A Darwinian Beverton-Holt model with immigration effect

In this study, we investigate the dynamics of a discrete-time evolutionary Beverton-Holt model under the immigration effect. The model tracks the dynamics of the population, coupled with that of the population mean trait. By using the center manifold theorem and bifurcation theory, we establish that the system undergoes Neimark-Sacker and period-doubling bifurcations when the immigration effect passes some critical values. Bifurcation diagrams, maximum Lyapunov exponents, and time series are examples of numerical simulations that not only illustrate our theoretical analysis but also show the complicated dynamical behaviors of this model. Furthermore, we applied the hybrid control method and the exponential method to achieve the asymptotic stability of an unstable equilibrium.

Automated summary

In this study, the dynamics of a discrete-time evolutionary Beverton-Holt model under the immigration effect are investigated. It is established that the system undergoes bifurcations when the immigration effect exceeds certain critical values. Numerical simulations, such as bifurcation diagrams and time series, are used to illustrate the theoretical analysis and demonstrate the complex dynamical behaviors of the model.