EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation.We obtain a critical value λ1^D(Ω0),and demonstrate that the existence of the predator inΩ0 only depends on the relationship of the growth rateμof the predator and λ1^D(Ω0),not on the prey.Furthermore,whenμ<λ1^D(Ω0),we obtain the existence and uniqueness of its positive steady state solution,while whenμ≥λ1^D(Ω0),the predator and the prey cannot coexist inΩ0.Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding regionΩ0,which is different from that of the classical Lotka-Volterra predator-prey model.

ON A MULTI-DELAY LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH FEEDBACK CONTROLS AND PREY DIFFUSION

This article is focusing on a class of multi-delay predator-prey model with feedback controls and prey diffusion. By developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the globally attractivity of positive solution for the predator-prey system.Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator-prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system.

The Stochastic Asymptotic Stability Analysis in Two Species Lotka-Volterra Model

The asymptotic stability of two species stochastic Lotka-Volterra model is explored in this paper. Firstly, the Lotka-Volterra model with random parameter is built and reduced into the equivalent deterministic system by orthogonal polynomial approximation. Then, the linear stability theory and Routh-Hurwitz criterion for nonlinear deterministic systems are applied to the equivalent one. At last, at the aid of Lyapunov second method, we obtain that as the random intensity or statistical parameter of random variable is changed, the stability about stochastic Lotka-Volterra model is different from the deterministic system.

Dynamic Analysis of a Predator-Prey Model with Holling-II Functional Response

A predator-prey model with linear capture term Holling-II functional response was studied by using differential equation theory. The existence and the stabilities of non-negative equilibrium points of the model were discussed. The results show that under certain limited conditions, these two groups can maintain a balanced position, which provides a theoretical reference for relevant departments to make decisions on ecological protection.

Canid Social Structure and Density Dependence Improve Predator-Prey Models of <i>Canis latrans</i>and <i>Lepus californicus</i>in Curlew Valley, UT

Prominent examples of predator-prey oscillations between prey-specific predators exist, but long-term data sets showing these oscillations are uncommon. We explored various models to describe the oscillating behavior of coyote (Canis latrans) and black-tailed jackrabbits (Lepus californicus) abundances in a sagebrush-steppe community in Curlew Valley, UT over a 31-year period between 1962 and 1993. We tested both continuous and discrete models which accounted for a variety of mechanisms to discriminate the most important factors affecting the time series. Both species displayed cycles in abundance with three distinct peaks at ten-year intervals. The coupled oscillations appear greater in the mid-seventies and a permanent increase in the coyote density seems apparent. Several factors could have influenced this predator-prey system including seasonality, predator satiation, density dependence, social structure among coyotes, and a change in the coyote bounty that took place during the course of data collection. Maximum likelihood estimation was used to obtain parameter values for the models, and Akaike Information Criterion (AIC) values were used to compare models. Coyote social structure and limiting resources in the form of density-dependence and satiation seemed to be important factors affecting population dynamics.

Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model

The Lotka-Volterra predator-prey model is widely used in many disciplines such as ecology and economics. The model consists of a pair of first-order nonlinear differential equations. In this paper, we first analyze the dynamics, equilibria and steady state oscillation contours of the differential equations and study in particular a well-known problem of a high risk that the prey and/or predator may end up with extinction. We then introduce exogenous control to reduce the risk of extinction. We propose two control schemes. The first scheme, referred as convergence guaranteed scheme, achieves very fine granular control of the prey and predator populations, in terms of the final state and convergence dynamics, at the cost of sophisticated implementation. The second scheme, referred as on-off scheme, is very easy to implement and drive the populations to steady state oscillation that is far from the risk of extinction. Finally we investigate the robustness of these two schemes against parameter mismatch and observe that the on-off scheme is much more robust. Hence, we conclude that while the convergence guaranteed scheme achieves theoretically optimal performance, the on-off scheme is more attractive for practical applications.

Stability Analysis and Hopf Bifurcation for ODE System of Predator-Prey Model with Mutual Interference

In this paper, the temporal and spatial patterns of a diffusive predator-prey model with mutual interference under homogeneous Neumann boundary conditions were studied. Specifically, first, taking the intrinsic growth rate of the predator as the parameter, we give a computational and theoretical analysis of Hopf bifurcation on the positive equilibrium for the ODE system. As well, we have discussed the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.

Existence of Four Periodic Solutions of a Ratio-Dependent Predator-Prey Model with Exploited Terms

We study a non-autonomous ratio-dependent predator-prey model with exploited terms. This model is of periodic coefficients, which incorporates the periodicity of the varying environment. By means of the coincidence degree theory, we establish sufficient conditions for the existence of at least four positive periodic solutions of this model.

Stability and Bifurcation Analysis of a Discrete Predator-Prey Model with Mixed Holling Interaction

In this paper,a discrete Lotka-Volterra predator-prey model is proposed that considers mixed functional responses of Holling types I and III.The equilibrium points of the model are obtained,and their stability is tested.The dynamical behavior of this model is studied according to the change of the control parameters.We find that the complex dynamical behavior extends from a stable state to chaotic attractors.Finally,the analytical results are clarified by some numerical simulations.

