Discrete Lotka–Volterra systems with time delay and its stability analysis

Abstract

We propose an extension of the discrete-time Lotka–Volterra (dLV) equations describing predator–prey dynamics with time delay $\tau$. Introducing time delay corresponds to considering multiple generations of each species and gives more expressive power to the model. For example, it becomes possible to model the situation where each individual is eaten only after it has grown up. In this paper, we focus on the system with minimal time delay ($\tau =1$) and analyze the stability of the system. In particular, we prove that when the number of species is three, the system exhibits the same asymptotic behavior as the original dLV system. For more general cases with an arbitrary odd number of species, we investigate the local stability of fixed points of the system with the help of the center manifold theory. It is shown that the fixed points that correspond to the asymptotic states of the original dLV system are locally stable.