Dynamical analysis of toxin-mediated deform biological population due to delayed toxicants uptake responses

Deform Biological Population

Abstract

In this prescribed model, we are proposing and analyzing to investigate the impact of externally emitted toxicants in a four-compartment
population model accompanied by the deformation delay($\tau_1$) and in toxicants uptake delay($\tau_2)$ through the biological population.
Results show that when the deformation delay ($\tau_1$)is absent, the system is locally stable for toxicants uptake delay ($\tau_2$) at
coexisting equilibrium points. But when both delays exist during the procedure, the biological population system becomes unstable,i.e., as
the deformation delay increases, stability of the system is distributed and after reaching a critical value of deformation delay, system
exhibits Hopf bifurcation. Therefore, population density oscillates concerning time. At the equilibrium point, the local asymptotic stable
nature forms through the region of attraction. Moreover, the Hopf bifurcation's stability and direction are discussed for the critical
values of delays. Additionally, to obtain the minimum cost, we are using optimal control theory through Pontryagin's maximum principle.
Finally, the numerical authentication of experimental results has been validated by the numerical simulation of the designed model.