Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.     

Existence and the dynamical behaviors of the positive solutions for a ratio-dependent predator–prey system with the crowing term and the weak growth

This paper deals with a ratio-dependent predator–prey system with the crowing term and the weak growth in the prey equation. Under the condition that the coefficient λ is less than a critical value ${\lambda }_1^D({\mathrm{\Omega }}_0)$, we obtain existence of multiple positive steady state solutions of the predator–prey system and the dynamical behaviors of its positive solutions. Our results show that the predator and the prey possess not only the common coexistence, but also the very weak coexistence which both of the predator and the prey are very low. Meantime, the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data. Therefore, our results may be used to explain some special phenomena which under some bad environment, the predator and the prey may still coexist.     

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.     

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.     

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.     

On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat

In this paper we study a system of reaction-diffusion equations arising from competition of two microbial populations for a single-limited nutrient with internal storage in an unstirred chemostat. The conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient. The reduced system (limiting system) generates a strongly monotone dynamical system in its feasible domain under a partial order. We construct suitable upper, lower solutions to establish the existence of positive steady-state solutions. Given the parameters of the reduced system, we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors. The primary conclusion is that the survival of species depends on species's intrinsic biological characteristics, the external environment forces and the principal eigenvalues of some scalar partial differential equations. We also lift the dynamics of the limiting system to the full system.     

Analysis of a prey-predator model with disease in prey

In this paper, a system of reaction-diffusion equations arising in eco-epidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the predator is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.     

Dynamics of an impulsively controlled Michaelis–Menten type predator–prey system

We study a predator–prey system with a Michaelis–Menten functional response and impulsive perturbations which contain chemical and biological control terms. By applying the Floquet theory, we establish conditions for the existence and stability of prey-free solutions of the system. We also show the existence of a positive periodic solution of the system by using the bifurcation theorem and find a sufficient condition that makes the system permanent. Moreover, numerical results on impulsive perturbations show that the system we consider can give birth to various kinds of dynamical behaviors.     

Stability,Steady‐State Bifurcations,and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.     

Dynamics of a stage-structured predator–prey model with prey impulsively diffusing between two patches

In this work, we propose a stage-structured predator–prey model, with prey impulsively diffusing between two patches. Using the discrete dynamical system determined by the stroboscopic map, we obtain a predator-extinction periodic solution. Further, the predator-extinction periodic solution is globally attractive. By the theory on the delay and impulsive differential equation, we prove that the investigated system is permanent. Our results indicate that the discrete time delay has influence to the dynamical behaviors of the investigated system.     

Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone

This paper is concerned with the stationary problem of a prey-predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity.     

Extinction and permanence of a two-prey two-predator system with impulsive on the predator

In this paper, the dynamic behaviors of a two-prey two-predator system with impulsive effect on the predator of fixed moment are investigated. By applying the Floquet theory of liner periodic impulsive equation, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is large than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining three species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.     

Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but contains an extra gestation delay term. Criteria for permanence and for predator extinction as well as the global attractiveness of the interior equilibrium are derived. The combined effects of the two delays and the degree of predator interference on the dynamical behaviors of the coexistence equilibrium are also studied both analytically and numerically. It is shown that complicated behaviors including chaotic and multi-periodic solutions may occur with the introduction of gestation delay, and that the predator interference can stabilize the system by simplifying the dynamical behaviors and enlarging the stability parameter fields.     

耗散对广义的WBK型耗散方程行波解的影响

运用平面动力系统理论对广义的WBK型耗散方程所对应的动力系统作了定性分析,给出了其在不同参数条件下的全局相图.研究了该方程行波解的性态与耗散系数r之间的关系,得到当耗散作用较大时行波表现为扭状孤波,当耗散作用较小时行波表现为衰减振荡解的结论.     

Turing pattern formation in a three species model with generalist predator and cross-diffusion

In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions.     

POSITIVE STEADY STATES AND DYNAMICS FOR A DIFFUSIVE PREDATOR-PREY SYSTEM WITH A DEGENERACY

In this article, we consider positive steady state solutions and dynamics for a spatially heterogeneous predator-prey system with modified Leslie-Gower and Holling-Type II schemes. The heterogeneity here is created by the degeneracy of the intra-specific pressures for the prey. By the bifurcation method, the degree theory, and a priori estimates, we discuss the existence and multiplicity of positive steady states. Moreover, by the comparison argument, we also discuss the dynamical behavior for the diffusive predator-prey system.     

On a predator-prey system with cross diffusion representing the tendency of predators in the presence of prey species

In this paper, a predator-prey system with cross-diffusion, representing the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species, under homogeneous Dirichlet boundary condition, is considered. Using the method of upper and lower solutions developed by Pao [C.V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal. 60 (2005) 1197-1217], sufficient conditions for the existence of positive solutions are provided when the induced cross diffusion coefficient is sufficiently small. Furthermore, the investigation of non-existence of positive solutions is also presented.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

A delay model for viral infection in toxin producing phytoplankton and zooplankton system

An eco-epidemiological delay model is proposed and analysed for virally infected, toxin producing phytoplankton (TPP) and zooplankton system. It is shown that time delay can destabilize the otherwise stable non-zero equilibrium state. The coexistence of all species is possible through periodic solutions due to Hopf bifurcation. In the absence of infection the delay model may have a complex dynamical behavior which can be controlled by infection. Numerical simulation suggests that the proposed model displays a wide range of dynamical behaviors. Different parameters are identified that are responsible for chaos.     

STABILITY PROPERTY OF A PREDATOR-PREY SYSTEM WITH A CONSTANT PROPORTION OF PREY REFUGE AND STAGE-STRUCTURE FOR PREY SPECIES

A predator-prey system with a constant proportion of prey refuge and stage-structure for prey species is proposed and studied in this paper. A set of conditions for the permanence of the system is obtained. The local stability of the system is discussed by the sign of eigenvalues. Furthermore, by using the iterative method, some suitable sufficient conditions for the global attractivity of the interior equilibrium is obtained. Our study shows that the constant proportion of prey refuge could lead to more complicate dynamic behaviors. Numerical simulations are also presented to illustrate the feasibility of the main results.