Stability Analysis of Some Population Models With Time Delay and Harvesting - Toaha, Syamsuddin (2006)
This research presents the development and extension of some models for the growth rates of population. The existing models, i.e., logistic population model for single population, predator – prey model, Wangersky – Cunningham model, competing model and symbiosis model for two interaction populations, are extended by considering time delay, harvesting function and time delay in harvesting term in the models to get some new population models. The time delay is considered in the model to make the model more accurate because the growth rate of population does not only depend on the present size of population but also depends on past information. The current size of the population does not immediately change the growth rate of the population, but there is a time delay. The population as a valuable stock, for example fish population, is then harvested. The considered harvesting functions in the new population models are constant effort and constant quota of harvesting. The new models are then analyzed to determine the stability of their equilibrium points. Before determining the stability of the equilibrium points, we provide the necessary and sufficient conditions for the existence of the equilibrium points. Since we consider population model, we just investigate the nonnegative equilibrium points. For some models, we determine only the sufficient conditions for the existence of the positive equilibrium points. The value of time delay, level of harvesting, initial size of populations, and parameters of the models need to be controlled so that the populations will not be extinct for a long time and also the populations give maximum profit. The methods used to study the stability of the equilibrium point are linearization model around the equilibrium point, eigenvalues method, phase plane analysis, and plotting trajectories around the equilibrium point. In order to determine the stability of the equilibrium point, we inspect the sign of real parts of the eigenvalues. The graphs of the trajectories are plotted to visualize the behavior of the trajectories. For the models with constant effort of harvesting, we determine the critical value of the effort that maximizes the profit and does not affect the stability of the equilibrium point. Some new theorems are constructed and proved to determine the time delay margin, stability switches and stability intervals. We find that there exists a certain condition so that the positive equilibrium point of the models becomes stable. From the analysis we find that the time delay can induce instability, stability switches and bifurcations in all the models except for the symbiosis model with time delay in harvesting term. The analysis also shows that for the models with constant effort of harvesting, there exists a critical value for the effort of harvesting that maximizes the profit function and maintains the stability of the equilibrium point. When we control the values of the parameters, level of harvesting, and time delay, the positive equilibrium point can be found and possibly stable. The existence of the populations also depends on the initial value of the population since we just consider local stability. For the models without time delay and harvesting, we find the global stability of the positive equilibrium point. For the models with a time delay, there exists either a time delay margin or some stability switches so that the positive equilibrium point remains stable on the stability interval. The maximum profit can be found without affecting the stability of the equilibrium point when the values of parameters and the level of constant efforts of harvesting are strictly controlled.