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

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

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.     

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.     

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.     

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.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

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.     

Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges

In this paper, the effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using the analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We will evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field.     

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.     

带有年龄结构的捕食者-食饵模型研究

研究带有年龄结构的捕食者-食饵模型的渐近行为.本文所研究的模型假定捕食者从幼年阶段到成年阶段的转变率依赖于幼年种群的密度,还假定幼年捕食者捕食食饵.本文最终给出了有年龄结构的捕食者-食饵模型的捕食者持久和灭绝的若干条件.     

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.     

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.     

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.     

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.     

Epidemiological models with age structure,proportionate mixing,and cross-immunity

Infection by one strain of influenza type A provides some protection (cross-immunity) against infection by a related strain. It is important to determine how this influences the observed co-circulation of comparatively minor variants of the H1N1 and H3N2 subtypes. To this end, we formulate discrete and continuous time models with two viral strains, cross-immunity, age structure, and infectious disease dynamics. Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age-specific, even if activity levels are independent of age. Sustained oscillations do not seem possible for a single-strain model, even in the presence of age-specific mortalities; and thus it is suggested that the interplay between cross-immunity and age-specific mortalities may underlie observed oscillations.     

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     

Hydrodynamic mediation of predator-prey interactions: differential patterns of prey susceptibility and predator success explained by variation in water flow

In most shallow water marine systems, fluid movements vary on scales that may influence local community dynamics both directly, through changes in the abundance of species, and indirectly, by modifying important behaviors of organisms. We examined how differences in current speed affect the outcome of predator-prey interactions for two species of marine benthic predators (knobbed whelks, Busycon carica, and blue crabs, Callinectes sapidus) foraging on two common prey species (bay scallops, Argopecten irradians, and hard clams, Mercenaria mercenaria). The predators differ in their foraging strategies and prey in their potential escape responses. Predation by blue crabs, highly mobile predators/scavengers that rely upon chemical odors transported in the water column to locate prey, could be strongly affected by changes in current speed and turbulent mixing because their foraging strategy relies on a high degree of spatial integration of prey odor plumes. Whelks, slow moving, predatory gastropods that often forage with their bodies buried in the sediment, may be less susceptible to flow-induced distortion of prey odor plumes because their sluggish movements result in a high degree of temporal integration of prey odors. Bay scallops, relatively mobile bivalves capable of rapid short-distance swimming burst, and hard clams, sedentary bivalves, have been shown to respond to varying degrees to predator odors that are dispersed in the water column. Flow regime for the predator-prey experiments was manipulated in situ using large channels. Predation by blue crabs on both juvenile hard clams and bay scallops decreased with increases in water flow (0-12 vs. 0-30 cm s⁻¹). Whelk predation on bay scallops increased with increases in water flow, whereas predation by whelks on hard clams did not differ between flow regimes. For blue crabs movement decreased at periods of high water flow. Because blue crabs locate prey through chemolocation of water-borne cues, which are diluted rapidly at higher flows, decreases in foraging may result from the inability to successfully detect prey at enhanced flows. Differences in predation by whelks could not be explained by a similar mechanism. Visual observations of foraging whelks revealed no differences in whelk behavior between the two flow regimes. The pattern of higher whelk predation on scallops at enhanced flow is likely to be related to a flow-inhibiting ability of scallops to detect predator approach. Thus, flow enhancement interferes with three of the predator-prey systems but the effect on predator success depends on whether the predator or prey is most affected.     

The dynamics of a Lotka–Volterra predator–prey model with state dependent impulsive harvest for predator

According to the economic and biological aspects of renewable resources management, we propose a Lotka–Volterra predator–prey model with state dependent impulsive harvest. By using the Poincaré map, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three under some conditions. Numerical results are carried out to illustrate the feasibility of our main results. The bifurcation diagrams of periodic solutions are obtained by using the numerical simulations, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations, which implies that the presence of pulses makes the dynamic behavior more complex.