Periodic Solutions of Periodic Delay Predator-Prey System with Nonmonotonic Functional Response

By using the continuation theorem of coincidence degree theory, sufficient conditions are obtained for the existence of positive periodic solutions of a delayed predator prey system with nonmonotonic functional response in a periodic environment.

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.

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。

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.

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.

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.

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.

A Leslie-Gower Holling Type-II Predator-Prey Mathematical Model with Disease in Prey Population Incorporating a Prey Refuge

We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits.

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.

GLOBAL ASYMPTOTIC STABILITY FOR A TWO-SPECIES DISCRETE RATIO-DEPENDENT PREDATOR-PREY SYSTEM

The dynamical behaviors of a two-species discrete ratio-dependent predator-prey sys- tem are considered. Some sufficient conditions for the local stability of the equilibria is obtained by using the linearization method. Further, we also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper [G. Chen, Z. Teng and Z. Hu, Analysis of stability for a discrete ratio-dependent predator-prey system, Indian J. Pure Appl. Math. 42（1） （2011） 1-26] has done. The method given in this paper is new and very resultful comparing with papers [H. F. Huo and W. T. Li, Existence and global stability of periodic solutions of a discrete predator--prey system with delays, Appl. Math. Comput. 153 （2004） 337-351; X. Liao, S. Zhou and Y. Chen, On permanence and global stability in a general Gilpin- Ayala competition predator prey discrete system, Appl. Math. Comput. 190 （2007） 500-509] and it can also be applied to study the global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present an open question.

Study of refuge use on a predator-prey system with a competitor for the prey

In this paper, we propose a predator-prey system with a competitor for the prey. The model incorporates a constant prey refuge and predation process follows Holling type II response function. Using the Routh-Hurwitz criterion, the sufficient conditions of locally asymptotically stable of all the equilibria are obtained. Furthermore, global stability of the positive equilibrium is investigated by constructing a suitable Lyapunov function. The occurrence of Hopf-bifurcation of the system is shown at a critical value ＂m＂ and the system can be stabilized by increasing amount of prey refuge. The result includes the sufficient conditions for uniform persistence. Numerical simulations are carried out to illustrate the obtained results and the dependence of the dynamic behavior on the prey refuge.

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.

Integrated search technique for parameter determination of SVM for speech recognition

Support vector machine(SVM)has a good application prospect for speech recognition problems;still optimum parameter selection is a vital issue for it.To improve the learning ability of SVM,a method for searching the optimal parameters based on integration of predator prey optimization(PPO)and Hooke-Jeeves method has been proposed.In PPO technique,population consists of prey and predator particles.The prey particles search the optimum solution and predator always attacks the global best prey particle.The solution obtained by PPO is further improved by applying Hooke-Jeeves method.Proposed method is applied to recognize isolated words in a Hindi speech database and also to recognize words in a benchmark database TI-20 in clean and noisy environment.A recognition rate of 81.5%for Hindi database and 92.2%for TI-20 database has been achieved using proposed technique.

Global stability of a stochastic predator-prey model with Allee effect

In this paper, we study a stochastic predator-prey model with Beddington-DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

ANALYSIS OF A PREDATOR-PREY MODEL WITH DISEASE IN THE PREY

In this paper, we discuss the behavior of a predator-prey model with disease in the prey with and without stochastic perturbation, respectively. First, we briefly give the dynamic of the deterministic system, by analyzing stabilities of its four equilibria. Then, we consider the asymptotic behavior of the stochastic system. By Lyapunov analysis methods, we show the stochastic stability and its long time behavior around the equi- librium of the deterministic system. We obtain there are similar properties between the stochastic system and its corresponding deterministic system, when white noise is small. But large white noise can make a unstable deterministic system to be stable.

Permanence of a diffusive Leslie-Gower predator-prey model incorporating a prey refuge

The diffusive Leslie-Clower predator-prey model incorporating a prey refuge is recon- sidered here. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained, and our results supplement earlier ones.

DYNAMICS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH ADDITIVE ALLEE EFFECT

In this paper, we investigate the dynamics of a diffusive predator prey model with Holling-II functional response and the additive Allee effect in prey. We show the local and global asymptotical stability of the positive equilibrium, and give the conditions of the existence of the Hopf bifurcation. By carrying out global qualitative and bifurcation analysis, it is shown that the weak and strong Allee effects in prey can induce different dynamical behavior in the predator-prey model. Furthermore, we use some numerical simulations to illustrate the dynamics of the model. The results may be helpful for controlling and managing the predator-prey system.

Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator prey model with harvesting

In this paper,a difference-algebraic predator prey model is proposed,and its complex dynamical behaviors are analyzed.The model is a discrete singular system,which is obtained by using Euler scheme to discretize a differential-algebraic predator-prey model with harvesting that we establish.Firstly,the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system the­ory.Further,by applying the new normal form of difference-algebraic equations,center manifold theory and bifurcation theory,the Flip bifurcation and Neimark-Sacker bifur­cation around the interior equilibrium point are studied,where the step size is treated as the variable bifurcation parameter.Lastly,with the help of Matlab software,some numerical simulations are performed not only to validate our theoretical results,but also to show the abundant dynamical behaviors,such as period-doubling bifurcations,period 2,4,8,and 16 orbits,invariant closed curve,and chaotic sets.In particular,the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.

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.