一类含扩散与时滞的Prey-Predator模型的周期解

利用上下解方法以及比较原理研究了一类含扩散与时滞的Prey-Predator模型,证明在一定条件下该模型的零平衡态及半平凡周期解的全局稳定性,并获得了这个系统具有一对周期拟解的充分条件.对任意的非负初值函数,这一对周期拟解构成的区间是此系统的一个吸引子.

一类食饵具有阶段结构的时滞Predator-Prey系统的周期解(英文)

研究了一类时滞Predator-Prey系统,其中Prey种群是具有两个生命阶段的种群,即幼年阶段和成年阶段。Predator种群只能捕食Prey幼年种群。通过应用Gaines和Mawhin重合度理论的连续函数定理,给出了系统正周期解存在的充分条件。

NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS

A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.

Experimental clearance rate and intraguild predation of jellyfish Cyanea nozakii

Cyanea nozakii,a common jellyfish distributed in offshore China,has a complex trophic relationship with other zooplankton groups.However,few studies have reported the predation rates and prey selection patterns of C.nozakii medusae on different prey items.Research is also lacking on the intraguild predation of Aurelia coerulea(another common bloom jellyfish in offshore China)by C.nozakii.To address the knowledge gaps,the clearance rates of C.nozakii for different prey items,including copepods(small<1000μm and large>1000μm),fish larvae,and gelatinous prey(hydromedusae,A.coerulea ephyrae,and chaetognaths),were measured.The influence of predator size on the clearance rate was also determined.Additionally,we examined the intraguild predation of C.nozakii on A.coerulea medusae.The clearance rates of C.nozakii varied widely with prey organisms,being independent of prey concentrations.Gelatinous organisms,except for chaetognaths,were captured with considerably high efficiency,followed by fish larvae and copepods,indicating the preferential prey selection of gelatinous organisms by C.nozakii.The clearance rate increased linearly with the cross-sectional area of C.nozakii.Body size in medusae may,to some extents,underpin their capacity to capture more prey by increasing the encounter rate and capture success through ontogeny.C.nozakii preyed voraciously on A.coerulea in high feeding efficiency,but the clearance rate decreased with increasing A.coerulea(as prey)size.This phenomenon of intraguild predation suggests a speculative hypothesis of potential population regulation of A.coerulea by C.nozakii.The information regarding the feeding ecology of C.nozakii reported in this study is important for understanding plankton dynamics in marine ecosystems with extensive occurrences of this jellyfish.

Stability of the Bifurcation Solutions for a Predator-Prey Model

The bifurcation solution of the nonnegative steady state of a reaction diffusion system was investigated. The combination of the sturm type eigenvalue and the theorem of bifurcation was used to study the local coexistence solutions, and obtain the stability of bifurcation solutions. The system model describes predator prey interaction in an unstirred chemostat.

Can an Invasive Prey Species Induce Morphological and Behavioral Changes in an Endemic Predator? Evidence from a South Korean Snake(Oocatochus rufodorsatus)

Introduction of an invasive prey species into an ecosystem may affect an endemic predator＇s fitness by altering the prey-predator system. Successful adaptation may allow the endemic predator to eat and control the invasive species, while unsuccessful adaptation may result in extinction of the predator. We examine the possible effects of the invasive North American bullfrog （Rana [Lithobates] catesbeiana） on the endemic Red-backed rat snake （Oocatochus rufodorsatus） in South Korea. We do so by comparing the morphology and behavior of adult and hatchling snakes from bullfrog-exposed （Taean） and bullfrog-unexposed （Hongcheon） populations. Among the seven morphological characteristics investigated, relative tail length （tail length/snout-vent length） of both adults and hatchlings from Taean was significantly greater than that of adults and hatchlings from Hongcheon. Also, adult snakes from Taean had a signiifcantly shorter latency of ifrst tongue lfick in response to prey compared to adults from Hongcheon. This difference was not observed in hatchlings. In other snake species, a longer relative tail length and shorter latency of ifrst tongue lfick are known to improve foraging efifciency, and these characters may be adaptations ofO. rufodorsatus to prey on bullfrogs. This study provides preliminary evidence that the presence of an invasive prey species may cause morphological and behavioral changes in an endemic predator.

A CONDITION OF THE EXISTENCE OF STABLE POSITIVE STEADY-STATE SOLUTIONS FOR A ONE PREDATOR TWO PREY SYSTEM

One predator two prey system is a research topic which has both the theoretical and practical values. This paper provides a natural condition of the existence of stable positive steady-state solutions for the one predator two prey system. Under this condition we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem, discuss the positive stable solution problem bifurcated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.

