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:
We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

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:
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:
A delayed ratio-dependent predator-prey discrete-time model with nonmonotone functional response is investigated in this paper. By using the continuation theorem of Mawhins coincidence degree theory, some new sufficient conditions are obtained for the existence of multiple positive periodic solutions of the discrete model. An example is given to illustrate the feasibility of the obtained result. 1. Introduction It is known that one of important factors impacted on a predator-prey system is the functional response. Holling proposed three types of functional response functions, namely, Holling I, Holling II, and Holling III, which are all monotonously nondescending [1]. But for some predator-prey systems, when the prey density reaches a high level, the growth of predator may be inhibited; that is, to say, the predator’s functional response is not monotonously increasing. In order to describe such kind of biological phenomena, Andrews proposed the so-called Holling IV functional response function [2] which is humped and declines at high prey densities . Recently, many authors have explored the dynamics of predator-prey systems with Holling IV type functional responses [3–11]. For example, Ruan and Xiao considered the following predator-prey model [5]: where and represent predator and prey densities, respectively. In (1.2), the functional response function is a special case of Holling IV functional response. The functional response functions mentioned previously only depend on the prey . But some biologists have argued that the functional response should be ratio dependent or semi-ratio dependent in many situations. Based on biological and physiological evidences, Arditi and Ginzburg first proposed the ratio-dependent predator-prey model [12] where the functional response function is ratio dependent. Many researchers have putted up a great lot of works on the ratio-dependent or semi-ratio-dependent predator-prey system [13–19]. Recently, some researchers incorporated the ratio-dependent theory and the inhibitory effect on the specific growth rate into the predator-prey model [3, 7, 11, 15]. Ding et al. considered a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay [11]; they obtained some sufficient conditions for the existence and global stability of a positive periodic solution to this system. Hu and Xia considered a functional response function [7, 15]: With the functional response function, Xia and Han proposed the following periodic ratio-dependent model with nonmonotone functional response [15]: where , , ,

Abstract:
The purpose of this work is to study the effect of harvesting on dynamics of prey- predator model with Holling Type III functional response. Mathematical analysis of the model equations with regard to the boundedness of solutions, existence of equilibria and their stabilities in both local and global manner are carried out. A combined harvesting policy for prey and predator species is discussed by using Pontrayagin’s Maximum principle and the effect of tax on prey-predator system is also shown. Butler-Mc Gehee lemma is used to identify the conditions which influence the persistence of the system. Finally, some numerical simulations are given to verify the mathematical conclusions.

Abstract:
In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given.

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

Abstract:
Cannibalism, which is the act of killing and at least partial consumption of conspecifics, is ubiquitous in nature. Mathematical models have considered cannibalism in the predator primarily, and show that predator cannibalism in two species ODE models provides a strong stabilizing effect. There is strong ecological evidence that cannibalism exists among prey as well, yet this phenomenon has been much less investigated. In the current manuscript, we investigate both the ODE and spatially explicit forms of a Holling-Tanner model, with ratio dependent functional response. We show that cannibalism in the predator provides a stabilizing influence as expected. However, when cannibalism in the prey is considered, we show that it cannot stabilise the unstable interior equilibrium in the ODE case, but can destabilise the stable interior equilibrium. In the spatially explicit case, we show that in certain parameter regime, prey cannibalism can lead to pattern forming Turing dynamics, which is an impossibility without it. Lastly we consider a stochastic prey cannibalism rate, and find that it can alter both spatial patterns, as well as limit cycle dynamics.

Abstract:
A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results. 1. Introduction As was pointed out by Berryman [1], the dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Furthermore, the study of the consequences of the hiding behavior of the prey on the dynamics of predator-prey interactions can be recognized as a major issue in both applied mathematics and theoretical ecology [2]. In general, the effects of prey refuges on the population dynamics are very complex in nature, but for modeling purposes, it can be considered as constituted by two components [2]. The first one, which affects positively the growth of prey and negatively that of predators, comprises the reduction of prey mortality due to the decrease in predation success. The second one may be the tradeoffs and by-products of the hiding behavior of prey which could be advantageous or detrimental for all the interacting populations [3]. Sih [4] obtained a set of general conditions which ensure that the refuge use has a stabilizing effect on Lotka-Volterra-type predator-prey systems; he also examined the effect of the cost of refuge use in decreased prey feeding or reproductive rate. In [5], González-Olivares and Ramos-Jiliberto investigated the dynamic behaviors of predator-prey system incorporating Holling type II functional response and a constant refuge: where denote the densities of prey and predator population at any time , respectively; are positive constants; here is the intrinsic per capita growth rate of prey; is the prey environmental carrying capacity; is the maximal per capita consumption rate of predators; is the amount of prey needed to achieve one-half of ; is the conversion factor denoting the number of newly born predators for each captured prey; is the death rate of the predator; is the number of prey that refuge can protect at time . Kar [6] also studies the dynamic behaviors of system (1.1). He obtained the conditions for the existence and stability of the equilibria and persistent criteria for the