Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat

In this paper, we introduce and study a model of a predator–prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

A numerical study of the formation of spatial patterns in twospotted spider mites

The aim of this paper is to study the formation of spatial patterns in a predator–prey system with Tetranychus urticae as prey and Phytoseiulus persimilis as predator. Logistic Lotka–Volterra predator–prey equations are solved numerically with two different response functions, two initial conditions and one data set. The spatial patterns are generated by introducing diffusion-driven instability in the predator–prey system. Among all parameters involved in predator–prey equations, only the predator interference parameter is varied to generate diffusion-driven instability leading to spatial patterns of population density. Spatial patterns are further generated with the inclusion of prey-taxis in the predator–prey system. Routh–Hurwitz’s conditions for stability are used to create instability with prey-taxis in the system. It is shown that it is possible to generate spatial patterns with zero flux boundary conditions even in a smaller domain with a suitable value of the predator interference parameter or prey-taxis.     

The global stability and pattern formations of a predator–prey system with consuming resource

The paper deals with a model that describes a predator–prey system with a common consuming resource. We use Lyapunov functions to prove the global stability of the kinetic system and the diffusive system. The existence of non-constant positive steady state solutions is shown to identify the range of parameters of spatial pattern formation for the cross-diffusion system.     

Dynamics of a predator–prey system with inducible defense and disease in the prey

Establishing and researching a population dynamical model based on the differential equation is of great significance. In this paper, a predator–prey system with inducible defense and disease in the prey is built from biological evolution and Eco-epidemiology. The effect of disease on population stability in the predator–prey system with inducible defense is studied. Firstly, we verify the positivity and uniform boundedness of the solutions of the system. Then the existence and stability of the equilibria are studied. There are no more than nine equilibrium points in the system. We use a sophisticated parameter transformation to study the properties of the coexistence equilibrium points of the system. A sufficient condition is established for the existence of Hopf bifurcation. Numerical simulations are performed to make analytical studies more complete.     

Global dynamic behavior of a predator–prey model under ratio-dependent state impulsive control

This paper studies the global dynamic behavior of a prey–predator model with square root functional response under ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. An order-k periodic orbit is obtained by employing the Brouwer’s fixed point theorem. Furthermore, the critical values are determined for the existence of orbitally asymptotically stable order-1 and order-2 periodic orbits in finite time. These critical values play an important role in determining different kinds of order-k periodic orbits and can also be used for designing the control parameters to obtain the desirable dynamic behavior of the controlled prey–predator system. Moreover, it is found that the local equilibrium point is also globally asymptotically stable under the control strategy. Numerical examples are provided to validate the effectiveness and feasibility of the theoretical results.     

A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate

We consider the dynamics of a general stage-structured predator–prey model which generalizes several known predator–prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.     

Global dynamics and controllability of a harvested prey–predator system with Holling type III functional response

In recent years overexploitation and collapse of several biological resources have been seen in fisheries, forestry and wildlife. This is due to unconventional and indiscrete harvesting of the resources. Consequently there is much current concern to find principles for the control and management of resources, particularly in those circumstances where both prey and predator species are being exploited. This work illustrates the use of harvesting efforts as a control to obtain strategies for the control of a prey–predator system with Holling type III functional response.     

Spatial propagation of a lattice predation–competition system with one predator and two preys in shifting habitats

This paper deals with a discrete diffusive predator–prey system involving two competing preys and one predator in a shifting habitat induced by the climate change. By applying Schauder's fixed-point theorem on various invariant cones via constructing several pairs of generalized super- and subsolutions, we establish four different types of supercritical and critical forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two competing aboriginal coexistent preys with an invading alien predator, a pair of aboriginal coexistent predator–prey and an invading alien prey, and the coexistence of three species to the extinction state, respectively. Meanwhile, the nonexistence of some subcritical forced waves is showed by contradiction. Furthermore, some numerical simulations are given to present and promote the theoretical results.     

Weak Allee effect in a predator–prey model involving memory with a hump

In this paper we consider a predator–prey system which has a factor that allows for a reduction in fitness due to declining population sizes, often termed an Allee effect. We study the influence of the weak Allee effect which is included in the prey equation and we determine conditions for the occurrence of Hopf bifurcation. The prey population is limited by the carrying capacity of the environment, and the predator growth rate depends on past quantities of the prey which is represented by a weight function that specifies a moment in the past when the quantity of food is the most important from the point of view of the present growth of the predator. The stability properties of the system and the biological issues of the memory and Allee effect on the coexistence of the two species are studied. Finally we present some simulations to verify the veracity of the analytical conclusions.     

Nonconstant Positive Steady States and Pattern Formation of 1D Prey-Taxis Systems

Prey-taxis is the process that predators move preferentially toward patches with highest density of prey. It is well known to have an important role in biological control and the maintenance of biodiversity. To model the coexistence and spatial distributions of predator and prey species, this paper concerns nonconstant positive steady states of a wide class of prey-taxis systems with general functional responses over 1D domain. Linearized stability of the positive equilibrium is analyzed to show that prey-taxis destabilizes prey–predator homogeneity when prey repulsion (e.g., due to volume-filling effect in predator species or group defense in prey species) is present, and prey-taxis stabilizes the homogeneity otherwise. Then, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis. Moreover, we provide detailed and thorough calculations to determine properties such as pitchfork and turning direction of the local branches. Our stability results also provide a stable wave mode selection mechanism for thee reaction–advection–diffusion systems including prey-taxis models considered in this paper. Finally, we provide numerical studies of prey-taxis systems with Holling–Tanner kinetics to illustrate and support our theoretical findings. Our numerical simulations demonstrate that the \(2\times 2\) prey-taxis system is able to model the formation and evolution of various striking patterns, such as spikes, periodic oscillations, and coarsening even when the domain is one-dimensional. These dynamics can model the coexistence and spatial distributions of interacting prey and predator species. We also give some insights on how system parameters influence pattern formation in these models.     