Turing Instability of Diffusive Predator⁃Prey System with Gompertz Growth

This paper mainly focus on the research of a predator⁃prey system with Gompertz growth of prey.When the system does not contain diffusion,the stability conditions of positive equilibrium and the occurring condition of the Hopf bifurcation are obtained.When the diffusion term of the system appears,the stable conditions of positive equilibrium and the Turing instability condition are also obtained.Turing instability is induced by the diffusion term through theoretical analysis.Thus,the region of parameters in which Turing instability occurs is presented.Then the amplitude equations are derived by the multiple scale method.The results will enrich the pattern dynamics in predator⁃prey systems.

Prey Predator Fishery Model with Stage Structure in Two Patchy Marine Aquatic Environment

In this paper, we propose and analyze a mathematical model to study the dynamics of a fishery resource system with stage structure in an aquatic environment that consists of two zones namely unreserved zone (fishing permitted) and reserved zone (fishing is strictly prohibited). In this model we introduce a stage structure in which predators are split into two kinds as immature predators and mature predators. It is assumed that immature predators cannot catch the prey and their foods are given by their parents (mature predators). It is also assumed that the fishing of immature predators prohibited in the unreserved zone and predator species are not allowed to enter inside the reserved zone. The local and global stability analysis has been specified. Biological and Bionomical equilibriums of the system are derived. Mathematical formulation of the optimal harvesting policy is given and its solution is derived in the equilibrium case by using Pontryagin’s maximum principle.

Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model

The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.

A Review of the Mathematical Models for the Impact of Seasonal Weather Variation and Infections on Prey Predator Interactions in Serengeti Ecosystem

Interaction between prey and predator species is a complex and non-linear process. Understanding various phenomena in the dynamics of prey-predator systems is vital to both mathematical ecology and conservation biology. Mathematical models on prey-predator systems have been the hot sport providing important information regarding the interactions of prey and predator species in various ecosystems. In this paper, a review of the available mathematical models on prey-predator systems was done. Our aim was to assess their structure, behaviour, available control strategies, population involved and their ability in predicting the future behaviour of the ecosystems. We observed diversities in the reviewed mathematical models, some model incorporated factors such as drought, harvesting and prey refuge as the factors that affect ecosystems, some ignored the contribution of environmental variations while others considered the variable carrying capacity. Most of the models reviewed have not considered the contribution of diseases and seasonal weather variation in the dynamics of prey predator systems. Some of the reviewed models do not match the real situation in most modelled ecosystems. Thus, to avoid unreliable results, this review reveals the need to incorporate seasonal weather variations and diseases in the dynamics of prey predator systems of Serengeti ecosystem.

ON A PERIODIC PREY-PREDATOR SYSTEM WITH INFINITE DELAYS

Based on the theory of coincidence degree,the existence of positive periodic solutions is established for a periodic prey predator system with infinite delays (t)=x(t)[α(t)-γ(t)y(t)-γ(t)∫ ∞ 0K 1(t,s)y(t-s) d s- ∫ ∞ 0∫ ∞ 0R 1(t,s,θ)y(t-s)y(t-θ) d θ d s], (t)=y(t)[-β(t)+μ(t)x(t)+μ(t)∫ ∞ 0K 2(t,s)x(t-s) d s+ ∫ ∞ 0∫ ∞ 0R 2(t,s,θ)x(t-θ)x(t-s) d θ d s],where α,γ,β,μ are positive continuous ω periodic functions, K i∈C (R×[0,∞),(0,∞))( i =1,2) are ω periodic with respect to their first arguments,.respectively, R i∈C (R×[0,∞)×[0,∞),(0,∞))( i =1,2) are ω periodic with respect to their first arguments,respectively.

Effect of random movement and cooperative hunting in the prey-predator system:A dynamical approach

Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.

Experimental clearance rates of Aurelia coerulea ephyrae and medusae,and the predation impact on zooplankton in Jiaozhou Bay

The population explosion of jellyfish Aurelia coerulea occurred in Jiaozhou Bay,China in 2009.The potential predation impact of A.coerulea on zooplankton was investigated.Population clearance potential and residence time(t 1/2)for copepods were calculated from laboratory clearance rates and measurements of jellyfish size and abundance from May to August 2009 in Jiaozhou Bay.Clearance rates varied widely with prey organisms,but they were not significantly different among various prey concentrations.Medusae captured rotifers,fish larvae and hydromedusae more efficiently than fish eggs,copepods and chaetognaths.Ephyrae captured rotifers and hydromedusae more efficiently than fish larvae and copepods.Clearance rate linearly increased with the cross sectional area of A.coerulea(size from 0.3 to 7.1 cm).Water temperature also had a marked effect on clearance rate and this was related to the increased beat frequency as water temperature increased.In early May 2009,A.coerulea potentially cleared the volume of water in the Bay less than 0.001 times a day,but this value was estimated to be more than 0.3 times a day in July.The t 1/2 for copepods was less than 6 d in June and July.Abundances of copepods,hydromedusae and chaetognaths were extremely low in 2009 compared to 2008 and 2010(jellyfish non-bloom years).Large predation pressure by the A.coerulea population occurred to control zooplankton communities in Jiaozhou Bay.A.coerulea,when present at a high population level,can be a keystone species in Jiaozhou Bay and control the trophic structure here.

