Gause-type Predator-prey Systems Research Articles

Global Hopf Bifurcation Of a Delayed Diffusive Gause-Type Predator-Prey System with the Fear Effect and Holling Type III Functional Response

In this paper, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G* of system (1.4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions' periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.

Bifurcation analysis in a predator–prey system with an increasing functional response and constant-yield prey harvesting

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis.

Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establisha prioriestimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

A predator-prey system with Holling-type functional response

We consider the Gause-type predator-prey system with a large class of growth and response functions, in the case where the response function is not smooth at the origin. We discuss the conditions under which this system has exactly one stable limit cycle or has a positive stable equilibrium point and we describe the basin of attraction of the stable limit cycle and the stable equilibrium point, respectively. Our results correct previous results of the existing literature obtained for the Holling response function x p /(a + x p), in the case where 0 < p < 1.

Neimark–Sacker bifurcation analysis on a numerical discretization of Gause-type predator–prey model with delay

This paper presents Neimark–Sacker bifurcation analysis for a kind of discrete Gause-type predator–prey system with time delay, which is obtained by using Euler discretization method. For biological reasons, we are only interested in positive solutions of system, thus some parameter conditions are given for the existence of a unique positive fixed point. Then by choosing delay as bifurcation parameter, and analyzing the associated characteristic equation, we obtain stability result of the positive fixed point. It is also demonstrated that Neimark–Sacker bifurcation occurs when the delay crosses some critical values. An explicit formula which determines the stability, direction and other properties of bifurcating periodic solution is derived by using the center manifold and normal form theory. Finally, a numerical example is given to support the analytic results.

Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.

Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

Periodic Solutions for Gause-Type Ratio-Dependent Predator-Prey Systems with Delays on Time Scales

This paper is devoted to periodic Gause-type ratio-dependent predator-prey systems with monotonic or nonmonotonic numerical responses on time scales. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solutions. In particular, our results improve and generalize some known ones.

Positive periodic solutions for impulsive Gause-type predator–prey systems

This paper is devoted to impulsive periodic Gause-type predator–prey systems with monotonic or non-monotonic numerical responses. With the help of a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions. As corollaries, some applications are listed. In particular, our results improve and generalize some known ones.

Existence of positive periodic solutions for neutral delay Gause-type predator–prey system

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions for neutral delay Gause-type predator–prey system. x ′ ( t ) = x ( t ) [ r ( t ) - a ( t ) x ( t - σ ( t ) ) - b ( t ) x ′ ( t - σ ( t ) ) ] - ϕ ( t , x ( t ) ) y ( t - τ 1 ( t ) ) , y ′ ( t ) = y ( t ) [ - d ( t ) + h ( t , x ( t - τ 2 ( t ) ) ) ] . In addition, our results are applicable to neutral delay predator–prey systems with different types of functional responses such as Holling-type II and Ivlev-type.

Positive periodic solutions for a class of neutral delay Gause-type predator–prey system

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions for neutral delay Gause-type predator–prey system { x ′ ( t ) = x ( t ) [ r ( t ) − a ( t ) x ( t − σ 1 ) − ρ x ′ ( t − σ 2 ) ] − ϕ ( t , x ( t ) ) y ( t − τ 1 ( t ) ) , y ′ ( t ) = y ( t ) [ − d ( t ) + h ( t , x ( t − τ 2 ( t ) ) ) ] . In addition, our results are applicable to neutral delay predator–prey systems with different types of functional responses such as Holling-type II and Ivlev-type.

Multiple periodic solutions in generalized Gause-type predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple positive periodic solutions in periodic Gause-type predator–prey systems with non-monotonic numerical responses and time delays. As corollaries, some applications are listed. In particular, our results improve and supplement those obtained by Chen [Y. Chen, Multiple periodic solutions of delayed predator–prey systems with type IV functional responses, Nonlinear Anal. Real World Appl. 5 (2004) 45–53].

POSITIVE PERIODIC SOLUTION FOR A GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DIFFUSION AND TIME DELAY

A two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solution for the system. As corollaries, some applications are listed. In particular, our results extend and improve some known results.

Positive periodic solutions in delayed Gause-type predator–prey systems

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of positive periodic solutions in delayed Gause-type predator–prey systems. Some known results are shown to be special cases of the presented paper.

Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems

The global stability of a multi-species interacting system has apparently important biological implications. In this paper we study the global stability of Gause-type predator-prey models by providing new criteria for the nonexistence of cycles and limit cycles. Our criteria have clear geometrical interpretations and are easier to apply than other methods employed in recent studies. Using these criteria and related techniques we are able to develop new results on the existence and uniqueness of cycles in Gause-type models with various growth and response functions.

A SURVEY OF CONSTRUCTING LYAPUNOV FUNCTIONS FOR MATHEMATICAL MODELS IN POPULATION BIOLOGY

In this paper we survey the construction of Lyapunov functions (or functionals) for various ecological models which take the form of ODE systems (or Reaction-Diffusion PDE systems). First we consider the resources-consumers type ecological models which study the competition of $n$ microorgansims for a single limiting resource or two complementary resources in the chemostat. Next we consider the Gause-type predator-prey systems and the Lesile-type predator-prey systems. From the Lyapunov functions of the predator-prey system we construct new Lyapunov functions for three-level food chain models and one prey two predators models. Suppose a Lyapunov function is known for an ecological model which takes the form of an ODE system. Then we construct a Lyapunov functional for the corresponding reaction-diffusion PDE systems. Open problems are indicated when there is gap in the theory.

Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response

The goal of this paper is to establish the uniqueness of limit cycles of the predator–prey systems with Beddington–DeAngelis functional response. Through a change of variables, the predator–prey system can be transformed into a better studied Gause-type predator–prey system. As a result, the uniqueness of limit cycles can be solved.

Persistence and Global Stability in a Delayed Gause-Type Predator–Prey System without Dominating Instantaneous Negative Feedbacks

A delayed Gause-type predator–prey system without dominating instantaneous negative feedbacks is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of constructing a suitable Lyapunov functional, sufficient conditions are derived for the local and global asymptotic stability of the positive equilibrium of the system.

Limit cycles in a generalized Gause-type predator-prey system

The qualitative behavior of solutions for a generalized Gause-type predator-prey system was studied. A large number of biological and bioeconomic models are special cases of this system. The system was investigated in the region D={(x, y)|x>0, y>0} because of the biological meaning of the system. The authors derived some sufficient conditions for the boundedness of the solutions and the existence of limit cycles of the system, which ensure that the system has at least one limit cycle. The theory of limit sets of autonomous plane systems and the theorem of cycle field of Poincare-Bendixson are efficiently employed in the research. The main results and their consequences presented not only generalize some known results, but also improve some corresponding results of other authors.

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system.

The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper. Through a simple change of variable, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.