This paper considers a Lotka-Volterra model with time delays and delay dependent parameters. The linear stability conditions are obtained with characteristic root method. The Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived. Finally,...

This paper examines the process and final result of extending the basic Lotka-Volterra predator-prey model to include different age groups within each population. Basic analysis of the results were performed to determine the relevance of the proposed extension. Calculations and graphing were handled with MatLab. The natural outcome of the model should be either a periodic cycle about an equilib...

Integrability and linearizability of the Lotka–Volterra systems are studied. We prove sufficient conditions for integrable but not linearizable systems for any rational resonance ratio. We give new sufficient conditions for linearizable Lotka–Volterra systems. Sufficient conditions for integrable Lotka–Volterra systems with 3 : q resonance are given. In the particular cases of 3 : 5 and 3 : 4 r...

We consider a three-dimensional discrete dynamical system that describes an application to economics of a generalization of the Lotka–Volterra prey–predator model. The dynamic model proposed is used to describe the interactions among industrial clusters (or districts), following a suggestion given by [23]. After studying some local and global properties and bifurcations in bidimensional Lotka–V...

In this paper we stochastically perturb the delay Lotka–Volterra model ẋ(t)= diag(x1(t), . . . , xn(t))[A(x(t)− x̄)+B(x(t − τ )− x̄)] into the stochastic delay differential equation (SDDE) dx(t)= diag(x1(t), . . . , xn(t)){[A(x(t)− x̄)+B(x(t − τ )− x̄)]dt + σ (x(t)− x̄)dw(t)}. The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially ...

We present a qualitative analysis of the Lotka-Volterra differential equation within rectangles that are transverse with respect to the flow. In similar way to existing works on affine systems (and positively invariant rectangles), we consider here nonlinear Lotka-Volterra n-dimensional equation, in rectangles with any kind of tranverse patterns. We give necessary and sufficient conditions for ...