THE MATHEMATICAL BEHAVIOR OF A NONAUTONOMOUS VOLTERRA PREDATOR-PREY SYSTEM WITH UNDERCROWDING EFFECT

This paper investigates a nonautonomous Volterra predator Prey system with undercrowding effect. A set of sufficient conditions for the existence and globally asymptotic stability of positive solution, which is easy to be verified, is obtained.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

Differences in predator-avoidance behavior between two invasive gobies and their native competitors

Globally,fish are frequently introduced beyond their native range.Some,like Ponto-Caspian gobies,are becoming invasive,achieving high colo-nization rates and constituting frequent prey for native predators.However,little is known about the effectiveness of antipredator behaviors of the invaders,which may shape their role in the invaded community and contribute to the invasion success.We compared antipredator behaviors of invasive gobies and native fish species after their detection by the predator,when the danger becomes direct.We studied 2 fish pairs,each consisting of an invasive and native species co-occurring in the environment and belonging to the same prey guild:(1)the racer goby Babka gymnotrachelus versus European bulhead Cottus gobio,(2)the monkey goby Neogobius fluviatilis versus gudgeon Gobio gobio,facing a naive predator(the Eurasian perch Perca fluviatilis).We analyzed behaviors of single prey individuals(escaping,staying in shelter,and activity)and sin-gle predators(activity,searching,following,capturing,and latency to prey consumption).In the predator presence,the bullhead was less active and more often managed to escape after capture than the racer goby.The gudgeon escaped before the capture more often than the monkey goby.The predator succeeded later with the bullhead compared to racer goby,whereas no differences in ingestion time occurred between the gudgeon and monkey goby.The results suggest that,in terms of hunting effort of native predators,the invasive gobies are equivalent to or more profitable prey than their native analogs,which can facilitate the integration of the gobies into local food webs.

ALMOST PERIODIC SOLUTION OF A NONAUTONOMOUS MODIFIED LESLIE-GOWER PREDATOR-PREY MODEL WITH NONMONOTONIC FUNCTIONAL RESPONSE AND A PREY REFUGE

A nonautonomous modified Leslie-Gower predator-prey model with non- monotonic functional response and a prey refuge is proposed and studied in this paper. Sufficient conditions which guarantee the permanence, extinction of the prey species and the global stability of the system are obtained, respectively. Also, by constructing a suitable Lyapunov function, some sufficient conditions are obtained for the existence of a unique globally attractive positive almost periodic solution of this model. Our results indicate that the prey refuge has positive effect on the coexistence of the species. Examples together with their numeric simulation show the feasibility of our main results.

Specialist predator in a multi-species prey community:boreal voles and weasels

Dissimilar vulnerabilities of different prey types and preferences of predators are factors likely to contribute to community dynamics.This may happen via differential individual properties of prey animals(e.g.vigilance,escape)or via habitat effects making hunting by a predator easier and more rewarding in some habitats,or both.Furthermore,community dynamics may be influenced by predator mediated apparent competition,in which an increase in one prey type has negative effects on another prey type indirectly via the shared predator.We summarize the current knowledge from the field in a model predator–prey system consisting of sympatric boreal vole species and their common specialist predator and review field studies using predator manipulation and studies on the responses of individuals in the laboratory and in outdoor enclosures.The vole species studied represent different prey types that are thought to have different vulnerabilities.Our observations on the main resident specialist predator,the least weasel(Mustela nivalis nivalis L.),show that it hunts according to prey availability and suitability of the hunting habitat.Prey voles respond to the presence of the predator behaviorally in various ways to avoid predation.We conclude that even if the least weasel is a specialized predator of small rodents it acts like a generalist predator within the small rodent guild and may facilitate the coexistence of prey species via predator switching.This may lead to interspecific synchrony between prey populations,which has often been observed.We suggest that the processes determining the community impact of predator–prey interactions are driven by the behavioral arms race between the predator and the prey,together with the habitat-dependent density of prey and net gain for the predator.

Canard explosion,homoclinic and heteroclinic orbits in singularly perturbed generalist predator-prey systems

This paper studies the dynamics of the generalist predator–prey systems modeled in[E.Alexandra,F.Lutscher and G.Seo,Bistability and limit cycles in generalist predator–prey dynamics,Ecol.Complex.14(2013)48–55].When prey reproduces much faster than predator,by combining the normal form theory of slow-fast systems,the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan[Relaxation oscillation and canard explosion,J.Differential Equations174(2)(2001)312–368;Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions,SIAM J.Math.Anal.33(2)(2001)286–314],we provide a detailed mathematical analysis to show the existence of homoclinic orbits,heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles.Moreover,on global stability of the unique positive equilibrium,we provide some new results.Numerical simulations are also carried out to support the theoretical results.