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.     

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.     

Unique coevolutionary dynamics in a predator-prey system

In this paper, we study the predator-prey coevolutionary dynamics when a prey's defense and a predator's offense change in an adaptive manner, either by genetic evolution or phenotypic plasticity, or by behavioral choice. Results are: (1) The coevolutionary dynamics are more likely to be stable if the predator adapts faster than the prey. (2) The prey population size can be nearly constant but the predator population can show very large amplitude fluctuations. (3) Both populations may oscillate in antiphase. All of these are not observed when the handling time is short and the prey's density dependence is weak. (4) The population dynamics and the trait dynamics show resonance: the amplitude of the population fluctuation is the largest when the speed of adaptation is intermediate. These results may explain experimental studies with microorganisms.     

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     

The effect of colored noise on spatiotemporal dynamics of biological invasion in a diffusive predator-prey system

Spatiotemporal dynamics of a predator-prey system is considered under the assumption that the predator is sensitive to colored noise. Mathematically, the model consists of two coupled diffusion-reactions. By means of extensive numerical simulations, the complex invasion pattern formations of the system are identified. The results show that a geographical invasion emerges without regional persistence when the intensity of colored noise is small. Remarkably, as the noise intensity increases, the species spreads via a patchy invasion only when the system is affected by red noise. Meanwhile, the relationship between local stability and global invasion is also considered. The predator, which becomes extinct in the system without diffusion, could invade locally when the system is affected by white noise. However, the local invasion is not followed by geographical spread.     

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.     

时标上一捕食二食饵系统的周期解（英文）

研究时标上一捕食二食饵系统.运用时标上Gaines和Mawhin的连续拓扑度定理,得到了系统存在周期解的新的充分条件.其研究方法可以广泛地运用来研究微分或者差分方程的周期解存在性问题.     

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.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect

Invasion of an exotic species initiated by its local introduction is considered subject to predator-prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion-reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.     

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.     

Differential impact of adults and nymphs of a generalist predator on an exotic invasive pest demonstrated by molecular gut-content analysis

Generalist predators have the capacity to regulate herbivore populations through a variety of mechanisms, but food webs are complex and defining the strength of trophic linkages can be difficult. Molecular gut-content analysis has revolutionized our understanding of these systems. Utilizing this technology, we examined the structure of a soybean food web, identified the potential for adult and immature Orius insidiosus (Hemiptera: Anthocoridae) to suppress Aphis glycines (Hemiptera: Aphididae), and tested the hypotheses that foraging behaviour would vary between life stages, but that both adults and immatures would exert significant predation pressure upon this invasive pest. We also identified the strength of trophic pathways with two additional food items: an alternative prey item, Neohydatothrips variabilis (Thysanoptera: Thripidae), and an intraguild predator, Harmonia axyridis (Coleoptera: Coccinellidae). A. glycines constituted a greater proportion of the diet of immature O. insidiosus, but N. variabilis DNA was found in greater frequency in adults. However, both life stages were important early-season predators of this invasive pest, a phenomenon predicted as having the greatest impact on herbivore population dynamics and establishment success. No adult O. insidiosus screened positive for H. axyridis DNA, but a low proportion (2.5%) of immature individuals contained DNA of this intraguild predator, thus indicating the existence of this trophic pathway, albeit a relatively minor one in the context of biological control. Interestingly, approximately two-thirds of predators contained no detectable prey and fewer than 3% contained more than one prey item, suggesting the possibility for food limitation in the field. This research implicates O. insidiosus as a valuable natural enemy for the suppression of early-season A. glycines populations.     

Role of infection on the stability of a predator-prey system with several response functions--a comparative study

In this paper, we have proposed and analyzed a mathematical model of an infected predator-prey system with different predators' functional response. The existence and uniqueness of solutions are established and solutions are shown to be uniformly bounded for all nonnegative initial values. Our overall mathematical and biological studies reveal that if the prey population is infected by a lethal disease, coexistence of all three species (i.e. host, parasite and predator) for any of three functional responses is never possible but different interesting dynamical behaviors are possible by varying two key parameters viz. the rate of infection and the attack rate on susceptible prey. Interplay between these two factors yields a diverse array of biologically relevant behavior, including switching of stability, extinction and oscillations.     

