Continuous- discrete mathematical model for control of growth of a plant population in a given time period under a given budget

Abstract

The paper discusses an economically viable way of controlling biomass of a vegetable plant population under the use of fertilizer in a given span of time under a given budget of expenditure. The span of time is divided in some suitable consecutive periods of equal duration P, called cohorts. The treatment is done on the available population of plant at the beginning of each cohort for a suitable time; the continuous dynamics in the change of biomass for this time period is governed by an ordinary differential equation involving total effort exerted in treating the initial population. Taking this improved value at the end of the time period as the initial value, the biomass of the population is allowed to move under its normal continuous dynamics given by Logistic growth equation for the rest of the time of that cohort.  The final concentration of the biomass at the end of the first cohort is obtained by following the above two types of dynamics. This is also considered as the starting biomass for the next cohort. The same process adopted for the first cohort is repeated for calculating the improvement of the biomass in the second cohort and the whole process is repeated till the end of the final cohort is reached. Next an objective function is formed for the given span of time. This measures the net profit in getting improvement in the weight of the biomass less the cost involved in the process of improving the weight of the biomass for the given period of time. As the analysis is done in considering different cohorts at regular intervals of time, so it is a discrete model. As within each cohort, the dynamics takes place continuously, so it is a continuous model too. As a whole, the model is found to be a continuous- discrete model. Hence method of optimal control for continuous-discrete model is used to determine how the treatment at the starting of each cohort be adjusted, depending on the allocated budget, so that the total net profit is maximum.