Tractable fish growth models considering individual differences with an application to the fish Plecoglossus altivelis altivelis

Modeling fish growth is an important research topic in ecological and fishery sciences because body weight statistics directly affect the total biomass of fish in a habitat, which in turn affects their population dynamics. Many models of fish growth assume that the fish population in a habitat is homogenous, meaning that there is no physiological spectrum and, therefore, no size spectrum. Moreover, models that account for the size spectrum are not always analytically tractable. We present novel mathematical models of fish growth in which the body weight of each fish is assumed to follow a von Bertalanffy-type model whose proportionality coefficient, representing the maximum body weight, may differ among individual fish. This probabilistic description introduces the size spectrum into the model, owing to which the time-dependent probability density of this model is obtained explicitly. We also consider a misspecified version and a stochastic version of the model as advanced cases. We apply the first model to the real growth data of Plecoglossus altivelis altivelis as a keystone fish species in Japan. The model successfully reproduces the skewed size spectrum of this fish species over multiple years. We further use the stochastic model to investigate how fish growth dynamics are affected by environmental fluctuations.