Positive Steady States of a Prey-predator Model with Diffusion and Non-monotone Conversion Rate

In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.     

Permanence and Global Attractivity of a Discrete Semi-Ratio-Dependent Predator-Prey System with Holling Ⅳ Type Functional Response

In this paper,we investigate a discrete semi-ratio dependent predator-prey system with Holling IV type functional response.For general nonautonomous case,sufficient conditions which ensure the permanence and the global stability of the system are obtained.Meanwhile,we discuss the existence of the positive periodic solution and global stability of the system.     

The Dynamics of a Predator-prey Model with Ivlev''''s Functional Response Concerning Integrated Pest Management

A mathematical model of a predator-prey model with Ivlev's functional response concerning inte-grated pest management(IPM)is proposed and analyzed.We show that there exists a stable pest-eradicationperiodic solution when the impulsive period is less than some critical values.Further more,the conditions forthe permanence of the system are given.By using bifurcation theory,we show the existence and stability ofa positive periodic solution.These results are quite different from those of the corresponding system withoutimpulses.Numerical simulation shows that the system we consider has more complex dynamical behaviors.Finally,it is proved that IPM stragey is more effective than the classical one.     

The Dynamics of a Predator-prey Model with Ivlev＇s Functional Response Concerning Integrated Pest Management

A mathematical model of a predator-prey model with Ivlev‘s functional response concerning integrated pest management (IPM) is proposed and analyzed. We show that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical values, Further more, the conditions for the permanence of the system are giverl. By using bifurcation theory, we show the existence and stability of a positive periodic solution. These results are quite different from those of the corresponding system without impulses. Numerical simulation shows that the system we consider has more complex dynamical behaviors.Finally, it is proved that IPM stragey is more effective than the classical one.     

On the Dynamics of Impulsive Predator-Prey Systems with Beddington–Deangelis-Type Functional Response

Ukrainian Mathematical Journal - We study a two-dimensional predator-prey system with Beddington–DeAngelis-type functional response with pulses in a periodic environment. In the analyzed...     

Periodic Solutions of a Delayed Predator-Prey Model with Stage Structure for Prey

A periodic predator-prey model with stage structure for prey and time delays due to negative feed-back and gestation of predator is proposed.By using Gaines and Mawhin's continuation theorem of coincidencedegree theory,sufficient conditions are derived for the existence of positive periodic solutions to the proposedmodel.Numerical simulations are presented to illustrate the feasibility of our main result.     

On the Predator-Prey System with Holling-（n ＋ 1） Functional Response

The qualitative properties of a predatorprey system with Holling-（n ＋ 1） functional response and a fairly general growth rate are completely investigated. The necessary and sufficient condition to guarantee the uniqueness of limit cycles is given. Our work extends the previous relevant results in the reference.     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Global Solvability for a Predator-Prey Model with Prey-Taxis and Rotational Flux Terms∗

In this paper the author investigates the following predator-prey model with prey-taxis and rotational flux terms ( ut = ?u ? ? · (uS(x, u, v)?v) + γuF(v) ? uh(u), x ∈ ?, t > 0, vt = D?v ? uF(v) + f(v), x ∈ ?, t > 0 (?) in a bounded domain with smooth boundary. He presents the global existence of generalized solutions to the model (?) in any dimension.     

Sufficient Stochastic Maximum Principle in a Regime-Switching Diffusion Model

We prove a sufficient stochastic maximum principle for the optimal control of a regime-switching diffusion model. We show the connection to dynamic programming and we apply the result to a quadratic loss minimization problem, which can be used to solve a mean-variance portfolio selection problem.     

Bifurcations in a Diffusive Predator–Prey Model with Beddington–DeAngelis Functional Response and Nonselective Harvesting

In this paper, we discuss the dynamics of a predator–prey model with Beddington–DeAngelis functional response and nonselective harvesting. By using the Lyapunov–Schmidt reduction, we obtain the existence of spatially nonhomogeneous steady-state solution. The stability and existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the change of a specific parameter are investigated by analyzing the distribution of the eigenvalues. We also get an algorithm for determining the bifurcation direction of the Hopf bifurcating periodic solutions near the nonhomogeneous steady-state solution. Finally, we show some numerical simulations to verify our analytical results.

     

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-Ⅳ Functional Response

In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳfunctionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.     

Hypoellipticity and Ergodicity of the Wonham Filter as a Diffusion Process

The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hörmander’s condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process.     

Structure of Coexistence States for a Class of Quasilinear Elliptic Systems

The structure of positive solutions of the p-Laplacian systems is discussed via bifurcation theory and monotone techniques.     

A Predator�CPrey Model with a Holling Type I Functional Response Including a Predator Mutual Interference

The most widely used functional response in describing predator–prey relationships is the Holling type II functional response, where per capita predation is a smooth, increasing, and saturating function of prey density. Beddington and DeAngelis modified the Holling type II response to include interference of predators that increases with predator density. Here we introduce a predator-interference term into a Holling type I functional response. We explain the ecological rationale for the response and note that the phase plane configuration of the predator and prey isoclines differs greatly from that of the Beddington–DeAngelis response; for example, in having three possible interior equilibria rather than one. In fact, this new functional response seems to be quite unique. We used analytical and numerical methods to show that the resulting system shows a much richer dynamical behavior than the Beddington–DeAngelis response, or other typically used functional responses. For example, cyclic-fold, saddle-fold, homoclinic saddle connection, and multiple crossing bifurcations can all occur. We then use a smooth approximation to the Holling type I functional response with predator mutual interference to show that these dynamical properties do not result from the lack of smoothness, but rather from subtle differences in the functional responses.     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

Positive Steady States of a Competitor- Competitor-Mutualist Model

In this paper we deal with the positive steady states of a Competitor-Competitor-Mutualist modelwith diffusion and homogeneous Dirichlet boundary conditions.We first give the necessary conditions,and thenestablish the sufficient conditions for the existence of positive steady states.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.