Migration Dynamics of Fish Schools in Heterothermal Environments

A mechanistic theory is presented to illustrate the ecologically scaled behavior of highly organized fish schools, which is principally controlled by thermal structure of the habitat. Starting from the equation governing the schooling dynamics of fish, and incorporating the hypothesized memory mechanism of thermal history into the model of klinokinesis, a mathematical formalism is developed which relates the individual processes of perception of temperature-gradients to the thermoregulatory migration of schools. The migration dynamics of schools in heterothermal environments is formulated as the modified Fickʹs law of diffusion in the presence of external potential field, which gives the advection-telegraph equation in combination with the continuity equation. A behavioral basis for the environmental potential function is proposed, which has been postulated ad hoc as a driving force of drift velocity in modeling biological aggregation patterns. It is predicted that the distribution of schools self-organizes into a critical state.