Classical Predator-prey System Research Articles

Dynamic behaviour of a predator-prey system with impulsive control strategy

In this paper, a class of predator-prey system with Beddington-DeAngelis functional response and Allee effect is studied, where impulsive effects are also considered in the model. By using small parameters perturbation skills, comparison theorem and Floquet theory for the impulsive equations, the globally asymptotical stability of the prey-eradication periodic solution and the persistence for the system are obtained. Finally, some numerical examples and simulations are presented by Matlab software to support the theoretical results. The influence of impulsive perturbation intensity and impulsive period on the persistence and periodic solution of the system are discussed.

Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

Numerical Solution of a Class of Predator-Prey Systems with Complex Dynamics Characters Based on a Sinc Function Interpolation Collocation Method

Although many kinds of numerical methods have been announced for the predator-prey system, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, in this paper, a new interpolation collocation method is proposed for a class of predator-prey systems with complex dynamics characters. Some complex dynamics characters and pattern formations are shown by using this new approach, and the results have a good agreement with theoretical results. Simulation results show the effectiveness of the method.

具有Holling Ⅳ型功能反应捕食系统的状态反馈控制

A class of predator-prey systems with Holling IV functional responses under state feedback control were studied. The sufficient conditions for the existence and stability of semi-trivial solutions and order-1 periodic solutions were obtained by means of the analogue of the Poincare criterion and the geometric theory for semi-continuous dynamical systems. The numerical simulation verifies the correctness of the conclusion and the effectiveness of the state feedback control, and reveals abundant dynamic behaviors of the state feedback control system, such as the fold bifurcation, the flip bifurcation and chaos.

Integrability and Multiple Limit Cycles in a Predator-Prey System with Fear Effect

In this paper, a class of predator-prey systems with fear effect is investigated, the integrability conditions of the origin and the positive equilibrium are obtained, and the fact that three limit cycles can be bifurcated from the positive equilibrium is proved, so bistable phenomenon can occur for this system.

Trophic and non-trophic interactions in binary links affect carbon flow in the soil micro-food web

Soil organisms constitute multitrophic food webs, which are important determinants for nutrient and carbon flow through terrestrial systems. However, these complex interactions are not well understood, particular in the most cryptic compartment – the link between microorganisms and microfauna. The present study investigates binary trophic links between microbial prey and nematodes in a controlled semi-natural soil environment. A microcosm system with circulating airflow was filled with γ-irradiated arable soil (Luvisol), inoculated with 13C labelled microbial diet and nematode grazers, and incubated for 72 h. Signatures of 13C and CO2 evolved from microcosms, soil phospholipid fatty acids and nematode densities were determined. Except for the model nematode Caenorhabditis elegans, all organisms joint in the 12 different investigated binary links represent dominant taxa at the arable field site. Studied were the fungal grazers Aphelenchoides saprophilus and Aphelenchus sp. in combination with the fungi Chaetomium globosum, Aspergillus terreus or Cryptococcus terreus as well as the bacterial grazers C. elegans and Acrobeloides buetschlii in combination with the bacteria Pseudomonas putida, Kitasatospora sp. or Sphingomonas trueperi.The results assigned trophic and non-trophic interactions and considerable variation between binary links. The biomass decline of S. trueperi in presence of C. elegans as well as the increased release of 13C in links of C. globosum and A. terreus with Aphelenchus sp. and S. trueperi with A. buetschlii assign nematode grazing and thus a classical predator-prey system. Such direct trophic interactions can induce distinct C fluxes within a binary link as assigned by the increased density (i.e. biomass) of C. elegans when feeding on S. trueperi. In contrast, the numbers of Aphelenchus sp. declined when feeding on A. terreus, and thus reduced C flow to the next trophic level. Grazing can also foster C allocation in microbial biomass via compensatory growth, as indicated e.g. for the link of Aphelenchus sp. with the fungi C. globosum and A. terreus. While these diverse responses to grazing were to be expected, the use of CO2 and 13C measurements further revealed that non-trophic interactions shape C flux in a binary link, particularly in the bacterial channel. Besides the fungal link of C. terreus with Aphelenchus sp., non-trophic interactions were obvious for the bacterial links Kitasatospora sp. with C. elegans or A. buetschlii, and P. putida with A. buetschlii. The presence of nematodes caused a reduction in respiration (evolved CO2), thus a decline in microbial activity. The formation of chemical and/or morphological defense is likely a reason for that reduced prey activity. For further studies the inclusion of such non-trophic interactions is necessary to achieve realistic models for C flow through soil food webs.

Stage-structured control on a class of predator-prey system in almost periodic environment

ABSTRACTIn this paper, we consider a kind of stage-structured predator-prey systems with two predators and general functional responses. By applying the comparison theorem, Lyapunov functional and Mawhin's continuation theorem of coincidence degree theory, this paper gives some new sufficient conditions for the uniform persistence, globally asymptotical stability and almost-periodic solutions for the above system. Two examples and numerical simulations are provided to illustrate the main results of this paper.

