A one dimensional model of population growth

• A model based on individual–individual distance-dependent interaction is proposed.
• The model presents full analytic solution.
• A rich phase diagram to which the population has access is observed.
• The phases are: exponential growth, convergence, divergence, and extinction.

In this work, a one dimensional population growth model is proposed. The model, based on the cooperative and competitive individual–individual distance-dependent interaction, allows us to get a full analytical solution. With this analytical approach, it was possible to investigate the dynamics of the population according to some parameters, as intrinsic growth rate, strength of the interaction between individuals, and the distance-dependent interaction. As a consequence of the individuals’ interaction, a rich phase diagram to which the population has access was observed. The phases observed are: convergence to carrying capacity, exponential growth, divergence at finite time, and extinction. Moreover, it was also observed that some phases are strictly dependent on the initial condition. For instance, in the cooperative regime with negative intrinsic growth rate, the population can diverge or become extinct according to the initial population size. The phases accessible to the population can be seen as a macroscopic behavior which emerges from the interaction among the individuals (the microscopic level).