Identifying the predator complex of <Emphasis Type="Italic">Homalodisca vitripennis</Emphasis> (Hemiptera: Cicadellidae): a comparative study of the efficacy of an ELISA and PCR gut content assay

A growing number of ecologists are using molecular gut content assays to qualitatively measure predation. The two most popular gut content assays are immunoassays employing pest-specific monoclonal antibodies (mAb) and polymerase chain reaction (PCR) assays employing pest-specific DNA. Here, we present results from the first study to simultaneously use both methods to identify predators of the glassy winged sharpshooter (GWSS), Homalodisca vitripennis (Germar) (Hemiptera: Cicadellidae). A total of 1,229 arthropod predators, representing 30 taxa, were collected from urban landscapes in central California and assayed first by means of enzyme-linked immunosorbent assay (ELISA) using a GWSS egg-specific mAb and then by PCR using a GWSS-specific DNA marker that amplifies a 197-base pair fragment of its cytochrome oxidase gene (subunit I). The gut content analyses revealed that GWSS remains were present in 15.5% of the predators examined, with 18% of the spiders and 11% of the insect predators testing positive. Common spider predators included members of the Salticidae, Clubionidae, Anyphaenidae, Miturgidae, and Corinnidae families. Common insect predators included lacewings (Neuroptera: Chrysopidae), praying mantis (Mantodea: Mantidae), ants (Hymenoptera: Formicidae), assassin bugs (Hemiptera: Reduviidae), and damsel bugs (Hemiptera: Nabidae). Comparison of the two assays indicated that they were not equally effective at detecting GWSS remains in predator guts. The advantages of combining the attributes of both types of assays to more precisely assess field predation and the pros and cons of each assay for mass-screening predators are discussed.     

Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.     

Impact of antimicrobial usage on the transmission dynamics of antimicrobial resistant bacteria among pigs

There is increasing evidence showing that antimicrobial consumption provides a powerful selective force that promotes the emergence of resistance in pathogenic, commensal as well as zoonotic bacteria in animals. The main aim of this study was to develop a modeling framework that can be used to assess the impact of antimicrobial usage in pigs on the emergence and transmission of resistant bacteria within a finisher pig farm. The transmission dynamics of drug-sensitive and drug-resistant bacteria among pigs in the herd were characterized by studying the local and global stability properties of steady state solutions of the system. Numerical simulations demonstrating the influence of factors such as initial prevalence of infection, presence of pre-existing antimicrobial resistant mutants, and frequency of treatment on predicted prevalence were performed. Sensitivity analysis revealed that two parameters had a huge influence on the predicted proportion of pigs carrying resistant bacteria: (a) the transmission coefficient between uninfected pigs and those infected with drug-resistant bacteria during treatment (β₂) and after treatment stops (β₃), and (b) the spontaneous clear-out rate of drug-resistant bacteria during treatment (γ₂) and immediately after treatment stops (γ₃). Control measures should therefore be geared towards reducing the magnitudes of β₂ and β₃ or increasing those of γ₂ and γ₃.     

Predator influence on the growth of a population with three genotypes

Summary A predator-prey system is proposed in which the prey species consists of three genotypes. It is shown that in the case of no predation, this system satisfies the Hardy-Weinberg principle. In the case where the genotype with the recessive gene has an advantage in its susceptibility to predation, it is shown that the other genotypes move toward extinction. Research for this paper was partially supported by the National Research Council of Canada Grant NRC A-4823, and was carried out while this author was visiting The University of Iowa.     

Top-down control in a patchy environment: revisiting the stabilizing role of food-dependent predator dispersal

In this paper, we revisit the stabilizing role that predator dispersal and aggregation have in the top-down regulation of predator-prey systems in a heterogeneous environment. We consider an environment consisting of sites interconnected by dispersal, and propose a novel mechanism of stabilization for the case with a non-sigmoid functional response of predators. We assume that the carrying capacity of the prey is infinitely large in each site, and show that successful top-down regulation of this otherwise globally unstable system is made possible through an interplay between the unevenness of prey fitness across the sites and the rapid food-dependent migration of predators. We argue that this mechanism of stabilization is different from those previously reported in the literature: in particular, it requires a high degree of synchronicity in local oscillations of species densities across the sites. Prey outbreaks take place synchronously, but the unevenness of prey growth rates across the sites results in a pronounced difference in the species densities, and so the predator quickly disperses to the sites with the highest prey abundances. For this reason, the consumption of prey mostly takes place in the sites with high densities of prey, which assures an efficient suppression of outbreaks. Furthermore, when the total size of prey population is low, the distribution of both species among the sites becomes more even, and this prevents overconsumption of the prey by the predator. Finally, we put forward the hypothesis that this mechanism, when considered in a tri-trophic plankton community in the water column, can explain the stability of the nutrient-rich low-chlorophyll open ocean regions.     

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.