Computational dynamics of the Nicholson-Bailey models

Abstract

In population dynamics, the Nicholson-Bailey model describes the host-parasitoid system which has been well studied since 1930 with a consideration that the parameters are all positive real numbers. In this article, the dynamics of the Nicholson-Bailey model $x_{n+1} = x_{n} (e^{r(1-\frac{x_{n}}{\kappa}) - ay_{n}})$ and $y_{n+1} = x_{n} (1-e^{-ay_{n}})$ is reinvestigated computationally where all the parameters are considered as real numbers. The model has all sorts of dynamical behavior such as chaotic, periodic and locally stable/unstable equilibriums. In addition, the dynamics of the scaled Nicholson-Bailey $x_{n+1}=(x_{n} + \alpha) (e^{r(1-\frac{(x_{n}+\alpha)}{\kappa})-a(y_{n}+\beta)})$ , $ y_{n+1}=(x_{n} +\alpha) (1-e^{-a(y_{n}+\beta)})$ where $\alpha$ and $\beta$ are scaling factors and of the noisy model $ x_{n+1}=x_{n} (e^{r(1-\frac{x_{n}}{\kappa})-ay_{n}})+\nu_{1}$ , $ y_{n+1}=x_{n} (1-e^{-ay_{n}})+\nu_{2}$ , where $(\nu_{1}, \nu_{2})$ is uniformly distributed noise over the interval (0,1), are also reconnoitered computationally.