Traveling wave for a time-periodic Lotka-Volterra model with bistable nonlinearity

This paper studies bistable wavefronts of a diffusive time-periodic Lotka-Volterra system.We obtain a new condition for the existence,uniqueness and stability of bistable timeperiodic traveling waves.This condition is sharp and greatly improves the result established in the reference(X.Bao and Z.Wang,Journal of Differential Equations,255(2013)2402-2435).An example is given to demonstrate our consequence.

Global Stability of A Stochastic Predator-prey Model with Stage-structure

In this paper, a stochastic predator-prey model with stage structure for predatorand ratio-dependent functional response is concerned. Sufficient conditions for the globalasymptotic stability of positive equilibrium are established. Some numerical simulations arecarried out to illustrate the theoretical results.

Spatial pattern formation of a ratio-dependent predator-prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator-prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.

Global Dynamics of a Predator-prey Model

In this paper, we consider a predator-prey model. A sufficient conditionis presented for the stability of the equilibrium, which is different from the one for themodel with Hassell-Varley type functional response. Furthermore, by constructing aLyapunov function, we prove that the positive equilibrium is asymptotically stable.

Canard Solutions in a Predator-Prey Model

The canard explosion phenomenon in a predator-prey model with Michaelis-Menten functional response is analyzed in this paper by employing the geometric singular perturbation theory. First, some turning points, such as, fold point, transcritical point, pitchfork point, canard point, are identified;then Hopf bifurcation, relaxation oscillation, together with the canard transition from Hopf bifurcation to relaxation oscillation are discussed.

Dynamic analysis of a predator-prey model of Gause type with Allee effect and non-Lipschitzian hyperbolic-type functional response

In this work,we study a predator-prey model of Gause type,in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response,which is neither differentiable nor locally Lipschitz at the predator axis.This kind of functional response is an extension of the so-called square root functional response,used to model systems in which the prey have a strong herd structure.We study the behavior of the solutions in the first quadrant and the existence of limit cycles.We prove that,for a wide choice of parameters,the solutions arrive at the predator axis in finite time.We also characterize the existence of an equilibrium point and,when it exists,we provide necessary and sufficient conditions for it to be a center-type equilibrium.In fact,we show that the set of parameters that yield a center-type equilibrium,is the graph of a function with an open domain.We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation.In particular,we guarantee the existence of a unique limit cycle,for small perturbations of the system.

Bifurcations,chaos analysis and control in a discrete predator-prey model with mixed functional responses

Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different functional responses(Holling type I and II functional responses)is discussed in this paper,which depicts a complex population relationship.The local dynamical behaviors of the interior fixed point of this system are studied.The detailed analysis reveals this system undergoes flip bifurcation and Neimark-Sacker bifurcation.Especially,we prove the existence of Marotto's chaos by analytical method.In addition,the hybrid control method is applied to control the chaos of this system.Numerical simulations are presented to support our research and demonstrate new dynamical behaviors,such as period-10,19,29,39,48 orbits and chaos in the sense of Li-Yorke.

Bifurcation analysis of a diffusive predator-prey model with hyperbolic mortality and prey-taxis

In this paper,we study a diffusive predator-prey model with hyperbolic mortality and prey-taxis under homogeneous Neumann boundary condition.We first analyze the influence of prey-taxis on the local stability of constant equilibria.It turns out that prey-taxis has influence on the stability of the unique positive constant equilibrium,but has no influence on the stability of the trivial equilibrium and the semi-trivial equilibrium.We then derive Hopf bifurcation and steady state bifurcation related to prey-taxis,which imply that the prey-taxis plays an important role in the dynamics.

Stability of a delayed predator prey model in a random environment

The stability of the first-order and second-order solution moments for a Harrison-type predator-prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Ito interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.

Moment stability for a predator–prey model with parametric dichotomous noises

In this paper, we investigate the solution moment stability for a Harrison-type predator-prey model with parametric dichotomous noises. Using the Shapiro-Loginov formula, the equations for the first-order and second-order moments are obtained and the corresponding stable conditions are given. It is found that the solution moment stability depends on the noise intensity and correlation time of noise. The first-order and second-order moments become unstable with the decrease of correlation time. That is, the dichotomous noise can improve the solution moment stability with respect to Gaussian white noise. Finally, some numerical results are presented to verify the theoretical analyses.

On the Dynamics of a Stochastic Ratio-Dependent Predator-Prey System with Infection for the Prey

In this paper, we investigate the dynamics of a stochastic predator-prey model with ratio-dependent functional response and disease in the prey. Firstly, we prove the existence and uniqueness of the positive solution for the stochastic model by using conventional methods. Then we obtain the threshold <img alt="" src="Edit_0a62b9be-7934-457b-aca3-af3420f5b5ee.png" /> for the infected prey population, that is, the disease will tend to extinction if <img alt="" src="Edit_e6cd63f6-de07-42be-a22a-8750d6c8aac9.png" />< 1, and it will exist in the long time if <img alt="" src="Edit_5964fdd8-a9fe-4dc2-b897-f4206f046f65.png" />> 1. Finally, the sufficient condition on the existence of a unique ergodic stationary distribution is obtained, which indicates that all the populations are permanent in the time mean sense. Numerical simulations are conducted to verify our analysis results.