一类捕食系统中的Hopf分岔

一类捕食系统中的Hopf分岔

Fast Sampling of Evolving Systems with Periodic Trajectories

We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are supposed to be of a special class, albeit admitting nonlinear parametrization and nonlinearities. The method enables to represent a given periodic solution as sums of computable integrals and functions that are explicitly dependent on parameters of interest and initial conditions. This allows invoking parallel computational streams in order to increase speed of calculations. Performance and practical implications of the method are illustrated with examples including classical predator-prey system and models of neuronal cells.

Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

Periodic Solution of a Class Predator-Prey System with Rate Stocking and Time Delay

In this paper, we investigate of a class of predator-prey system with rate stocking and time delay, the existence positive periodic solution by using coincidence degree theory. We obtain the sufficient conditions which guarantee existence of the positive periodic solution of the periodic system. Some new results obtained.

Deterministic Continuation of Stochastic Metastable Equilibria via Lyapunov Equations and Ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.

The dynamical complexity of a Ivlev-type prey–predator system with impulsive effect

Abstract Based on the classical predator–prey system with Ivlev-type functional response, an impulsive differential equations to model the process of periodic perturbations on the predator at different fixed time is established. It proves that there exists a locally asymptotically stable prey-eradication periodic solution when the impulse period is less than some critical value, and otherwise, the system can be permanent. Numerical results show that the system considered has more complicated dynamics. such as quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, symmetry-breaking pitchfork bifurcation and crises, etc.

Complicated dynamics of a predator–prey system with Watt-type functional response and impulsive control strategy

Based on the classical predator–prey system with Watt-type functional response, an impulsive differential equations to model the process of periodic perturbations on the predator at different fixed time for pest control is proposed and investigated. It proves that there exists a globally asymptotically stable prey-eradication periodic solution when the impulse period is less than some critical value, and otherwise, the system can be permanent. Numerical results show that the system considered has more complicated dynamics involving quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and “crises”, etc. It will be useful for studying the dynamic complexity of ecosystems.

Role of standard incidence in an eco-epidemiological system: A mathematical study

Ecology and epidemiology are two major fields of study in their own right, but they have some common features. [Chattopadhyay, J., Pal, S., El Abdllaoui, A., 2003. Classical predator–prey system with infection of prey population—a mathematical model. Math. Meth. Appl. Sci. 26, 1211–1222] considered a predator–prey model with disease in the prey population. They analyzed the system based on the assumption that horizontal incidence follows simple mass action incidence. Mass action incidence is appropriate for a constant population, but for a large population, standard incidence is more appropriate. The complicated dynamics around (0, 0, 0) arise because of standard incidence. The conditions under which the population reaches the origin either by following the axis or in a spiral pattern were determined. Numerical experiments were performed to observe the dynamics of the system with mass action incidence and standard incidence. This investigation showed that the infection rate plays a crucial role for predicting the behavior of the dynamics in the long run.

Classical predator–prey system with infection of prey population—a mathematical model

AbstractThe present paper deals with the problem of a classical predator–prey system with infection of prey population. A classical predator–prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical findings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic. Copyright © 2003 John Wiley & Sons, Ltd.

The global stability of predator-prey system of Gause-type with Holling III functional response

This paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey system of Gause-type with Holling III functional response. The Dulac's criterion is applied and liapunov functions are constructed to establish the global stability.

Population dynamics of a parasite and hyperparasite in a closed system: model analysis and parameter estimation

In this paper, we introduce and test models of the dynamics of a parasite and hyperparasite in a closed system from which the parasite's host is excluded. The models are tested against data for the temporal dynamics of a fungal plant parasite of lettuce, Sclerotinia minor, and a fungal hyperparasite, Sporidesmium sclerotivorum. The models are developed by treating the hyperparasite-parasite dynamics as a classical predator-prey system, but then expanding the prey component into a compartmental system of susceptible (uninfected) and infected parasites. The resulting hybrid model enables the separation of the dynamics of susceptible parasites, which are capable of causing disease, and infected parasites, which support reproduction of the hyperparasite. The influence of different functional and numerical responses are compared by analytical results and by parameter estimation.

On constant effort harvesting and stocking in a class of predator-prey systems

We analyze the global behaviour of predator-prey systems under constant effort harvesting and stocking of either or both species. Mathematically this is equivalent to the behaviour of an unharvested system with different parameters; in particular every solution tends either to an equilibrium or to a limit cycle, and there is no possibility of finite-time extinction of either species if both populations are positive initially. The dependence of the behaviour on the harvesting efforts is discussed.