Invasion dynamics of a population growth model with the Allee effect in a one-dimensional patchy structure

In this work, an ecological population model with the Allee effect is studied, and the dispersal rate in a one-dimensional patch is estimated. The population is observed to grow in accordance with the logistic-type map with the Allee effect, and the dynamics of the population growth is observed to be controlled by the growth rate and the threshold of the Allee effect. The spread of the species in a one-dimensional patch structure can be described by using the discrete diffusion equation, and the time evolution is continuous. In discrete space and continuous-time dynamics, the expansion velocity of the population is observed to depend strongly on the Allee threshold and the dispersal rate. Without the Allee effect, the invasion velocity increases as a function of the dispersal rate whereas, with the Allee effect, the invasion velocity depends on the dispersal rate, Allee threshold, and growth rate. Three regions are observed in the plane of the dispersal rate and growth rate: namely, the pinning state, the invading state, and the absorbing state.