An impulsive ratio-dependent predator–prey system with diffusion

We investigate the predator–prey system with diffusion, when biological and environmental parameters are assumed to change in periodical manner over time. The system is affected by impulses which can be considered as a control. Conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solutions are obtained.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

Extinction and permanence in delayed stage-structure predator–prey system with impulsive effects

Based on the classical stage-structured model and Lotka–Volterra predator–prey model, an impulsive delayed differential equation to model the process of periodically releasing natural enemies at fixed times for pest control is proposed and investigated. We show that the conditions for global attractivity of the ‘pest-extinction’ (‘prey-eradication’) periodic solution and permanence of the population of the model depend on time delay. We also show that constant maturation time delay and impulsive releasing for the predator can bring great effects on the dynamics of system by numerical analysis. As a result, the pest maturation time delay is considered to establish a procedure to maintain the pests at an acceptably low level in the long term. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy-pest) model with age structure, exhibit a new modelling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay

Two-dimensional delayed continuous time dynamical system modeling a predator–prey food chain, and based on a modified version of Holling type-II scheme is investigated. By constructing a Liapunov function, we obtain a sufficient condition for global stability of the positive equilibrium. We also present some related qualitative results for this system.     

Eco-epidemiology model with age structure and prey-dependent consumption for pest management

In this paper, a prey-dependent consumption predator–prey (natural enemy-pest) model with age structure for the predators and infectious disease in the prey, is considered. Infectious pests, immature natural enemies and mature natural enemies are released impulsively. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we obtain both the sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. The results provide a reliable theoretical tactics for pest management.     

Uniform persistence of asymptotically periodic multispecies competition predator–prey systems with Holling III type functional response

In this paper, we studied the persistence of the asymptotically periodic multispecies competition predator–prey system with Holling III type functional response. Further, by use of the Standard Comparison Theorem, we improved the results of paper [C. Chen, F. Chen, Conditions for global attractivity of multispecies ecological competition-predator system with Holling III type functional response, Journal of Biomathematics 19(2) (2004) 136–140].     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

A periodic and delayed ratio-dependent predator–prey system with Holling type III functional response and stage structure for both prey and predator is investigated. It is assumed that immature predator and mature individuals of each species are divided by a fixed age, and immature predator do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solution of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

The dynamic complexity of a host–parasitoid model with a Beddington–DeAngelis functional response

This paper investigates the complex dynamics in a discrete-time model of predator–prey interaction with a Beddington–DeAngelis functional response. Local stability analysis of this model is carried out and many forms of complexities are observed using ecology theories and numerical simulation of the global behavior. Furthermore, the existence of a strange attractor and computation of the largest Lyapunov exponent also demonstrate the chaotic dynamic behavior of the model. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics.     

Multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with numerical responses. As corollaries, some applications are listed.     

Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

In this paper, we study a predator–prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey population’s intrinsic growth rate and find the conditions for the stability and for the permanence of this system.     

An impulsive predator–prey model with disease in the prey for integrated pest management

In this paper, an impulsive predator–prey model with disease in the prey is investigated for the purpose of integrated pest management. In the first part of the main results, we get the sufficient condition for the global stability of the susceptible pest-eradication periodic solution. This means if the release amount of infective prey and predator satisfy the condition, then the pest will be doomed. In the second part of the main results, we also get the sufficient condition for the permanence of the system. This means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. In the last section, we interpret our mathematical results. We also point out some possible future work.     

Predator–prey interaction with harvesting: mathematical study with biological ramifications

In this article, we study a simple predator–prey interaction where predator population is subjected to harvesting. Our qualitative analysis shows different outcomes including switching of stability, oscillations and deterministic extinction. Theoretically observed results have been tested with the parameter values of Paramecium aurelia and its predator Didinium nasutum. Study reveals that harvesting effort and predator’s attack rate may be used as control parameters for the system. Simulation results also indicate that the system may exhibit bistability for some parametric region. Our study also gives the possible answer to the question – why do we frequently observe coexisting predator–prey system in natural system? The study may be helpful to design control strategy for harvested predator–prey system.     

Bifurcation analysis and control of a class of hybrid biological economic models

This paper systematically studies a hybrid predator–prey economic model, which is formulated by differential-difference-algebraic equations. It shows that this model exhibits two bifurcation phenomena at the intersampling instants. One is saddle–node bifurcation, and the other is singular induced bifurcation which indicates that economic profit may bring impulse at some critical value, i.e., rapid expansion of biological population in terms of ecological implications. On the other hand, for the sampling instants, the system undergoes Neimark–Sacker bifurcation at a critical value of economic profit, i.e., the increase of economic profit destabilizes the system and generates a unique closed invariant curve. Moreover, the state feedback controller is designed so that singular induced bifurcation and Neimark–Sacker bifurcation can be eliminated and the population can be driven to steady states by adjusting harvesting costs and the economic profit. At the same time, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here.     

A Legendre spectral element method on a large spatial domain to solve the predator–prey system modeling interacting populations

A Legendre spectral element method is developed for solving a one-dimensional predator–prey system on a large spatial domain. The predator–prey system is numerically solved where the prey population growth is described by a cubic polynomial and the predator’s functional response is Holling type I. The discretization error generated from this method is compared with the error obtained from the Legendre pseudospectral and finite element methods. The Legendre spectral element method is also presented where the predator response is Holling type II and the initial data are discontinuous.     

Impulsive state feedback control of a predator–prey model

The dynamics of a predator–prey model with impulsive state feedback control, which is described by an autonomous system with impulses, is studied. The sufficient conditions of existence and stability of semi-trivial solution and positive period-1 solution are obtained by using the Poincaré map and analogue of the Poincaré criterion. The qualitative analysis shows that the positive period-1 solution bifurcates from the semi-trivial solution through a fold bifurcation. The bifurcation diagrams of periodic solutions are obtained by using the Poincaré map, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations.     

Disease control in a food chain model supplying alternative food

Necessity to find a non-chemical method of disease control is being increasingly felt due to its eco-friendly nature. In this paper the role of alternative food as a disease controller in a disease induced predator–prey system is studied. Stability criteria and the persistence conditions for the system are derived. Bifurcation analysis is done with respect to rate of infection. The main goal of this study is to show the non-trivial consequences of providing alternative food in a disease induced predator–prey system. Numerical simulation results illustrate that there exists a critical infection rate above which disease free system cannot be reached in absence of alternative food whereas supply of suitable alternative food makes the system disease free up to certain infection level. We have computed the disease free regions in various parametric planes. This study is aimed to introduce a new non-chemical method for controlling disease in a predator–prey system.     

Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.     

Stability and Hopf bifurcation in a predator–prey model with stage structure for the predator

A predator–prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.     

Positive periodic solution for a semi-ratio-dependent predator–prey system with diffusion and time delays

A two-species semi-ratio-dependent predator–prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solution for the system. As corollaries, some applications are listed.