Dynamic complexities of predator–prey system in a pulsed chemostat

In this paper, we study a food chain model with Holling III and Monod type functional response under periodic pulsed conditions, which contains with predator, prey and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in prey and predator. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Bifurcations in a Leslie–Gower model with constant and proportional prey refuge at high and low density

Assuming that the prey refuge is proportional to the prey density if its population size is below a critical threshold, or constant if its size is above the threshold, this paper proposes, and qualitatively analyzes, a Leslie–Gower predator–prey model assuming alternative feeding and harvesting in predators, and a Holling II function as the predator functional response. From the results of the mathematical analysis to the predator–prey models with proportional or constant prey refuge, the proposed model retains the same bifurcation cases obtained for each model analyzed. However, appropriate alterations of the parameters representing the critical threshold of prey population size and harvest in predators allows the formation of at least one limit cycle, stable or unstable, that lives in both vector fields of the proposed model.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Dynamics of predator–prey models with a strong Allee effect on the prey and predator-dependent trophic functions

The complex dynamics of a two-trophic chain are investigated. The chain is described by a general predator–prey system, in which the prey growth rate and the trophic interaction functions are defined only by some properties determining their shapes. To account for undercrowding phenomena, the prey growth function is assumed to model a strong Allee effect; to simulate the predator interference during the predation process, the trophic function is assumed predator-dependent. A stability analysis of the system is performed, using the predation efficiency and a measure of the predator interference as bifurcation parameters. The admissible scenarios are much richer than in the case of prey-dependent trophic functions, investigated in Buffoni et al. (2011). General conditions for the number of equilibria, for the existence and stability of extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated. Numerical results illustrate the qualitative behaviours of the system, in particular the presence of limit cycles, of global bifurcations and of bistability situations.     

Global existence in a food chain model consisting of two competitive preys,one predator and chemotaxis

A model for the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition is considered. Besides diffusing, the predator species moves toward higher concentrations of a chemical substance produced by the prey. The prey, in turn, moves away from high concentrations of a substance secreted by the predators. The resulting reaction–diffusion system consists of three parabolic equations along with three elliptic equations describing the diffusion of the chemical substances. The local existence of nonnegative solutions is proved. Then uniform estimates in Lebesgue spaces are provided. These estimates lead to boundedness and global well-posedness for the system. Numerical simulations are presented and discussed.     

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.     

自全国各地100多位理事Road Machinery Branch of China Highway and Transportation Society He

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution (xs(t), 0, zs(t)) is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period τ, the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.     

A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect

In this paper, the predator–prey system with the Beddington–DeAngelis functional response is developed, by introducing a proportional periodic impulsive catching or poisoning for the prey populations and a constant periodic releasing for the predator. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. This model has the potential to protect predator from extinction, but under some conditions may also lead to extinction of the prey. That is, the system exists a locally stable prey-eradication periodic solution when the impulsive period satisfies an inequality. The condition for permanence is established via the method of comparison involving multiple Liapunov̀ functions. Further, by numerical simulation method the influences of the impulsive perturbations and mutual interference by predators on the inherent oscillation are investigated. With the increasing of releasing for the predator, the system appears a series of complex phenomenon, which include (1) period-doubling, (2) chaos attractor, (3) period-halfing. (4) non-unique dynamics (meaning that several attractors coexist).     

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

An attempt has been made to understand the role of top predator interference and gestation delay on the dynamics of a three species food chain model involving intermediate and top predator populations. Interaction between the prey and an intermediate predator follows the Volterra scheme (with Holling type IV functional response), while that between the top predator and its prey depends on Beddington–DeAngelis type functional response. Stability switches and Hopf-bifurcation occurs when the delay crosses some critical value. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Furthermore, the direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined using the center manifold theorem and normal form theory. Computer simulations have been carried out to illustrate the analytical findings. Different diagnostic tests, like, initial sensitivity, Lyapunov exponent, recurrence plot tests ensure the complex dynamical behavior of the model system. Finally, we observed the subcritical Hopf-bifurcation phenomena in the designed model system and the bifurcating periodic solution is unstable for the considered set of parameter values.     

Adaptive control of nonlinear complex Holling II predator–prey system with unknown parameters

In recent years, control of nonlinear complex predator–prey systems has attracted the attention of many researchers. The previous works have some weaknesses such as neglecting the consideration of the effects of both model uncertainties and unknown parameters and having an infinite time of convergence. To overcome the mentioned shortages, this article solves the problem of robust control of nonlinear complex Holling type II predator–prey system in a given finite time. It is assumed that the parameters of the system are fully unknown in advance and some uncertainties perturb the system's dynamics. To tackle the system unknown parameters, some adaptation laws are introduced. Thereafter, a robust switching controller is proposed to finite‐timely stabilize the predator–prey system. An illustrative example demonstrates the efficiency and usefulness of the proposed control strategy. © 2015 Wiley Periodicals, Inc. Complexity 21: 260–266, 2016