Abstract:
This paper aims to study the effect of Harvesting on predator species with time-delay on a Holling type-IV prey-predator model. Harvesting has a strong impact on the dynamic evolution of a population. Two delays are considered in the model of this paper to describe the time that juveniles of prey and predator take to mature. Dynamics of the system is studied in terms of local and Hopf bifurcation analysis. Finally, numerical simulation is done to support the analytical findings.

Abstract:
A modified Holling-Tanner predator-prey system with multiple delays is investigated. By analyzing the associated characteristic equation, the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are established. Direction and stability of the periodic solutions are obtained by using normal form and center manifold theory. Finally, numerical simulations are carried out to substantiate the analytical results.

Abstract:
The dynamic behavior of a predator-prey model with Holling type IV functional response is investigated with respect to impulsive control strategies. The model is analyzed to obtain the conditions under which the system is locally asymptotically stable and permanent. Existence of a positive periodic solution of the system and the boundedness of the system is also confirmed. Furthermore, numerical analysis is used to discover the influence of impulsive perturbations. The system is found to exhibit rich dynamics such as symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics.

Abstract:
In this paper, we focus on a spatial Holling-type IV predator-prey model which contains some important factors, such as diffusion, noise (random fluctuations) and external periodic forcing. By a brief stability and bifurcation analysis, we arrive at the Hopf and Turing bifurcation surface and derive the symbolic conditions for Hopf and Turing bifurcation in the spatial domain. Based on the stability and bifurcation analysis, we obtain spiral pattern formation via numerical simulation. Additionally, we study the model with colored noise and external periodic forcing. From the numerical results, we know that noise or external periodic forcing can induce instability and enhance the oscillation of the species, and resonant response. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

Abstract:
A discrete semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay is investigated. It is proved the general nonautonomous system is permanent and globally attractive under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence and global attractivity of positive periodic solutions. We show that the conditions for the permanence of the system and the global attractivity of positive periodic solutions depend on the delay, so, we call it profitless.

Abstract:
In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions to the model

Abstract:
We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria. 1. Introduction One of the powerful and effective methods on the existence of periodic solutions to periodic systems is the continuation method, which gives easily verifiable sufficient conditions. See Gaines and Mawhin [1] for detailed description of this method. In [2], Chen studied the following periodic predator-prey system with a Holling type IV functional response: where , , , , and , , are -periodic functions with and , and , are positive constants. The results on the existence of multiple periodic solutions have been obtained by employing the continuation method. There are some works following this direction. See, for example, [3–6]. To generalize Chen’s results, Ding and Jiang [4] considered the following periodic Gause-type predator-prey system with time delays: where , , , and are continuous -periodic functions with . They also afforded verifiable criteria for the existence of multiple positive periodic solutions for the system (2) when the numerical response function is nonmonotonic. Their results improve and supplement those in [2]. As we know, in population dynamics, many evolutionary processes experience short-time rapid change after undergoing relatively long smooth variation. For example, the harvesting and stocking occur at fixed time, and some species usually immigrate at the same time every year. Incorporating these phenomena gives us impulsive differential equations. For theory of impulsive differential equations, we refer to [7–16]. Based on the previous ideas, in [17], Wang, Dai, and Chen considered the following impulse predator-prey system with a Holling type IV functional response: where the assumptions on , , , , , , , , , are the same as (1), ( , , ), is a strictly increasing sequence with , and . Further, there exist a such that ( , ) and for . By employing the continuation theorem, they presented sufficient conditions on the existence of two positive periodic solutions to system (3). In this paper, we will consider the following Gause-type predator-prey systems with impulse and time delays: where the assumptions on , , , , , and are the same as (3). and are the prey and the predator population size, respectively. The function is the growth

Abstract:
By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response , . 1. Introduction The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature [1]. In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, and sparrow and sparrow hawk, described by Tanner [2] and Wollkind et al. [3], May [4] developed the Holling-Tanner prey-predator model In system (1.1), and stand for prey and predator density at time . , , , , , are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half-capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively. Nowadays attention have been paid by many authors to Holling-Tanner predator-prey model (see [5–7]). Recently, there is a growing explicit biological and physiological evidence [8–10] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations [11, 12]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of Liang and Pan [13] obtained results for the global stability of the positive equilibrium of (1.2). However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing [14], Gopalsamy [15], Kuang [16], and MacDonald [17] for general delayed biological systems. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays. Recently, Saha and Chakrabarti [18] considered the following delayed ratio-dependent Holling-Tanner

Abstract:
This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included. 1. Introduction Predator-prey dynamics continues to draw interest from both applied mathematicians and ecologists due to its universal existence and importance. Many kinds of predator-prey models have been studied extensively [1–6]. It is well known that there are many species whose individual members have a life history that takes them through immature stage and mature stage. To analyze the effect of a stage structure for the predator or the prey on the dynamics of a predator-prey system, many scholars have investigated predator-prey systems with stage structure in the last two decades [7–15]. In [7], Wang considered the following predator-prey system with stage structure for the predator and obtained the sufficient conditions for the global stability of a coexistence equilibrium of the system: where represents the density of the prey at time . and represent the densities of the immature predator and the mature predator at time , respectively. For the meanings of all the parameters in system (1.1), one can refer to [7]. Considering the gestation time of the mature predator, Xu [8] incorporated the time delay due to the gestation of the mature predator into system (1.1) and considered the effect of the time delay on the dynamics of system (1.1). There has also been a significant body of work on the predator-prey system with stage structure for the prey. In [12], Xu considered a delayed predator-prey system with a stage structure for the prey: where and denote the population densities of the immature prey and the mature prey at time , respectively. denotes the population density of the predator at time . All the parameters in system (1.2) are assumed positive. is the birth rate of the immature prey. is the transformation rate from immature individual to mature individuals. is the intraspecific competition coefficient of

Abstract:
We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings. 1. Introduction Predator-prey dynamics has long been and will continue to be of interest to both applied mathematicians and ecologists due to its universal existence and importance [1]. Although the early Lotka-Volterra model has given way to more sophisticated models from both a mathematical and biological point of view, it has been challenged by ecologists for its functional response suffers from paradox of enrichment and biological control paradox. The ratio-dependent models are discussed as a solution to these difficulties and found to be a more reasonable choice for many predator-prey interactions [2–4]. One type of the ratio-dependent models which plays a special role in view of the interesting dynamics it possesses is the ratio-dependent Holling-Tanner predator-prey system [5, 6]. A ratio-dependent Holling-Tanner predator-prey system takes the form of where and represent the population of prey species and predator species at time . It is assumed that in the absence of the predator, the prey grows logistically with carrying and intrinsic growth rate . The predator growth equation is of logistic type with a modification of the conventional one. The parameter represents the maximal predator per capita consumption rate, and is the half capturing saturation constant. The parameter is the intrinsic growth rate of the predator and is the number of prey required to support one predator at equilibrium, when equals . All the parameters are assumed to be positive. Liang and Pan [6] established the sufficient conditions for the global stability of positive equilibrium of system (1.1) by constructing Lyapunov function. Considering the effect of time delays on the system, Saha and Chakrabarti [7] considered the following delayed system where is the negative feedback delay of the prey. Saha and Chakrabarti [7] proved that the system (1.2) is permanent under certain conditions and obtained the conditions for the local and global stability of the positive equilibrium. It is well known that studies on dynamical systems not only involve a discussion of stability and persistence, but also involve many dynamical behaviors