takes into account that predators and prey in experimental systems often do not search randomly but prefer some parts of the test area. The model is applied to the mite system Phytoseiulus persimilis/Tetranychus urticae under simple artificial conditions.     

Dynamical behaviour of a two-predator model with prey refuge

A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.     

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.     

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.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.     

Some results on global stability of a predator-prey system

In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Holling's type-3 functional response in some sense. We also discuss the global stability of Kolmogorov's model. Some examples are presented in the discussion section.Works partially supported by the National Science Council of the Republic of China     

Small prey size offers immunity to predation: a case study on two species of <Emphasis Type="Italic">Asplanchna</Emphasis> and three brachionid prey (Rotifera)

We tested the hypothesis that small prey can coexist with large predators. For this we confronted two predators (smaller Asplanchna brightwellii: 900 μm and larger A. sieboldi: 1400 μm) with three prey rotifers (smaller: Anuraeopsis fissa (70 μm); larger: Brachionus calyciflorus (200 μm) and intermediate: B. patulus (120 μm)) using functional response, prey preference, population growth and life table demography. Regardless of prey type, A. sieboldi was able to consume more prey than A. brightwellii and it consumed higher number of B. patulus than of B. calyciflorus or A. fissa. Prey preference experiments showed that A. brightwellii had no preference for B. calyciflorus regardless of prey density, while A. sieboldi preferred B. calyciflorus and avoided A. fissa. Data on population growth showed that A. brightwellii was always numerically more abundant than A. sieboldi. Prey type had a significant effect on peak abundances of A. sieboldi but not of A. brightwellii. Life table demography data revealed a significantly lower lifespan in A. brightwellii fed B. calyciflorus, compared to B. patulus, but not when compared to A. fissa. A. sieboldi lifespan was not affected by prey type. Depending on prey type and predator species, generation time varied from 2 to 3 days. Both lowest (0.38 d⁻¹) and highest (0.98 d⁻¹) population growth rates were observed in A. sieboldi. We suggest that reduced reproductive output in Asplanchna was caused by either large (B. calyciflorus) or small (A. fissa) prey. At natural densities of Anuraeopsis, it is unlikely that Asplanchna reaches abundances high enough to exterminate this prey. By its extremely small size (combining low energetic profitability with low encounter rates with predators) A. fissa may coexist with Asplanchna in nature. Dedicated to H. J. Dumont for his 65th year. Guest editors: S. S. S. Sarma, R. D. Gulati, R. L. Wallace, S. Nandini, H. J. Dumont and R. Rico-Martínez Advances in Rotifer Research     

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

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

The effects of refuges on predator-prey interactions: a reconsideration

Prey refuges are widely believed to prevent prey extinction and damp predator-prey oscillations. A review of the empirical evidence suggests that refuges are indeed capable of playing the former role. But the conditions under which they do so are not understood, nor is there any solid evidence for an effect on population fluctuations. The intuitive view that refuges act to stabilize equilibria and damp predator-prey oscillations is based in several theoretical studies of extremely simple models. Using a more realistic model, I show that several kinds of refuges can exert a locally destabilizing effect and create stable, large-amplitude oscillations which would damp out if no refuge was present. This finding contrasts sharply with the usual view. I argue that current evidence is tol weak, and the range of theoretically possible effects is too broad, to justify any simple characterization of refuge effects in nature. Manipulative empirical studies are an important first step toward correcting this situation, and I discuss some important factors to consider in their design.