Fractional order model for yield through diagnosed/undiagnosed soil
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Fractional order model for yield through diagnosed/ undiagnosed soil Nita H. Shah1 · Ekta N. Jayswal1 · Purvi M. Pandya1 Accepted: 16 November 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract Farming is the basic economy of any country. Soil fertility and water are key resources for yielding. The fractional order compartmental model is prepared in Caputo sense to improve the density of yield through diagnosed and undiagnosed soil. To measure the growing intensity of yield, the basic reproduction number is formulated using the next generation matrix method for integer order non-linear dynamical system. Local stability analysis is described for both the equilibrium points; undiagnosed soil free and optimum equilibrium point. Global stability is exhaustively computed by generating a Lyapunov function. With reference to the basic reproduction number, bifurcation analysis has been defined which expresses the chaotic nature of soil fertility. Moreover, optimal control theory is applied in the present fractional model to optimize yield. And optimality conditions are calculated with the help of Pontryagin maximum principle. The numerical simulation for different fractional orders is performed concerning validated data to analyze the behavior with respect to the order. Keywords Fractional-order system · Caputo derivative · Bifurcation · Optimal control · Chaos
Communicated by José Alberto Cuminato. * Nita H. Shah [email protected] Ekta N. Jayswal [email protected] Purvi M. Pandya [email protected] 1
Department of Mathematics, Gujarat University, Ahmedabad, Gujarat 380009, India
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
1 Introduction The agricultural sector is the principal source of employment in almost all of the developing countries. In other words, agriculture plays a strategic and effective role in economic development of a country. One of the major properties of the environment which is needed for yielding is soil. In order to develop yielding, we precisely need concrete information about which factor must be considered and how it can be determined to improve the quality of soil for a particular crop. Hence, the concepts of nutrient diagnosis of soil and its importance are led to be considered by growers and it has been systematized in many countries and laboratories around the world. Diagnosed soil means one should identify the nature of land by investigation of its symptoms. Sometimes this soil may be too fertile and yield production is not feasible in that area. Diagnosis of soil helps to warn about pH imbalance, potential nutrient insufficiencies or extra damaging soluble salts. This supports, to assist the analysis of plant culture problems, to protect the environment from contamination by the overspill of excess fertilizers, to improve the nutritional balance of the soil and to avoid waste of money and energy by knowing the proper quantity of fertilizer needed according to the nutrient requirement of crops. Therefore, the s
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