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We study some qualitative behaviour of a modified discrete-time host–parasitoid model. Modification of classical Nicholson–Bailey model is considered by introducing Pennycuick growth function for the host population. Furthermore, the existence and uniqueness of positive equilibrium point of proposed system is investigated. We prove that the positive solutions of modified system are uniformly bounded and the unique positive equilibrium point is locally asymptotically stable under certain parametric conditions. Moreover, it is also investigated that system undergoes Neimark–Sacker bifurcation by using standard mathematical techniques of bifurcation theory. Complexity and chaotic behaviour are confirmed through the plots of maximum Lyapunov exponents. In order to stabilise the unstable steady state, the feedback control strategy is introduced. Finally, in order to support theoretical discussions, numerical simulations are provided.

In this paper, we study the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form x n+1 = αx n-3 β + γ y n y n-1 y n-2 y n-3

We study qualitative behavior of a modified prey–predator model by introducing density-dependent per capita growth rates and a Holling type II functional response. Positivity of solutions, boundedness and local asymptotic stability of equilibria were investigated for continuous type of the prey–predator system. In order to discuss the rich dynamics of the proposed model, a piecewise constant argument was implemented to obtain a discrete counterpart of the continuous system. Moreover, in the case of a discrete-time prey–predator model, the boundedness of solutions and local asymptotic stability of equilibria were investigated. With the help of the center manifold theorem and bifurcation theory, we investigated whether a discrete-time model undergoes period-doubling and Neimark–Sacker bifurcation at its positive steady-state. Finally, two novel generalized hybrid feedback control methods are presented for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. In order to illustrate the effectiveness of the proposed control strategies, numerical simulations are presented.

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In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, and global behavior of equilibrium points of a discrete Lotka-Volterra model given bywhere parameters α, β, γ , δ, , η ∈ R + , and initial conditions x 0 , y 0 are positive real numbers. Moreover, the rate of convergence of a solution that converges to the unique positive equilibrium point is discussed. Some numerical examples are given to verify our theoretical results. MSC: 39A10; 40A05

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