A theoretical and computational dynamical analysis of a deformed biological population due to prolongation in internally emitted toxicants

Biological species have their structure and functions. But in the last few decades, due to toxicants emitted by biological species through domestic wastage, industrialization, pesticides, burning garbage, etc., there have been rapidly abnormal changes in their function like productivity, change in shape, necrosis, etc. The effect of these abnormal changes is not instantaneous, i.e., it takes their own time. The time-consuming process creates more complications for the survival of these biological species. Due to these time-consuming abnormal changes, we have developed and analyzed a deformed biological population model with population deformation delay <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>τ</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and toxicant's depletion delay <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>τ</mml:mi> <mml:mn>2</mml:mn></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> for the internally emitted toxicants. Therefore, a model is designed and focused on analyzing the impact of the delay of these abnormal changes due to internally emitted toxicants on the biological population. The contribution is summarized as follows:•Examined the system's stability and explored the possibility of the Hopf bifurcation with respect to deformation and depletion delays. In this process, the critical threshold for delays is determined.•Additionally, with the help of the optimal control method and using Pontryagin's maximum principle, we minimize the cost of control strategies.•Provided the numerical authentication of our theoretical findings, using MATLAB simulations.