ZizhengYang/Phase_transition

This work expands the application of Gibb's canonical ensemble to ecological modeling. Previously, the ensemble was used to model species competing for resources in an environment with a fixed carrying capacity. Now, we extend the model to also cover the population growth before the carrying capacity is reached, by assuming the existence of an imaginary species, and observe a phase transition to the previously studied resource-limited situation. Standard quantities from statistical mechanics are reinterpreted in our biological context. For example, energies become growth rates, temperature becomes time, and the continuous thermodynamic potential gives a normalized logarithmic total population count before the phase transition, and a mortality count afterwards. In addition, when a phase space interpretation is given to the quasilinearization used for a classical solution of the differential equations governing the resource-limited situation, it is possible to extend the interpretation regularly to the period of unconstrained growth that precedes the phase transition.