Leslie-Gower Type Predator-prey Model Research Articles

BIFURCATIONS IN A LESLIE-GOWER TYPE PREDATOR-PREY MODEL WITH A RATIONAL NON-MONOTONIC FUNCTIONAL RESPONSE

A Leslie-Gower type predator-prey model including group defense formation is analyzed. This phenomenon, described by a non-monotonic function originates interesting dynamics; positiveness, boundedness, permanence of solutions, and existence of up to three positive equilibria are established. The solutions are highly sensitive to initial conditions since there exists a separatrix curve dividing their behavior. Two near trajectories can have far omega-limit sets. The weakness of a singularity is established showing two limit cycles can exist. Numerical simulations endorse the analytical outcomes.

Estabilidad de un modelo depredador-presa tipo Leslie Gower con un efecto Allee fuerte con retardo

In this paper, a modified Leslie-Gower type predator-prey model introducing in prey population growth a delayed strong Allee effect is studied. The Leslie-Gower model with Allee effect has none, one or two positive equilibrium points but the incorporation of a time delay in the growth rate destabilizes the system, breaking the stability when the delay cross a critical point. The existence of a Hopf bifurcation is studied in detail and the numerical simulations confirm the theoretical results showing the different scenarios. We present biological interpretations for species prey-predator type.

A Leslie–Gower type predator-prey model considering herd behavior

A Leslie–Gower type predator-prey model considering herd behavior

Spatiotemporal complexity in a Leslie-Gower type predator-prey model near Turing-Hopf point

The Leslie-Gower type predator-prey system with the ratio-dependent Holling III functional response and Neumann boundary conditions is investigated in this paper. First, the boundedness results of both parabolic and elliptic equations are presented. Hereafter, the existence of the codimension-two Turing-Hopf point (C2THP) is identified, where the Turing and the Hopf modes intersect. To further explore the spatiotemporal dynamics near the C2THP, it is necessary to derive the amplitude equations, however, there are few results about that in the two-dimensional domain. Here the method of weakly nonlinear analysis is adopted to derive the amplitude equations. The temporal patterns, hexagonal patterns, and plane wave patterns, as well as the sufficient conditions of their existence and stability, can be presented through amplitude equations.

Permanence and periodic solution for a modified Leslie-Gower type predator-prey model with diffusion and non constant coefficients

In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T-periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results.

Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting

Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.

The Effect of Delay on A Diffusive Predator–Prey System with Modified Leslie–Gower Functional Response

In this paper, a delayed diffusive modified Leslie–Gower-type predator–prey model with Beddington–DeAngelis functional response subject to Neumann boundary condition is considered. The stability/instability of the non-negative equilibria and a detailed Hopf bifurcation analysis are investigated. And explicit formulas for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived by the theory of normal form and center manifold. In addition, some numerical simulations are carried out.

A Leslie–Gower-type predator–prey model with sigmoid functional response

In this work, a continuous-time predator–prey model of Leslie–Gower type considering a sigmoid functional response is analysed. Using the MatLab package some simulations of the dynamics are shown. Conditions for the existence of equilibrium points, their nature and the existence of at least one limit cycle in phase plane are established. The existence of a separatrix curve dividing the behaviour of trajectories is proved. Thus, two closed trajectories can have different ω-limits being highly sensitive to initial conditions. Moreover, for a subset of parameter values, it can be possible to prove that the point (0,0) can be globally asymptotically stable. So, both populations can go to extinction, but simulations show that this situation is very difficult. According to our knowledge no previous work exists analysing the model presented here. A comparison of the model here studied with the May–Holling–Tanner model shows a difference on the quantity of limit cycles.

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.

Coexistence states for a modified Leslie-Gower type predator-prey model with diffusion

This paper is concerned with a modified Leslie-Gower predator-prey model with general functional response under homogeneous Robin boundary conditions. We establish the existence of coexistence states by the fixed index theory on positive cones. As an example, we apply the obtained results to this model with Holling-type II functional response. Our results show that the intrinsic growth rates and the principle eigenvalues of the corresponding elliptic problems with respect to the Robin boundary conditions play more important roles than other parameters for the existence of positive solutions.MSC: 35J55, 37B25, 92D25.