A theoretical model with time-delay, competition and transitions among cohorts to study tilapia populations

Abstract

A tilapia culture system can have many cohorts mixed over a growing time and these coexist with some competitive pressure that affect the populational structure from the beginning to harvesting time. This paper presents the first model that integrates competition, transition rates among cohorts, and time-delay, which aims to know the biomass proportions corresponding to each cohort at the end of a period time in regards to the initial populational structure. Through several simulations with different value intervals, the sensitivity of model to its parameters was analyzed. It was observed that the transition rates among cohorts could define the final populational structure. The highest values provoked a fall of biomass proportions of all cohorts about zero, while that when the medium values were used the solutions could be closer to natural phenomenon. When the values of competition rates between cohorts were tested, there was an inverse relationship with the maximum point reached by the small and medium fish cohorts because the large individuals have a greater impact on the small class. Such solutions presented oscillation periods that increased with the highest values assigned to this variable. As the time-delay values were varied, in high competition conditions, the solutions approximated a periodic orbit that was far from the equilibrium. It was due to the negative effect of the large fish cohort on the other cohorts was possible in short and more frequent time periods or long and infrequent time periods. In this model the positive effect of competition towards the large fish cohort depended on the time-delay. Based on this one might suggest that the time-delay could also be incorporated into competition and growth rates of small and medium fish cohorts. Despite that the model still has the disadvantage of incorporating constant transition rates among cohorts, it was possible observe its sensitivity to the changes of these variables. The model provides final proportional biomass once the solutions have been stabilized. This permits us to predict the changes in the biomass of a population at time of harvesting.